Algebras, hyperalgebras, nonassociative bialgebras and loops

José M. Pérez–Izquierdo^{1}

Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain E-mail: jm.perez@dmc.unirioja.es

Abstract

Sabinin algebras are a broad generalization of Lie algebras that include Lie, Malcev and Bol algebras as very particular examples. We present a construction of a universal enveloping algebra for Sabinin algebras, and the corresponding Poincaré– Birkhoff–Witt Theorem. A nonassociative counterpart of Hopf algebras is also introduced. Loop algebras and universal enveloping algebras of Sabinin algebras are natural examples of these nonassociative Hopf algebras. Identities of loops move to identities of nonassociative Hopf algebras by a linearizing process.

1 Introduction

Hyperalgebras were introduced by L. Sabinin and P. O. Miheev in [15,16]. At the V International Conference “Nonassociative Algebra And Its Applications” celebrated in Oaxtepec (Mexico) from July 27th to August 2nd (2003), and after a public proposal of L. Bokut there was a general agreement in renaming these structures as Sabinin algebras. With the kind agreement^{2} of professor Miheev, in this paper we shall follow this convention.

Roughly speaking, a quasigroup is a nonempty set Q with a binary operation ·: Q×Q → Q such that for any a ∈ Q the left and right multiplication operators by a are bijective. In case that there exists e ∈ Q, the identity element, with ea = a = ae for any a ∈ Q then (Q, ·, e) is called a loop. Therefore, a loop is the nonassociative counterpart of a group.

1 Supported by MCYT (BFM 2001-3239-C03-02) and the Comunidad Autónoma de La Rioja (ANGI 2001/26). 2 Private correspondence with J. Mostovoy

Preprint submitted to Elsevier Science 29 January 2004

However, in the same way that the inverse map is considered as an essential part of the group structure, for any quasigroup or loop it is natural to incorporate the left and right divisions \: Q × Q → Q and /: Q × Q → Q given by a\b = L_{a}^{−1}(b) and a/b = R_{b}^{−1}(a) in the definition. Hence we arrive at the formal definition of quasigroups and loops:

Definition 1 A quasigroup (Q, ·, \, /) is a nonempty set with three binary operations · (multiplication), \ (left division) and / (right division) such that

a\(ab) = b, a(a\b) = b, (ab)/b = a and (a/b)b = a.

for any a, b ∈ Q. In case that a\a = b/b for any a, b ∈ Q we say that the quasigroup is a loop.

The definition of loop is equivalent to imposing the existence of an identity element in the underlying quasigroup. To emphasize the existence of this element it is usual to write (Q, ·, \, /, e) for a loop.

One of the most celebrated results in Mathematics is the correspondence between Lie algebras and local Lie groups. However, in [6] Malcev showed that this correspondence may appear even when the associativity is removed. Malcev studied the relationship between local analytic Moufang loops and Malcev algebras and proved a correspondence between them similar to the known correspondence between local Lie groups and Lie algebras. These results were extended by Sabinin and Miheev [14] to local analytic Bol loops and Bol algebras. At this point it came up that two operations on the tangent space were needed to study local Bol loops, and a more general task emerged: to define an algebraic structure on the tangent space of any local analytic loop so that the Lie correspondence holds. This was achieved in [15,16] and the algebraic structures considered there are what now we call Sabinin algebras.

A vector space V is called a Sabinin algebra if it is endowed with multilinear operations

\begin{matrix}⟨x_{1}, x_{2}, …, x_{m}; y, z⟩ m ≥ 0 and \\ Φ(x_{1}, x_{2}, …, x_{m}; y_{1}, y_{2}, …, y_{n}), m ≥ 1, n ≥ 2 \end{matrix}

which satisfy the identities

⟨x_{1}, x_{2}, …, x_{m}; y, z⟩ = −⟨x_{1}, x_{2}, …, x_{m}; z, y⟩,

⟨x_{1}, x_{2}, …, x_{r}, a, b, x_{r+1}, …, x_{m}; y, z⟩ − ⟨x_{1}, x_{2}, …, x_{r}, b, a, x_{r+1}, …, x_{m}; y, z⟩

+ ∑_{k=0}^{r} ∑_{α}⟨x_{α_{1}}, …, x_{α_{k}}, ⟨x_{α_{k+1}}, …, x_{α_{r}}; a, b⟩, …, x_{m}; y, z⟩ = 0,

σ_{x,y,z}(⟨x_{1}, …, x_{r}, x; y, z⟩ + ∑_{k=0}^{r} ∑_{α}⟨x_{α_{1}}, …, x_{α_{k}}; ⟨x_{α_{k+1}}, …, x_{α_{r}}; y, z⟩, x⟩) = 0

and

Φ(x_{1}, …, x_{m}; y_{1}, …, y_{n}) = Φ(x_{τ(1)}, …, x_{τ(m)}; y_{δ(1)}, …, y_{δ(n)}),

where α runs the set of all bijections of the type α: {1, 2, …, r} → {1, 2, …, r}, i ↦ α_{i}, α_{1} < α_{2} < ⋯ < α_{k}, α_{k+1} < ⋯ < α_{r}, k = 0, 1, …, r, r ≥ 0, σ_{x,y,z} denotes the cyclic sum by x, y, z; τ ∈ S_{m}, δ ∈ S_{n} and S_{l} is the symmetric group on l symbols. The operations ⟨; ⟩ and the so called multioperator Φ are independent and sometimes the term “Sabinin algebra” is used for a vector space equipped only with operations ⟨; ⟩ satisfying the corresponding properties. In the basic examples (Lie, Malcev and Bol algebras) the multioperator vanishes.

This unappealing object turned out to be quite natural after the work of Shestakov and Umirbaev [19] where they showed that in the same way that an associative algebra gives rise to a Lie algebra after skew–symmetrization, over fields of characteristic zero a nonassociative algebra C originates a Sabinin algebra G(C) after adequate manipulations. Moreover, in case that C is a (not necessarily associative) bialgebra then Prim(C), the primitive elements of C, is a Sabinin subalgebra of G(C).

The basic question to face is the following [19]: does any Sabinin algebra appear as a Sabinin subalgebra of G(C) for some algebra C?

In this paper we provide an affirmative answer to this question. For any Sabinin algebra (V, ⟨; ⟩, Φ) over a field of characteristic zero we construct an algebra, its universal enveloping algebra, U((V, ⟨; ⟩, Φ)) such that (V, ⟨; ⟩, Φ) is a Sabinin subalgebra of G(U((V, ⟨; ⟩, Φ))). The algebra U((V, ⟨; ⟩, Φ)) is a (not necessarily associative) bialgebra and Prim(U((V, ⟨; ⟩, Φ))) = V. However, in solving this problem a nonassociative counterpart of Hopf algebras comes into play.

An H–bialgebra is a bialgebra with two extra operations \ and / satisfying

\begin{matrix}∑ x_{(1)}\(x_{(2)}y) = ϵ(x)y = ∑ x_{(1)}(x_{(2)}\y) and \\ ∑(yx_{(1)})/x_{(2)} = ϵ(x)y = ∑(y/x_{(1)})x_{(2)}. \end{matrix}

Natural H–bialgebras are the loop algebra of a loop and the universal enveloping algebras of Sabinin algebras. Hopf algebras are exactly the associative and coassociative unital H–bialgebras. The associative law (xy)z = x(yz) is only one example of identity we may impose to an H–bialgebra. However, in working with universal enveloping algebras other not so trivial, but quite well placed in a “Hopf context”, identities appear.

The universal enveloping algebras of some special Sabinin algebras have precursors. The generalized alternative nucleus N_{alt}(A) of an algebra A is defined as

N_{alt}(A) = {a ∈ A | (a, x, y) = −(x, a, y) = (x, y, a) ∀x, y ∈ A}

where (x, y, z) stands for the associator. Regardless of the algebra A we start with, the generalized alternative nucleus is a Malcev algebra with the commutator product. Moreover, in [11] it is proved that given any Malcev algebra M over a field of characteristic ≠ 2, 3 there exists a universal enveloping algebra U(M) such that M is a subalgebra of N_{alt}(U(M)). This universal enveloping algebra is isomorphic to the universal enveloping algebra constructed in this paper when considering the Malcev algebra as a Sabinin algebra. Probably, the most distinguished property of U(M) as H–bialgebra is that it satisfies the identity

∑ x_{(1)}(y(x_{(2)}z)) = ∑((x_{(1)}y)x_{(2)})z.

In fact, if this identity holds in a unital coassociative H–bialgebra then the primitive elements form a Malcev algebra (they lay in the generalized alternative nucleus) and the group–like elements form a Moufang loop. The methods used for Malcev algebras extend to Bol algebras. The left generalized alternative nucleus of an algebra A is defined as

LN_{alt}(A) = {a ∈ A | (a, x, y) = −(x, a, y) ∀x, y ∈ A}

and it is a Lie triple system with the triple product [a, b, c] = a(bc) − b(ac) − c(ab−ba). Any subspace of V closed by this triple product and the commutator product [a, b] = ab−ba forms a Bol algebra, and any Bol algebra (V, [ , , ], [ , ]) over a field of characteristic not 2 is obtainable in this way from its universal enveloping algebra U(V) [10]. Moreover, U(V) is an H–bialgebra which satisfies the identity

∑ a_{(1)}(b(a_{(2)}c)) = ∑(a_{(1)}(ba_{(2)}))c.

Again, for any coassociative unital H–bialgebra satisfying this identity the primitive elements form a Bol algebra and the group–like elements form a Bol loop.

The universal enveloping algebras of Malcev and Bol algebras are nice examples of H–bialgebras satisfying certain identities, so the corresponding loop algebras are. In this way, nonassociative algebras and Hopf algebras interlace smoothly.

The definition of H–bialgebra is very much oriented to model the existing examples (universal enveloping algebras and loop algebras) and these are naturally cocommutative. Up to some extend, very roughly one might consider a “quantum loop” as a noncocommutative H–bialgebra. A sharper point of view arises from the study of identities. However, concrete examples and formulation deserve further research.

This paper is organized as follows. In Section 2 we prove some basic properties about H–bialgebras. In Section 3 we study in which sense identities on loops give rise to identities on H–bialgebras allowing us to translate known results on loops to H–bialgebras. We also check that the universal enveloping algebras of Malcev and Bol algebras satisfy the identities previously mentioned. In Section 4 we properly introduce Sabinin algebras and the basic constructions of them from Lie algebras. The approach we adopt shows that the surprising construction of Shestakov and Umirbaev is quite natural indeed. The underlying vector space of the universal enveloping algebra is presented in Section 5. In this section we also prove the Poincaré–Birkhoff–Witt Theorem for the universal enveloping algebras although at this stage the product is not defined yet! One proof using Gröbner (or Gröbner–Shirshov) bases is also given. The needed product for the universal enveloping algebras is obtained in Section 6 and a nice Lie algebra is shown to be compatible with this product. The analogous of the Milnor–Moore Theorem is also presented in this section. Finally, we compare the existing universal enveloping algebras of Malcev algebras with the ones introduced in this paper.

2 Nonassociative bialgebras

A coalgebra (C, Δ, ϵ) is a vector space C equipped with two maps

Δ: C → C ⊗ C and ϵ: C → F

such that

(Id ⊗ϵ)Δ = Id = (ϵ ⊗ Id)Δ,

where the natural identification C⊗F ≅ C ≅ F ⊗C is assumed. The coalgebra (C, Δ, ϵ) is called coassociative if (Δ ⊗ Id)Δ = (Id ⊗Δ)Δ, and it is called cocommutative if τΔ = Δ where τ(x ⊗ y) = y ⊗ x. It is customary to write

Δ(x) = ∑_{(x)} x_{(1)} ⊗ x_{(2)} or Δ(x) = ∑ x_{(1)} ⊗ x_{(2)}.

A (nonunital) bialgebra (B, Δ, ϵ, ·) is a coalgebra (B, Δ, ϵ) with a bilinear product

·: C × C → C
(x, y) ↦ xy

such that

Δ(xy) = ∑ x_{(1)}y_{(1)} ⊗ x_{(2)}y_{(2)} and ϵ(xy) = ϵ(x)ϵ(y).

A unital bialgebra (B, Δ, ϵ, ·, u) is a bialgebra (B, Δ, ϵ, ·) equipped with a linear map u: F → B, the unit, such that

1x = x = x1, Δ(1) = 1 ⊗ 1 and ϵ(1) = 1

where 1 = u(1) ∈ B.

Definition 2 An H–bialgebra (H, Δ, ϵ, ·, \, /) is a bialgebra (H, Δ, ϵ, ·) with two extra bilinear operations, the left and right division,

\begin{matrix}\: H × H & → & H & /: H × H & → & H \\ (x, y) & ↦ & x\y & (x, y) & ↦ & x/y \end{matrix}

such that

∑ x_{(1)}\(x_{(2)}y) = ϵ(x)y = ∑ x_{(1)}(x_{(2)}\y) and  (1)
∑(yx_{(1)})/x_{(2)} = ϵ(x)y = ∑(y/x_{(1)})x_{(2)}.  (2)

A unital H–bialgebra (H, Δ, ϵ, ·, u, \, /) is a unital bialgebra (H, Δ, ϵ, ·, u) such that (H, Δ, ϵ, ·, \, /) is an H– bialgebra.

Proposition 3 An associative and coassociative unital H–bialgebra is a Hopf algebra. Moreover, in this case x\y = S(x)y and x/y = xS(y), where S denotes the antipode.

PROOF. The natural candidate to be the antipode is S(x) = x\1. In fact, by (1) we have that

∑ x_{(1)}S(x_{(2)}) = ϵ(x)1,

and by the associativity and coassociativity

∑ x_{(1)}\(x_{(2)} · S(x_{(3)})y) = \{ \begin{matrix}∑ x_{(1)}\(ϵ(x_{(2)})y) = x\y \\ ∑ ϵ(x_{(1)})S(x_{(2)})y = S(x)y \end{matrix}

so

S(x)y = x\y.

In particular,

∑ S(x_{(1)})x_{(2)} = ∑ x_{(1)}\x_{(2)} = ∑ x_{(1)}\(x_{(2)} · 1) = ϵ(x)1,

hence S is the antipode. The last relation x/y = xS(y) follows easily. □

We make the linear operators to act from the left, however from Section 4 on we will make them to act from the right for coherence with [19].

Given a coalgebra (C, Δ, ϵ), the vector space Hom(C, End(C)) is a unital algebra with the product given by the convolution

(φ ∗ ψ)(x) = ∑ φ(x_{(1)})ψ(x_{(2)})

and unit element ι: x ↦ ϵ(x) Id. In case that the coalgebra C is coassociative then Hom(C, End(C)) is an associative algebra. The multiplicative structure of a bialgebra (B, Δ, ϵ, ·) is determined by the elements of Hom(B, End(B))

\begin{matrix}L: B & → & End(B) & R: B & → & End(B) \\ x & ↦ & L_{x} & x & ↦ & R_{x} \end{matrix}

Although obviously it suffices one of these maps to determine the multiplicative structure, we prefer to present both of them to emphasize their importance on the existence of left and right division respectively.

Proposition 4 Let (B, Δ, ϵ, ·) be a coassociative bialgebra. There exist maps \: B × B → B and /: B × B → B such that (B, Δ, ϵ, ·, \, /) is an H–bialgebra if and only if the elements L and R are invertible in Hom(B, End(B)). In that case the operations \ and / are uniquely determined.

PROOF. By hypothesis Hom(B, End(B)) is associative. The element L is invertible in this algebra if and only if there exists LD: B → End(B) such that

∑ LD(x_{(1)})L(x_{(2)}) = ϵ(x) Id = ∑ L(x_{(1)})LD(x_{(2)}),

or equivalently

∑ LD(x_{(1)})(x_{(2)}y) = ϵ(x)y = ∑ x_{(1)}LD(x_{(2)})(y),

and in that case the inverse LD is unique. Comparing this identity with (1) we obtain the desired result about the existence and uniqueness of \. One proceeds with / similarly. □

Recall that a coalgebra is called connected if the dimension of the coradical is one.

Proposition 5 Let (H, Δ, ϵ, ·, u) be a coassociative unital bialgebra. If the coalgebra (H, Δ, ϵ) is connected then we may define on (H, Δ, ϵ, ·, u) a (unique) structure of H-bialgebra.

PROOF. Let us show that R, L are invertible in Hom(H, End(H)). By a result of Takeuchi [20] we only need to check that their restriction to H_{0} = F 1 (the coradical of H) are invertible in Hom(H_{0}, End(H)), but this is obvious. □

Proposition 6 Let (H, Δ, ϵ, ·, \, /) be a coassociative H–bialgebra. We have that

Δ(x\y) = ∑ x_{(2)}\y_{(1)} ⊗ x_{(1)}\y_{(2)} and Δ(x/y) = ∑ x_{(1)}/y_{(2)} ⊗ x_{(2)}/y_{(1)}.

PROOF. Consider A = Hom(H, End(H ⊗ H)). With the convolution A is an associative algebra with unit element ι: x ↦ ϵ(x) Id where Id denotes the identity map on H ⊗ H. Let V = End(H ⊗ H) and define

A × V → V
(φ, f) ↦ φf: x ⊗ y ↦ ∑(φ_{x_{(1)}}f)(x_{(2)} ⊗ y).

With this action V becomes a unital A–module. Let us now define φ ∈ A and f, g ∈ V by

φ: x ↦ φ_{x} = ∑ L_{x_{(1)}} ⊗ L_{x_{(2)}},
f: x ⊗ y ↦ Δ(x\y) and
g: x ⊗ y ↦ ∑ x_{(2)}\y_{(1)} ⊗ x_{(1)}\y_{(2)}.

The first formula in the statement will follow once we had proved that φ is invertible in A and that φf = φg. To show that φ is invertible we define φ^{−1}: x ↦ φ_{x}^{−1}: y ⊗ z ↦ ∑ x_{(2)}\y ⊗ x_{(1)}\z. We have that

(φ ∗ φ^{−1})_{x}(y ⊗ z) = ∑(φ_{x_{(1)}}φ_{x_{(2)}}^{−1})(y ⊗ z)
= ∑(L_{x_{(1)}} ⊗ L_{x_{(2)}})(x_{(4)}\y ⊗ x_{(3)}\z) = ϵ(x)y ⊗ z

and similarly (φ^{−1} ∗ φ)_{x} = ϵ(x) Id, therefore φ^{−1} is the inverse of φ in A. Finally,

(φg)(x ⊗ y) = ∑ φ_{x_{(1)}}g(x_{(2)} ⊗ y) = ∑(x_{(1)} ⊗ x_{(2)})(x_{(4)}\y_{(1)} ⊗ x_{(3)}\y_{(2)})
= ∑ x_{(1)}(x_{(4)}\y_{(1)}) ⊗ x_{(2)}(x_{(3)}\y_{(2)}) = ϵ(x)y_{(1)} ⊗ y_{(2)}
= ϵ(x)Δ(y)

and

(φf)(x ⊗ y) = ∑(x_{(1)} ⊗ x_{(2)})Δ(x_{(3)}\y) = ∑ Δ(x_{(1)}(x_{(2)}\y))
= ϵ(x)Δ(y).

This proves that Δ(x\y) = ∑ x_{(2)}\y_{(1)} ⊗ x_{(1)}\y_{(2)}. The proof of the second identity in the statement is similar. □

Proposition 7 In any H–bialgebra (H, Δ, ϵ, ·, \, /)

ϵ(x\y) = ϵ(x)ϵ(y) = ϵ(x/y).

PROOF. The first identity follows from

ϵ(x\y) = ϵ((∑ ϵ(x_{(1)})x_{(2)})\y) = ∑ ϵ(x_{(1)})ϵ(x_{(2)}\y)
= ∑ ϵ(x_{(1)}(x_{(2)}\y)) = ϵ(ϵ(x)y) = ϵ(x)ϵ(y).

Similarly ϵ(x/y) = ϵ(x)ϵ(y). □

Corollary 8 Let (H, Δ, ϵ, ·, \, /) be a coassociative H–bialgebra such that set of group elements

G(H) = {a ∈ H | Δ(a) = a ⊗ a and ϵ(a) = 1}

is nonempty. Then (G(H), ·, \, /) is a quasigroup.

PROOF. By Proposition 6 and Proposition 7, G(H) is closed under \ and /. Moreover, (1) and (2) imply that

a\(ax) = x = a(a\x) and (xa)/a = x = (x/a)a

for any a ∈ G(H), therefore (G(H), ·, \, /) is a quasigroup. □

Corollary 9 For any coassociative unital H–bialgebra (H, Δ, ϵ, ·, u, \, /) the set (G(H), ·, \, /, u) is a loop.

3 Linearizing identities

3 The study of varieties of quasigroups is one cornerstone of the theory of quasigroups. Attached to any set of identities for quasigroups there exists a set of “identities” for cocommutative coassociative H–bialgebras. For instance, corresponding to the left Moufang identity

x(y(xz)) = ((xy)x)z

on quasigroups we have the “linearization”

∑ x_{(1)}(y(x_{(2)}z)) = ∑((x_{(1)}y)x_{(2)})z (3)

for H–bialgebras, and corresponding to the right Moufang identity

((zx)y)x = z(x(yx))

on quasigroups we have the linearization

∑((zx_{(1)})y)x_{(2)} = ∑ z(x_{(1)}(yx_{(2)})) (4)

3 The presentation we adopt is a great simplification of the original arguments. It was written while the author was collaborating with Professor J. Mostovoy.

for H–bialgebras. However, and this is the motivation for this section, it is known that for any quasigroup the left and right Moufang identities are equivalent, so one may wonder whether for any cocommutative coassociative H– bialgebra, (3) and (4) are equivalent. The goal of this section is to prove that if an identity for quasigroups is a consequence of other identities then the linearization of the former is a consequence of the linearizations of the later.

3.1 A linearizing process

In this subsection F will denote a type of algebras (see [1]) and (C, Δ, ϵ) will be a coassociative and cocommutative coalgebra such that C is an F–algebra and any operation {f, k} ∈ F induces a homomorphism of coalgebras f: C^{⊗k} → C (if k = 0 then f: F → C).

Since C is an F–algebra, then for any coassociative coalgebra V, the vector space Hom(V, C) is an F–algebra by

f(α_{1}, …, α_{n})(v) = ∑ f(α_{1}(v_{(1)}), …, α_{n}(v_{(n)})) v ∈ V

in case that n ≥ 1, and f(1)(v) = ϵ(v)f(1) if n = 0. If we consider the coalgebra C^{⊗N} (N ≥ 1) then we have the distinguished maps

ϵ_{i}(a_{1} ⊗ ⋯ ⊗ a_{N}) = ϵ_{i}(a_{1}, …, a_{N}) = ϵ(a_{1}) ⋯ ϵ̂(a_{i}) ⋯ ϵ(a_{N})a_{i}

where ϵ̂(a_{i}) means that this factor is omitted and 1 ≤ i ≤ N. Therefore, if n ≤ N there exists a homomorphism l_{n,N} (the linearizing map) from the term algebra T (X) (see [1]) on X = {x_{1}, x_{2}, …, x_{n}} over F to Hom(C^{⊗N}, C) sending x_{i} to ϵ_{i}.

Definition 10 Given p, q ∈ T (X) with X = {x_{1}, x_{2}, …, x_{n}}, we will say that C satisfies the linearization of the identity p ≈ q if l_{n,N}(p) = l_{n,N}(q) for some N ≥ n.

Note 1 It is convenient to remark that l_{n,N}(p) = l_{n,N}(q) for some N ≥ n iff l_{n,N}(p) = l_{n,N}(q) for all N ≥ n. So, if we set l_{n} = l_{n,n} then C satisfies the linearization of the identity p ≈ q iff l_{n}(p) = l_{n}(q).

Example 11 Consider F = {{·, 2}, {\, 2}, {/, 2}}, p = x_{1}(x_{2}(x_{1}x_{3})) and q = ((x_{1}x_{2})x_{1})x_{3}. We have that l_{3}(p)(a ⊗ b ⊗ c) = ∑ a_{(1)}(b(a_{(2)}c)) and l_{3}(q)(a ⊗ b ⊗ c) = ∑((a_{(1)}b)a_{(2)})c. Thus, C satisfies the linearization of the Moufang identity iff

∑ a_{(1)}(b(a_{(2)}c)) = ∑((a_{(1)}b)a_{(2)})c

holds on C.

We will denote the set of all coalgebra maps from a coalgebra V to C by Coalg(V, C). This set is an F–subalgebra of Hom(V, C).

Lemma 12 Let V be a coassociative and cocommutative coalgebra. Given p(x_{1}, …, x_{n}) ∈ T (X), n ≥ 1, and α_{1}, …, α_{n} ∈ Coalg(V, C)we have that

p(α_{1}, …, α_{n}) = l_{n}(p) ∘ (α_{1} ⊗ ⋯ ⊗ α_{n}) ∘ Δ^{n}.

PROOF. We proceed by induction on the length of p, the number of operation symbols in p from F (see [1] for details). If p(x_{1}, …, x_{n}) = x_{i} (1 ≤ i ≤ n) then

∑ l_{n}(p)(α_{1}(v_{(1)}) ⊗ ⋯ ⊗ α_{n}(v_{(n)})) = α_{i}(v) = p(α_{1}, …, α_{n})(v).

In general, given an expression p(x_{1}, …, x_{n}) ∈ T (X) with p(x_{1}, …, x_{n}) = f(r_{1}(x_{1}, …, x_{n}), …, r_{k}(x_{1}, …, x_{n})) for some r_{1}, …, r_{k} ∈ T (X) and {f, k} ∈ F then

∑ l_{n}(p)(α_{1}(v_{(1)}) ⊗ ⋯ ⊗ α_{n}(v_{(n)}))
=^{⟨1⟩} ∑ f(l_{n}(r_{1}), …, l_{n}(r_{k}))(α_{1}(v_{(1)}) ⊗ … ⊗ α_{n}(v_{(n)}))
=^{⟨2⟩} ∑ f(l_{n}(r_{1})(α_{1}(v_{(1)}) ⊗ ⋯ ⊗ α_{n}(v_{(n)})), …
…, l_{n}(r_{k})(α_{1}(v_{(n(k−1)+1)}) ⊗ … ⊗ α_{n}(v_{(kn)})))
=^{⟨3⟩} ∑ f(r_{1}(α_{1}, …, α_{n})(v_{(1)}), …, r_{k}(α_{1}, …, α_{n})(v_{(k)}))
= p(α_{1}, …, α_{n})(v)

where in ⟨1⟩ we have used that l_{n} is a homomorphism of F–algebras, in ⟨2⟩ we use that α_{1}, …, α_{n} are coalgebra maps, the structure of F–algebra on Hom(V, C) and the cocommutativity of V, and ⟨3⟩ follows from the hypothesis of induction. □

Theorem 13 Let Σ be a set of identities and p ≈ q a consequence of Σ. If C satisfies the linearization of all identities in Σ, then C satisfies the linearization of p ≈ q.

PROOF. Since p ≈ q may be derived from a finite subset of Σ (see [1]) we may assume that all the terms involved in the identities we use belong to T ({x_{1}, …, x_{n}}) for some n big enough. Then by Lemma 12 we have that Coalg(C^{⊗n}, C) satisfies the identities in Σ, so it also satisfies the identity p ≈ q.

Since (ϵ_{1} ⊗ ⋯ ⊗ ϵ_{n}) ∘ Δ^{n}(c_{1} ⊗ ⋯ ⊗ c_{n}) = c_{1} ⊗ ⋯ ⊗ c_{n} for any c_{1}, …, c_{n} ∈ C, then again Lemma 12 implies that l_{n}(p) = l_{n}(q). □

We close this subsection with an example of application of Theorem 13 to obtain new results about H–bialgebras. Many other results may be obtained with the same techniques.

A quasigroup is an F–algebra, with F = {{·, 2}, {\, 2}, {/, 2}}, that satisfies the identities

a\(ab) ≈ b, a(a\b) ≈ b, (ba)/a ≈ b and (b/a)a ≈ b.

By Proposition 6 and Proposition 7 any cocommutative and coassociative H–bialgebra is an F–algebra, the operations being homomorphisms of coalgebras, and by definition it satisfies the linearizations of these identities. V. Shcherbacov and V. Izbash [18] (the result was announced in 1993) and independently K. Kunen [5] with the help of the automated deduction tool OTTER proved that the identities

\begin{matrix}M1 : (a(bc))a ≈ (ab)(ca) & M2 : (ab)(ca) ≈ a((bc)a) \\ N1 : ((ba)c)a ≈ b(a(ca)) & N2 : ((ab)a)c ≈ a(b(ac)) \end{matrix} (5)

are equivalent for quasigroups. Moreover, any of them implies the existence of two–sided unit element, which expressed in terms of identities means that

a\a ≈ b/b (6)

holds, a\a being the identity element for any a. Therefore, they are Moufang loops. For long time it has been known that these loops satisfy

a\b ≈ (a\(c\c))b, b/a ≈ b(a\(c\c)) and (ab)\(c\c) ≈ (b\(c\c))(a\(c\c)),

that with the notation a^{−1} = a\(c\c) become the familiar identities

a\b ≈ a^{−1}b, b/a ≈ ba^{−1} and (ab)^{−1} ≈ b^{−1}a^{−1} (7)

Let us now fix a cocommutative and coassociative nonunital H–bialgebra H.

The linearizations of (5) are

\begin{matrix}M1 : ∑(a_{(1)}(bc))a_{(2)} = ∑(a_{(1)}b)(ca_{(2)}) \\ M2 : ∑(a_{(1)}b)(ca_{(2)}) = ∑ a_{(1)}((bc)a_{(2)}) \\ N1 : ∑((ba_{(1)})c)a_{(2)} = ∑ b(a_{(1)}(ca_{(2)})) \\ N2 : ∑((a_{(1)}b)a_{(2)})c = ∑ a_{(1)}(b(a_{(2)}c)). \end{matrix} (8)

By Theorem 13 and the result from Shcherbacov, Izbash and Kunen, these identities are equivalent on H. Moreover, if H satisfies any of these identities then it also satisfies the linearization of (6)

ϵ(b) ∑ a_{(1)}\a_{(2)} = ϵ(a) ∑ b_{(1)}/b_{(2)} (9)

and the element 1 = ∑ c_{(1)}\c_{(2)} with ϵ(c) = 1 is the identity of H. Therefore, H is a unital H–bialgebra. Furthermore, if we define S(a) = a\1 then the linearization of (7) becomes

a\b = S(a)b, b/a = bS(a) and S(ab) = S(b)S(a). (10)

In particular,

∑ S(a_{(1)})a_{(2)} = ϵ(a)1 = ∑ a_{(1)}S(a_{(2)})

and we have an analogue of the antipode of Hopf algebras.

3.2 Envelopes of Malcev and Bol algebras (I)

Recall that a Malcev algebra (M, [ , ]) over a field of characteristic ≠ 2 is a vector space with a skew– symmetric product such that

[J(x, y, z), x] = J(x, y, [x, z])

where J(x, y, z) = [[x, y], z] − [[x, z], y] − [x, [y, z]] is the jacobian of x, y and z. These algebras appear as tangent spaces of smooth Moufang loops.

A Lie triple system (V, [ , , ]) is a vector space with a trilinear operation such that

[a, a, b] = 0,
[a, b, c] + [b, c, a] + [c, a, b] = 0 and
[x, y[a, b, c]] = [[x, y, a], b, c] + [a, [x, y, b], c] + [a, b, [x, y, c]].

A (left) Bol algebra (V, [ , , ], [ , ]) is a Lie triple system (V, [ , , ]) with an additional bilinear skew–symmetric operation [a, b] satisfying

[a, b, [c, d]] = [[a, b, c], d] + [c, [a, b, d]] + [c, d, [a, b]] + [[a, b], [c, d]].

Left Bol algebras appear as tangent spaces of smooth left Bol loops, that is, loops that satisfy the left Bol identity

a(b(ac)) = (a(ba))c.

Malcev and Bol algebras are very much related since any Malcev algebra (over a field of characteristic ≠ 2, 3) is a left Bol algebra with its bilinear product and the triple product given by

[a, b, c] = [[a, b], c] − \frac{1}{3}J(a, b, c).

In [10], for any left Bol algebra (V, [ , , ], [ , ]) over a field of characteristic ≠ 2 a universal enveloping algebra U(V) is constructed in such a way that V ⊆ U(V), the operations in V are recovered as

\begin{matrix}[a, b] = ab − ba, \\ [a, b, c] = a(bc) − b(ac) − c(ab − ba) \end{matrix}

and V ⊆ LN_{alt}(U(V)) = {a ∈ U(V) | (a, x, y) = −(x, a, y) ∀x, y ∈ U(V)}.

A similar result was previously proved for Malcev algebras [11]. Given a Malcev algebra (M, [ , ]) over a field of characteristic ≠ 2, 3 then there exists an algebra U(M) such that M ⊆ U(M), the operation on M is recovered as

[a, b] = ab − ba

and M ⊆ N_{alt}(U(M)) = {a ∈ U(M) | (a, x, y) = −(x, a, y) = (x, y, a) ∀x, y ∈ U(M)}. If we consider M as a left Bol algebra then the envelopes as Malcev and Bol algebra are isomorphic. These envelopes are in fact connected cocommutative and coassociative unital bialgebras. Therefore, by Proposition 5 they are H–bialgebras. Moreover,

Theorem 14 Let (V, [ , , ], [ , ]) be a left Bol algebra over a field of characteristic ≠ 2 and U(V) its universal enveloping algebra. Then U(V) satisfies

∑ a_{(1)}(b(a_{(2)}c)) = ∑(a_{(1)}(ba_{(2)}))c.

PROOF. Recall that U(V) is a filtered algebra U(V) = ⋃_{n=0}^{∞} U(V)_{n} with U(V)_{n} = span⟨a_{i_{1}}(⋯ (a_{i_{n−1}}a_{i_{n}}) ⋯) | a_{i_{1}}, …, a_{i_{n}} ∈ V, n ∈ ℕ⟩ and the corresponding graded algebra Gr(U(V)) is commutative and associative.

Given 0 ≠ a ∈ U(V), there exists n such that a ∈ U(V)_{n} but a ∉ U(V)_{n−1}. We will prove the result using induction on n. For n = 0 the result is obvious. If n = 1 then a ∈ V and the formula follows from the fact that a ∈ LN_{alt}(U(V)). Let us now assume that we have proved the result for any element on U(V)_{n} and let v ∈ U(V)_{n+1}. Since Gr(U(V)) is commutative and associative we may assume that v = u ∘ a = ua + au with u ∈ U(V)_{n} and a ∈ V. We have

∑ L_{v_{(1)}}L_{b}L_{v_{(2)}} = ∑ L_{(a∘u)_{(1)}}L_{b}L_{(a∘u)_{(2)}}
= ∑ L_{a∘u_{(1)}}L_{b}L_{u_{(2)}} + L_{u_{(1)}}L_{b}L_{a∘u_{(2)}}
= ∑(L_{a} ∘ L_{u_{(1)}})L_{b}L_{u_{(2)}} + L_{u_{(1)}}L_{b}(L_{a} ∘ L_{u_{(2)}})
= ∑ L_{a}L_{u_{(1)}}L_{b}L_{u_{(2)}} + L_{u_{(1)}}L_{a}L_{b}L_{u_{(2)}}
+L_{u_{(1)}}L_{b}L_{u_{(2)}}L_{a} + L_{u_{(1)}}L_{b}L_{aL_{u_{(2)}}}
= L_{∑ a∘(u_{(1)}(bu_{(2)}))+u_{(1)}((a∘b)u_{(2)})}

Evaluating these operators on 1 we obtain that ∑ a ∘ (u_{(1)}(bu_{(2)})) + u_{(1)}((a ∘ b)u_{(2)}) = ∑ v_{(1)}(bv_{(2)}), so

∑ L_{v_{(1)}}L_{b}L_{v_{(2)}} = ∑ L_{v_{(1)}(bv_{(2)})}

as desired. □

A similar result holds for Malcev algebras.

Theorem 15 Let (M, [ , ]) be a Malcev algebra over a field of characteristic ≠ 2, 3 and U(M) its universal enveloping algebra. Then U(M) satisfies

∑ a_{(1)}(b(a_{(2)}c)) = ∑((a_{(1)}b)a_{(2)})c.

Finally, we should remark that some properties of U(V) and U(M) follow by linearizing the corresponding properties of Bol and Moufang loops as seen in the previous subsection.

4 Sabinin algebras

Recall that a vector space V is called a Sabinin algebra if it is endowed with multilinear operations

\begin{matrix}⟨x_{1}, x_{2}, …, x_{m}; y, z⟩,  m ≥ 0 \\ Φ(x_{1}, x_{2}, …, x_{m}; y_{1}, y_{2}, …, y_{n}),  m ≥ 1, n ≥ 2 \end{matrix}

which satisfy the identities

⟨x_{1}, x_{2}, …, x_{m}; y, z⟩ = −⟨x_{1}, x_{2}, …, x_{m}; z, y⟩,

⟨x_{1}, x_{2}, …, x_{r}, a, b, x_{r+1}, …, x_{m}; y, z⟩ − ⟨x_{1}, x_{2}, …, x_{r}, b, a, x_{r+1}, …, x_{m}; y, z⟩

+ ∑_{k=0}^{r} ∑_{α}⟨x_{α_{1}}, …, x_{α_{k}}, ⟨x_{α_{k+1}}, …, x_{α_{r}}; a, b⟩, …, x_{m}; y, z⟩ = 0,

σ_{x,y,z}⟨x_{1}, …, x_{r}, x; y, z⟩ + ∑_{k=0}^{r} ∑_{α}⟨x_{α_{1}}, , …, x_{α_{k}}; ⟨x_{α_{k+1}}, …, x_{α_{r}}; y, z⟩, x⟩ = 0,

and

Φ(x_{1}, …, x_{m}; y_{1}, …, y_{n}) = Φ(x_{τ(1)}, …, x_{τ(m)}; y_{δ(1)}, …, y_{δ(n)}),

where α runs the set of all bijections of the type α: {1, 2, …, r} → {1, 2, …, r}, i ↦ α_{i}, α_{1} < α_{2} < ⋯ < α_{k}, α_{k+1} < ⋯ < α_{r}, k = 0, 1, …, r, r ≥ 0, σ_{x,y,z} denotes the cyclic sum by x, y, z; τ ∈ S_{m}, δ ∈ S_{n} and S_{l} is the symmetric group.

Let T(V) be the tensor algebra over V endowed with its natural structure of bialgebra, that is, V ⊆ Prim(T(V)). Looking at ⟨; ⟩ as a map

⟨; ⟩: T(V) ⊗ V ⊗ V → V

we may write the definition of a Sabinin algebra very shortly as

⟨x; a, b⟩ + ⟨x; b, a⟩ = 0  (11)

⟨x[a, b]y; c, e⟩ + ∑⟨x_{(1)}⟨x_{(2)}; a, b⟩; c, e⟩ = 0  (12)
σ_{a,b,c} (⟨xc; a, b⟩ + ∑⟨x_{(1)}; ⟨x_{(2)}; a, b⟩, c⟩) = 0  (13)
Φ(x_{1}, …, x_{m}; y_{1}, …, y_{n}) = Φ(x_{τ(1)}, …, x_{τ(m)}; y_{δ(1)}, …, y_{δ(n)})  (14)

where as always we have used the Sweedler’s sigma notation for the comultiplication [19].

4.1 Sabinin algebras from primitive elements (I)

The set of primitive elements of a bialgebra is not a subalgebra in the usual sense, however, in their striking work [19] Shestakov and Umirbaev proved that this set is closed under certain operations and it naturally becomes a Sabinin algebra. We devote this subsection to reviewing the, at this point, surprising construction of Shestakov and Umirbaev. The reader should be aware that some modifications have been made.

Let B be a free unital nonassociative algebra over X ∪ Y ∪ {z} where X = {x_{1}, x_{2}, …} and Y = {y_{1}, y_{2}, …}. By the universal property of B there exists a unique homomorphism of unital algebras

Δ: B → B ⊗ B
w ↦ ∑ w_{(1)} ⊗ w_{(2)}

such that X ∪ Y ∪ {z} ⊆ Prim(B). Given x_{1}, …, x_{m} ∈ X, y_{1}, …, y_{n} ∈ Y, let

u = ((x_{1}x_{2}) ⋯)x_{m} and v = ((y_{1}y_{2}) ⋯)y_{n}.

From q_{0,0}(1, 1, z) = q_{0,n}(1, v, z) = q_{m,0}(u, 1, z) = 0 one recursively defines q_{m,n}(u, v, z), m, n ≥ 1 by

(u, v, z) = ∑ u_{(1)}q(u_{(2)}, v_{(2)}, z) · v_{(1)} (15)

where (u, v, z) denotes the associator of u, v and z. So,

\begin{matrix}q(x_{1}, y_{1}, z) = (x_{1}, y_{1}, z) and \\ q(u, v, z) = (u, v, z) − ∑_{|u_{(1)}|+|v_{(1)}|≥1} u_{(1)}q(u_{(2)}, v_{(2)}, z) · v_{(1)} \end{matrix}

where |u| denotes the total degree of the monomial u.

The importance of this construction is that, as shown in [19], the set of primitive elements of any bialgebra is closed under [ , ] (the usual commutator) and q_{m,n}(x_{1}, …, x_{m}; y_{1}, … y_{n}, z) m, n ≥ 1. Therefore, given a nonassociative algebra C over a field of characteristic zero one may consider the operations

⟨y, z⟩ = −[y, z]
⟨x_{1}, …, x_{m}; y, z⟩ = −q_{m,1}(x_{1}, …, x_{m}, y, z) + q_{m,1}(x_{1}, …, x_{m}, z, y)

and

Φ(x_{1}, …, x_{m}; y_{1}, …, y_{n}) =

\frac{1}{m!} \frac{1}{n!} ∑_{τ∈S_{m},δ∈S_{n}} q_{m,n−1}(x_{τ(1)}, …, x_{τ(m)}, y_{δ(1)}, …, y_{δ(n)}))

for all m ≥ 1 and n ≥ 2. Let G(C) denote the vector space C endowed with the operations ⟨; ⟩ and Φ. Shestakov and Umirbaev proved in [19] that G(C) is a Sabinin algebra, and in case that C is a bialgebra then Prim(C) is a Sabinin subalgebra of G(C).

Finally, we also should mention that although we started from a slightly different definition of the primitive operations considered in [19], however the operations ⟨x_{1}, …, x_{n}; y, z⟩ and ⟨y, z⟩ remain the same.

4.2 Sabinin algebras from Lie algebras

Sabinin and Miheev shown that given a Lie algebra L, a subalgebra H and a vector space V with L = H ⊕V, then V inherits a structure of Sabinin algebra from the product on L. In this subsection we present a different approach to this construction providing a new interpretation of the operations ⟨; ⟩. One important example of this construction arises when starting with a unital algebra C, L the Lie algebra generated by the right multiplication operators, H = {φ ∈ L | 1φ = 0} and V = {R_{a} | a ∈ C} since we naturally recover the Sabinin algebra G(C) defined by Shestakov and Umirbaev. In the following we will assume that linear maps act from the right.

We first fix some notation. Let L be a Lie algebra, H a subalgebra and V a vector space with L = H ⊕ V. Given d_{1}, …, d_{n} ∈ L we will use the notation

[d_{1}] = d_{1},
[d_{1}, …, d_{n}] = [d_{1}, [⋯, [d_{n−1}, d_{n}]]] n ≥ 2 and
{d_{1}, …, d_{n}} = π_{V} ([d_{1}, …, d_{n}]) n ≥ 1,

where π_{V} denotes the projection on V parallel to H. Given a ∈ V we will consider

{; a}: T(L) → V

defined by

{1; a} = 0, and {d_{1} ⋯ d_{n}; a} = {d_{1}, …, d_{n}, a},

and the projection

T(L) → U(L)/H U(L)
d_{1} ⋯ d_{n} ↦ \overline{d_{1} ⋯ d_{n}}

where U(L) denotes the universal enveloping algebra of L.

Lemma 16 Given d_{1}, …, d_{n} ∈ L and w = d_{1} ⋯ d_{n} ∈ T(L), then

\overline{wd} = − ∑_{i=1}^{∞} ∑(−1)^{i}\overline{w_{(1)}\{w_{(2)}; \{⋯; \{w_{(i)}; \{w_{(i+1)}d\} ⋯\}}

for any d ∈ L.

PROOF. We will proceed by induction on n. If n = 0 then w = 1 so the right hand side of the formula is − ∑_{i=1}^{∞}(−1)^{i}\overline{1\{1; \{⋯, \{1d\} ⋯\}} = −(−\overline{\{d\}}) = \overline{d} = \overline{wd}. Let us assume that the formula holds for any w = d_{1} ⋯ d_{m} with m < n. Given w = d_{1} ⋯ d_{n}, let w′ = d_{1} ⋯ d_{n−1}, then

\overline{wd} = \overline{w′d_{n}d} = \overline{w′dd_{n}} + \overline{w′[d_{n}, d]}
= − ∑_{i=1}^{∞} ∑(−1)^{i}\overline{w′_{(1)}\{w′_{(2)}; \{⋯; \{w′_{(i+1)}d\} ⋯\}d_{n}}

− ∑_{i=1}^{∞} ∑ \overline{w′_{(1)}\{w′_{(2)}; \{⋯; \{w′_{(i)}; \{w′_{(i+1)}[d_{n}, d]\} ⋯\}}.

With Z = − ∑_{i=1}^{∞} ∑(−1)^{i}\overline{w′_{(1)}\{w′_{(2)}; \{⋯; \{w′_{(i)}; \{w′_{(i+1)}[d_{n}, d]\} ⋯\}} and Y = − ∑_{i=1}^{∞} ∑(−1)^{i}\overline{w′_{(1)}d_{n}\{w′_{(2)}; \{⋯; \{w′_{(i)}; \{w′_{(i+1)}d\} ⋯\}} we have

\overline{wd} = Z + Y − ∑_{i=1}^{∞} ∑(−1)^{i}\overline{w′_{(1)}[\{w′_{(2)}; \{⋯; \{w′_{(i)}; \{w′_{(i+1)}d\} ⋯\}, d_{n}]}
= Z + Y − ∑_{i=1}^{∞} ∑_{j=1}^{∞} ∑(−1)^{i+j}\overline{w′_{(1)}\{w′_{(2)}; ⋯}

\overline{⋯; \{w′_{(j)}; \{w′_{(j+1)}d_{n}\{w′_{(j+2)}; ⋯; \{w′_{(i+j+1)}d\} ⋯\}}
= Z + Y − ∑_{k=2}^{∞} ∑_{j=1}^{k−1} ∑(−1)^{k}\overline{w′_{(1)}\{w′_{(2)}; \{⋯}
\overline{⋯; \{w′_{(j+1)}d_{n}; \{w′_{(j+2)}; \{⋯; \{w′_{(k+1)}d\} ⋯\}}
= − ∑_{k=1}^{∞} ∑_{j=1}^{k−1} ∑(−1)^{k}\overline{w′_{(1)}\{w′_{(2)}; ⋯; \{w′_{(j)}d_{n}; \{⋯; \{w′_{(k+1)}d\} ⋯\}}
= − ∑_{k=1}^{∞}(−1)^{k} ∑ \overline{w_{(1)}\{w_{(2)}; ⋯; \{w_{(j)}; \{⋯; \{w_{(k+1)}d\} ⋯\}}

as desired. □ Given w ∈ T(L) we define

⟨w; d⟩ = ∑_{i=1}^{∞} ∑(−1)^{i}{w_{(1)}; {⋯; {w_{(i)}d} ⋯} ∈ V (16)

for any d ∈ L. So, in U(L)/H U(L) the relation

\overline{wd} = − ∑ \overline{w_{(1)}⟨w_{(2)}; d⟩} (17) holds. Given x_{1}, …, x_{n}, a, b ∈ V with n ≥ 1, let us define

\begin{matrix}⟨x_{1} ⋯ x_{n}; a, b⟩ = ⟨x_{1} ⋯ x_{n}; [a, b]⟩ and \\ ⟨1; a, b⟩ = ⟨a, b⟩ = ⟨1; [a, b]⟩. \end{matrix} (18)

Proposition 17 The vector space V with the operations ⟨; ⟩ defined by (18) is a Sabinin algebra. PROOF. First, in order to prove (13), we observe that in U(L)/H U(L) we have

− ∑ \overline{x_{(1)}⟨x_{(2)}c; a, b⟩} − ∑ \overline{x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; a, b⟩, c⟩} =
\overline{xc[a, b]} + ∑ \overline{x_{(1)}c⟨x_{(2)}; a, b⟩} + ∑ \overline{x_{(1)}[⟨x_{(2)}; a, b⟩, c]} =
\overline{xc[a, b]} + ∑ \overline{x_{(1)}⟨x_{(2)}; a, b⟩c} =
\overline{xc[a, b]} − \overline{x[a, b]c} = −\overline{x[[a, b], c]}

so the cyclic sum on a, b, c of this element vanishes.

We may use induction on the degree of x ∈ T(L) to prove (13). If x = 1 then σ_{a,b,c}(\overline{⟨xc; a, b⟩} + ∑ \overline{⟨x_{(1)}; ⟨x_{(2)}; a, b⟩, c⟩}) = 0, so

σ_{a,b,c}(⟨xc; a, b⟩ + ∑⟨x_{(1)}; ⟨x_{(2)}; a, b⟩, c⟩) ∈ H U(L) ∩ V.

Since by the Poincaré-Birkhoff-Witt Theorem for Lie algebras H U(L)∩V = 0 then we obtain the result in this case. Let us assume that we have proved (13) for any element of degree lower than |x|,

0 = σ_{a,b,c}(∑ \overline{x_{(1)}⟨x_{(2)}c; a, b⟩} + ∑ \overline{x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; a, b⟩, c⟩})
= σ_{a,b,c}(\overline{⟨xc; a, b⟩} + ∑ \overline{⟨x_{(1)}; ⟨x_{(2)}; a, b⟩, c⟩})

+σ_{a,b,c}( ∑_{|x_{(2)}|<|x|} x_{(1)}(⟨x_{(2)}c; a, b⟩ + ∑⟨x_{(2)}; ⟨x_{(3)}; a, b⟩, c⟩))

= σ_{a,b,c}(\overline{⟨xc; a, b⟩} + ∑ \overline{⟨x_{(1)}; ⟨x_{(2)}; a, b⟩, c⟩})

and the result follows as in the case x = 1.

Given y = y_{1} ⋯ y_{r} ∈ T(L) with y_{1}, …, y_{r} ∈ V,

∑ \overline{x_{(1)}y_{(1)}⟨x_{(2)}[a, b]y_{(2)}; c, d⟩} + ∑ \overline{x_{(1)}[a, b]y_{(1)}⟨x_{(2)}y_{(2)}; c, d⟩}

= −\overline{x[a, b]y[c, d]} = −(⋯ (\overline{x[a, b]}y_{1}) ⋯)y_{r})[c, d]
= ∑ \overline{x_{(1)}⟨x_{(2)}; a, b⟩y[c, d]}
= − ∑ \overline{x_{(1)}⟨x_{(2)}; a, b⟩y_{(1)}⟨x_{(3)}y_{(2)}; c, d⟩}
− ∑ \overline{x_{(1)}y_{(1)}⟨x_{(2)}⟨x_{(3)}; a, b⟩y_{(2)}; c, d⟩}
= ∑ \overline{x_{(1)}[a, b]y_{(1)}⟨x_{(2)}y_{(2)}; c, d⟩} − ∑ \overline{x_{(1)}y_{(1)}⟨x_{(2)}⟨x_{(3)}; a, b⟩y_{(2)}; c, d⟩}

so

∑ \overline{x_{(1)}y_{(1)}⟨x_{(2)}[a, b]y_{(2)}; c, d⟩} = − ∑ \overline{x_{(1)}y_{(1)}⟨x_{(2)}⟨x_{(3)}; a, b⟩y_{(2)}; c, d⟩}

and we can conclude (12) by induction on |x| + |y|. □

Example 18 (Shestakov-Umirbaev) Let C be an algebra with 1, L the Lie algebra generated by the right multiplication operators on C, and H = {φ ∈ L | 1φ = 0}. Since L = H ⊕ V with V = span⟨R_{x} | x ∈ C⟩, then the structure of Sabinin algebra on V is inherited by C by

⟨R_{x_{1}} ⋯ R_{x_{n}}; R_{a}, R_{b}⟩ = R_{⟨x_{1},...,x_{n};a,b⟩}.

If we denote R_{x_{1}} ⋯ R_{x_{n}} by R then in U(L)/H U(L) we have \overline{R[R_{a}, R_{b}]} = − ∑ \overline{R_{(1)}⟨R_{(2)}; R_{a}, R_{b}⟩}. Since H U(L) kills the unit element then we may evaluate this equality on 1 to obtain

(ua)b − (ub)a = − ∑ u_{(1)}⟨u_{(2)}; a, b⟩ (19)

with u = ((x_{1}x_{2}) ⋯)x_{n}, so

(u, a, b) − (u, b, a) + u[a, b] = − ∑ u_{(1)}⟨u_{(2)}; a, b⟩.

Using that

⟨1; a, b⟩ = 1⟨R_{1}; R_{a}, R_{b}⟩ = −1{R_{a}, R_{b}} = −1[R_{a}, R_{b}] = −[a, b]

then we finally obtain that

(u, b, a) − (u, a, b) = ∑_{|u_{(2)}|>1} u_{(1)}⟨u_{(2)}; a, b⟩.

This is the relation (17) in [19], so ⟨; ⟩ is the same Sabinin algebra structure on V as the one in [19].

Given a Lie algebra L = H ⊕ V with H a subalgebra, one may induce multilinear operations ⟨; ⟩ on V by the following recurrence:

{xab} + ∑{x_{(1)}⟨x_{(2)}; a, b⟩} = 0. (20)

The Lie algebra L is called a Lie envelope of (V, ⟨; ⟩). Sabinin and Miheev [17] proved that (V, ⟨; ⟩) is a Sabinin algebra and that any Sabinin algebra admits a Lie envelope. This construction of (V, ⟨; ⟩) agrees with the one in (18).

Proposition 19 Let ⟨; ⟩ be defined by (20), then

⟨x; a, b⟩ = ∑_{i=1}^{∞} ∑(−1)^{i}{x_{(1)}; {⋯; {x_{(i)}ab} ⋯}.

PROOF. For x = 1 the statement follows from

⟨1; a, b⟩ + {ab} = 0.

Let us assume that the statement is true for any element of degree < |x| then

⟨x; a, b⟩ = −{xab} − ∑_{|x_{(1)}|≠1}{x_{(1)}⟨x_{(2)}; a, b⟩}

= −{xab} − ∑_{|x_{(1)}|≠1} ∑_{i=1}^{∞}(−1)^{i}{x_{(1)}{x_{(2)}; {⋯; {x_{(i+1)}ab} ⋯}

= −{xab} + ∑_{|x_{(1)}|≠1} ∑_{i=2}^{∞}(−1)^{i}{x_{(1)}{x_{(2)}; {⋯; {x_{(i)}ab} ⋯}

= −{xab} + ∑_{i=2}^{∞} ∑(−1)^{i}{x_{(1)}; {x_{(2)}; {⋯; {x_{(i)}ab} ⋯}
= ∑_{i=1}^{∞} ∑(−1)^{i}{x_{(1)}; {⋯; {x_{(i)}ab} ⋯}

□

5 A Poincaré–Birkhoff–Witt Theorem for Sabinin algebras

Let (V, ⟨; ⟩) be a Sabinin algebra and

S̃(V) = T(V)/ span⟨x[a, b]y + ∑ x_{(1)}⟨x_{(2)}; a, b⟩y | x, y ∈ T(V), a, b ∈ V ⟩.

The aim of this section is to prove that for any basis {a_{i}}_{i∈Λ} of V with Λ a totally ordered set then

\{\overline{a_{i_{1}} ⋯ a}_{i_{n}} | i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n} and n ∈ ℕ\}

is a basis of S̃(V).

The name Poincaré–Birkhoff–Witt Theorem for this result is justified by the following example.

Example 20 Any Lie algebra L is a Sabinin algebra with

⟨x_{1}, …, x_{n}; a, b⟩ = \{ \begin{matrix}0 & n ≥ 1 \\ −[a, b] & n = 0. \end{matrix}

For this structure of Sabinin algebra we have

S̃(L) = T(L)/⟨x(ab − ba)y − x[a, b]y | x, y ∈ T(L), a, b ∈ L⟩ = U(L)

and the Poincaré-Birkhoff-Witt Theorem for Sabinin algebras will imply the usual Poincaré-Birkhoff-Witt Theorem for Lie algebras.

There are several ways to achieve our goal, and it is a matter of taste which one to follow. We first derive the result very quickly from the usual Poincaré– Birkhoff–Witt Theorem for Lie algebras using a Lie envelope of (V, ⟨; ⟩) (see paragraph previous to Proposition 19). However we also present a proof based on Gröbner bases so that the result for Lie algebras becomes a particular case.

Theorem 21 (Poincaré-Birkhoff-Witt) Let {a_{i} | i ∈ Λ} be a totally ordered basis of V. Then \{\overline{a_{i_{1}} ⋯ a}_{i_{n}} | i_{1} ≤ ⋯ ≤ i_{n} and n ≥ 0\} is a basis of S̃(V).

PROOF. Let E = H ⊕ V be a Lie envelope. By Proposition 19, (16) and (17) in U(E)/H U(E) we have that \overline{x[a, b]} = − ∑ \overline{x_{(1)}⟨x_{(2)}; a, b⟩}. Thus the natural map from T(V) to U(E)/H U(E) factors through S̃(V). By the usual Poincaré–Birkhoff–Witt Theorem the image of \{\overline{a_{i_{1}} ⋯ a}_{i_{n}} | i_{1} ≤ ⋯ ≤ i_{n} and n ≥ 0} is linearly independent in U(E)/H U(E), so this set is linearly independent in S̃(V). Since it obviously generates the whole S̃(V) then it is a basis. □

We devote the remainder of this section to present a proof of the Poincaré– Birkhoff–Witt Theorem for Sabinin algebras based on Gröbner bases. We restrict ourselves to the case dim V ≤ ℵ_{0}.

In the following we will use the word monomial as synonymous of (associative) monomial in the basis {a_{1}, a_{2}, …} of V. It will be convenient to order these monomials by the deglex order. In this order u < v if and only if either deg(u) < deg(v) or deg(u) = deg(v) and u = wa_{i}u′, v = wa_{j}v′ with i < j. The main feature of this order to our purposes is that it is multiplicative and it satisfies the descending chain condition allowing us to use induction (see [2] from which we borrow the main ideas in our arguments). The leading monomial LM(f) of a nonzero element f in T(V) is defined in the obvious way, but it will always be taken with coefficient 1.

Let

I = span ⟨x[a, b]y + ∑ x_{(1)}⟨x_{(2)}; a, b⟩y | x, y ∈ T(V), a, b ∈ V ⟩

with [a, b] = ab − ba and

G = {x[a, b] + ∑ x_{(1)}⟨x_{(2)}; a, b⟩ |

x is a monomial ordered in a non decreasing way and a > b}.

Lemma 22 The right ideal I of T(V) is generated by G.

PROOF. Given a monomial w we define

Î_{w} = span⟨gu | g ∈ G, u is a monomial, and LM(g)u ≤ w⟩.

By definition Î_{w} ⊆ I (in fact Î_{w} is contained in the right ideal generated by G) for any w so, we will finish once we prove the following claim:

f = x[a, b] + ∑ x_{(1)}⟨x_{(2)}; a, b⟩ ∈ Î_{LM(f)}. (21)

This element might not belong to G since no order is assumed on x. We will prove this claim by using induction on the degree of x. Without lost of generality we may assume that a > b. If x is ordered in a non decreasing way, and this happens if deg(x) ≤ 1 for instance, then f belongs to G and the result follows. Assume that we have proved the claim for any x with deg(x) < n. Given any monomial x of degree n we define O(x) = max{deg(x′) | x = x′y for some monomials x′ and y with x′ ordered}. We will use backwards induction (starting from n) on O(x) to show that the claim also holds if deg(x) = n. If O(x) = n then x is ordered and we know that the claim is true in this case. Therefore we may assume that x = x′dcx′′ with d > c and x′ ordered. We have that

f = x′dcx′′[a, b] + ∑ x_{(1)}⟨x_{(2)}; a, b⟩
= x′cdx′′[a, b] + x′[d, c]x′′[a, b] + ∑ x_{(1)}⟨x_{(2)}; a, b⟩.

Since O(x′cdx′′) > O(x′dcx′′) then by the backwards induction x′cdx′′[a, b] + ∑(x′cdx′′)_{(1)}⟨(x′cdx′′)_{(2)}; a, b⟩ ∈ Î_{LM(x′cdx′′[a,b])} ⊆ Î_{LM(f)} (note that Î_{w′} ⊆ Î_{w} if w′ ≤ w). Thus, modulo Î_{LM(f)},

f ≡ − ∑(x′cdx′′)_{(1)}⟨(x′cdx′′)_{(2)}; a, b⟩ + x′[d, c]x′′[a, b] + ∑ x_{(1)}⟨x_{(2)}; a, b⟩
=^{⟨1⟩} − ∑ x′_{(1)}cdx′′_{(1)}⟨x′_{(2)}x′′_{(2)}; a, b⟩ − ∑ x′_{(1)}cx′′_{(1)}⟨x′_{(2)}dx′′_{(2)}; a, b⟩
− ∑ x′_{(1)}dx′′_{(1)}⟨x′_{(2)}cx′′_{(2)}; a, b⟩ − ∑ x′_{(1)}x′′_{(1)}⟨x′_{(2)}cdx′′_{(2)}; a, b⟩
+x′[d, c]x′′[a, b] + ∑ x_{(1)}⟨x_{(2)}; a, b⟩
=^{⟨2⟩} ∑ x′_{(1)}[d, c]x′′_{(1)}⟨x′_{(2)}x′′_{(2)}; a, b⟩ + ∑ x′_{(1)}x′′_{(1)}⟨x′_{(2)}[d, c]x′′_{(2)}; a, b⟩
+x′[d, c]x′′[a, b]

where ⟨1⟩ follows by expanding ∑(x′cdx′′)_{(1)}⟨(x′cdx′′)_{(2)}; a, b⟩ (recall that c and d are primitive elements) and ⟨2⟩ follows by expanding ∑ x_{(1)}⟨x_{(2)}; a, b⟩ and simplifying terms. Since x′[d, c] + ∑ x′_{(1)}⟨x′_{(2)}; d, c⟩ ∈ G then by the very definition of Î_{w} we obtain that x′[d, c]x′′[a, b]+∑ x′_{(1)}⟨x′_{(2)}; d, c⟩x′′[a, b] ∈ Î_{LM(f)}. Therefore, modulo Î_{LM(f)},

f ≡ ∑ x′_{(1)}[d, c]x′′_{(1)}⟨x′_{(2)}x′′_{(2)}; a, b⟩ + ∑ x′_{(1)}x′′_{(1)}⟨x′_{(2)}[d, c]x′′_{(2)}; a, b⟩
− ∑ x′_{(1)}⟨x′_{(2)}; d, c⟩x′′[a, b]
≡^{⟨3⟩} ∑ x′_{(1)}[d, c]x′′_{(1)}⟨x′_{(2)}x′′_{(2)}; a, b⟩ + ∑ x′_{(1)}x′′_{(1)}⟨x′_{(2)}[d, c]x′′_{(2)}; a, b⟩
+ ∑ x′_{(1)}⟨x′_{(2)}; d, c⟩x′′_{(1)}⟨x′_{(3)}x′′_{(2)}; a, b⟩
+ ∑ x′_{(1)}x′′_{(1)}⟨x′_{(2)}⟨x′_{(3)}; d, c⟩x′′_{(2)}; a, b⟩

where in ⟨3⟩ we have used that the claim holds for elements of degree < n and that Î_{LM(x′_{(1)}⟨x′_{(2)};d,c⟩x′′[a,b])} ⊆ Î_{LM(f)} (compare the degree of the subindices). Finally, since

∑ x′_{(1)}[d, c]x′′_{(1)}⟨x′_{(2)}x′′_{(2)}; a, b⟩ + ∑ x′_{(1)}⟨x′_{(2)}; d, c⟩x′′_{(1)}⟨x′_{(3)}x′′_{(2)}; a, b⟩ =
∑ (x′_{(2)}[d, c] + ∑ x′_{(2)}⟨x′_{(3)}; d, c⟩)x′′_{(1)}⟨x′_{(1)}x′′_{(2)}; a, b⟩

and x′_{(2)}[d, c]+ ∑ x′_{(2)}⟨x′_{(3)}; d, c⟩ ∈ G then modulo Î_{LM(f)} this sum vanishes. So,

f ≡ ∑ x′_{(1)}x′′_{(1)}⟨x′_{(2)}[d, c]x′′_{(2)}; a, b⟩ + ∑ x′_{(1)}x′′_{(1)}⟨x′_{(2)}⟨x′_{(3)}; d, c⟩x′′_{(2)}; a, b⟩

which by (12) gives f ∈ Î_{LM(f)}. □

Theorem 23 (Poincaré-Birkhoff-Witt) Let {a_{1}, a_{2}, …} be a basis of V. Then

\{\overline{a_{i_{1}} ⋯ a}_{i_{n}} | i_{1} ≤ ⋯ ≤ i_{n} and n ≥ 0\}

is a basis of S̃(V).

PROOF. As in [2] it is enough to prove that whenever LM(g)u = LM(g′)v then gu−g′v ∈ span⟨g′′u | g′′ ∈ G, u is a monomial and LM(g′′)u < LM(g)u⟩. For short we will denote LM(g)u by t and span⟨g′′u | g′′ ∈ G, u is a monomial and LM(g′′)u < LM(g)u⟩ by I_{t}.

Let x, x′ be two ordered monomials, a > b, a′ > b′ and u, v such that

xabu = x′a′b′v.

We have that either xab = x′a′b′ and u = v (there is nothing to prove in this case) or xab is a prefix of x′a′b′ with xab ≠ x′a′b′. We will focus on this later case. Since a > b and x′ is ordered then x′ = xa and a′ = b, so u = b′v. That is,

x′ = xa, a′ = b and u = b′v with a > b > b′ and the factors of x are ≤ a.

On the other hand,

gu − g′v = (x[a, b] + ∑ x_{(1)}⟨x_{(2)}; a, b⟩)u − (x′[a′, b′] + ∑ x′_{(1)}⟨x′_{(2)}; a′, b′⟩)v
= xab′bv − xbab′v + ∑ x_{(1)}⟨x_{(2)}; a, b⟩b′v − ∑ x_{(1)}a⟨x_{(2)}; b, b′⟩v
− ∑ x_{(1)}⟨x_{(2)}a; b, b′⟩v

Therefore, if we prove that the element C defined by

xab′b − xbab′ + ∑ x_{(1)}⟨x_{(2)}; a, b⟩b′
− ∑ x_{(1)}a⟨x_{(2)}; b, b′⟩ − ∑ x_{(1)}⟨x_{(2)}a; b, b′⟩

belongs to I_{xabb′} then gu − g′v will belong to I_{xabu} as desired.

Since

C = xab′b − xbab′ + ∑ x_{(1)}⟨x_{(2)}; a, b⟩b′ − ∑ x_{(1)}⟨x_{(2)}; b, b′⟩a
+ ∑ x_{(1)}[⟨x_{(2)}; b, b′⟩, a] − ∑ x_{(1)}⟨x_{(2)}a; b, b′⟩

and x[b, b′]+∑ x_{(1)}⟨x_{(2)}; b, b′⟩, ∑ x_{(1)}[⟨x_{(2)}; b, b′⟩, a]+∑ x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; b, b′⟩, a⟩ ∈ span⟨G⟩ then modulo I_{xabb′} we have

C ≡ xab′b − xbab′ + ∑ x_{(1)}⟨x_{(2)}; a, b⟩b′ + x[b, b′]a
− ∑ x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; b, b′⟩, a⟩ − ∑ x_{(1)}⟨x_{(2)}a; b, b′⟩
= xab′b − xbab′ + ∑ x_{(1)}b′⟨x_{(2)}; a, b⟩ − ∑ x_{(1)}[b′, ⟨x_{(2)}; a, b⟩] + x[b, b′]a
− ∑ x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; b, b′⟩, a⟩ − ∑ x_{(1)}⟨x_{(2)}a; b, b′⟩

By (21) (xb′)[a, b] + ∑(xb′)_{(1)}⟨(xb′)_{(2)}; a, b⟩ ∈ Î_{xb′ab} (the leading monomial of this element is xb′ab) and since b′ < a then by definition Î_{xb′ab} ⊆ I_{xabb′}, so

C ≡ xab′b − xbab′ − xb′[a, b] − ∑ x_{(1)}⟨x_{(2)}b′; a, b⟩ − ∑ x_{(1)}[b′, ⟨x_{(2)}; a, b⟩]
+x[b, b′]a − ∑ x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; b, b′⟩, a⟩ − ∑ x_{(1)}⟨x_{(2)}a; b, b′⟩.

Since ∑ x_{(1)}[b′, ⟨x_{(2)}; a, b⟩] + ∑ x_{(1)}⟨x_{(2)}; b′, ⟨x_{(3)}; a, b⟩⟩ ∈ span⟨G⟩ then

C ≡ xab′b − xbab′ − xb′[a, b] − ∑ x_{(1)}⟨x_{(2)}b′; a, b⟩

+ ∑ x_{(1)}⟨x_{(2)}; b′, ⟨x_{(3)}; a, b⟩⟩ + x[b, b′]a − ∑ x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; b, b′⟩, a⟩

− ∑ x_{(1)}⟨x_{(2)}a; b, b′⟩
= xab′b − xbab′ − xb′[a, b] + x[b, b′]a + ∑ x_{(1)}⟨x_{(2)}b; b′, a⟩
+ ∑ x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; b′, a⟩, b⟩

where the equality follows from (13). Collecting terms we obtain

C ≡ x[a, b′]b − xb[a, b′] + ∑ x_{(1)}⟨x_{(2)}b; b′, a⟩ + ∑ x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; b′, a⟩, b⟩.

By (21) we have that xb[a, b′]−∑ x_{(1)}⟨x_{(2)}b; b′, a⟩+∑ x_{(1)}b⟨x_{(2)}; a, b′⟩ ∈ Î_{xbab′} ⊆ I_{xabb′} so

C ≡ x[a, b′]b + ∑ x_{(1)}b⟨x_{(2)}; a, b′⟩ + ∑ x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; b′, a⟩, b⟩.

Again, by (21) x[a, b′]b + ∑ x_{(1)}⟨x_{(2)}; a, b′⟩b ∈ Î_{xab′b} ⊆ I_{xabb′} thus

C ≡ − ∑ x_{(1)}⟨x_{(2)}; a, b′⟩b + ∑ x_{(1)}b⟨x_{(2)}; a, b′⟩ + ∑ x_{(1)}⟨x_{(2)}; ⟨x_{(3)}; b′, a⟩, b⟩
= ∑ x_{(1)}[b, ⟨x_{(2)}; a, b′⟩] + ∑ x_{(1)}⟨x_{(2)}; b, ⟨x_{(3)}; a, b′⟩⟩
≡ 0

where the last congruence is a consequence of

∑ x_{(1)}[b, ⟨x_{(2)}; a, b′⟩] + x_{(1)}⟨x_{(2)}; b⟨x_{(3)}; a, b′⟩ ∈ span⟨G⟩.

□

The vector space S̃(V) is filtered by S̃(V)_{n} = span⟨x̄ ∈ S̃(V) | |x| ≤ n⟩ and we may define the degree |x̄| of a nonzero element x̄ ∈ S̃(V) as the minimum n such that x̄ ∈ S̃(V)_{n} (the degree of ¯0 is set to −∞ as usual). We may consider the map

ϕ: T(V) → Gr(S̃(V))
x_{1} ⊗ ⋯ ⊗ x_{n} ↦ \overline{x_{1} ⋯ x}_{n} + S̃(V)_{n−1}

where Gr(S̃(V)) = ⨁_{n=0}^{∞} S̃(V)_{n}/S̃(V)_{n−1} denotes the graded vector space associated to the filtration S̃(V) = ⋃_{n=0}^{∞} S̃(V)_{n}. By definition of S̃(V) this map induces a map

ϕ: S(V) → Gr(S̃(V))

from the symmetric algebra on V to Gr(S̃(V)). The Poincaré–Birkhoff–Witt Theorem amounts to saying that this map is an isomorphism.

Since in T(V)

Δ(x[a, b]y + ∑ x_{(1)}⟨x_{(2)}; a, b⟩y) =
= ∑_{(x),(y)}(x_{(2)}[a, b]y_{(2)} + ∑_{(x_{(2)})} x_{(2)}⟨x_{(3)}; a, b⟩y_{(2)}) ⊗ x_{(1)}y_{(1)}

+ ∑_{(x),(y)} x_{(1)}y_{(1)} ⊗ (x_{(2)}[a, b]y_{(2)} + ∑_{(x_{(2)})} x_{(2)}⟨x_{(3)}; a, b⟩y_{(2)})

then S̃(V) inherits the structure of coalgebra from T(V), i.e.

Δ(x̄) = ∑ x̄_{(1)} ⊗ x̄_{(2)} = ∑ \overline{x_{(1)}} ⊗ \overline{x_{(2)}} and ϵ(x̄) = ϵ(x). (22)

Using the Poincaré–Birkhoff–Witt Theorem it is easy to check that S̃(V) and S(V) are isomorphic coalgebras.

6 A universal enveloping algebra for Sabinin algebras

The universal enveloping algebra of a Sabinin algebra (V, ⟨; ⟩, Φ) will be define in subsection 6.2 as a particular case of algebra (S̃(V), l). While (S̃(V), l) is defined over arbitrary fields, the particular choice of l involved in the definition of the universal enveloping algebra of a Sabinin algebra only makes sense over fields of characteristic zero. We follow our convention of making linear maps to act from the right in this part of the paper. To avoid annoying notation, given x̄ ∈ S̃(V) and a ∈ V we will write x̄a instead of \overline{xa}. After defining the product on (S̃(V), l) this notation will become even more natural.

6.1 Algebras (S̃(V), l)

Given d ∈ Hom(S̃(V), V) we will use the notation

⟨x̄; d⟩ = (x̄)d.

It will become natural in our context to consider E = Hom(S̃(V), V) with the product

⟨x̄; [d, d′]⟩ = ∑⟨x̄_{(1)}; ⟨x̄_{(2)}; d⟩, ⟨x̄_{(3)}; d′⟩⟩ − ⟨x̄_{(1)}⟨x̄_{(2)}; d⟩; d′⟩
+⟨x̄_{(1)}⟨x̄_{(2)}; d′⟩; d⟩

and the action

x̄d = − ∑ x̄_{(1)}⟨x̄_{(2)}; d⟩.

Proposition 24 The algebra E = (Hom(S̃(V), V), [ , ]) is a Lie algebra and S̃(V) is a right E–module.

PROOF. Since

x̄dd′ − x̄d′d = − ∑(x̄_{(1)}⟨x̄_{(2)}; d⟩)d′ + ∑(x̄_{(1)}⟨x̄_{(2)}; d′⟩)d
= ∑(x̄_{(1)}⟨x̄_{(2)}; d⟩)⟨x̄_{(3)}; d′⟩ + ∑ x̄_{(1)}⟨x̄_{(2)}⟨x̄_{(3)}; d⟩; d′⟩
− ∑(x̄_{(1)}⟨x̄_{(2)}, d′⟩)⟨x̄_{(3)}; d⟩ − ∑ x̄_{(1)}⟨x̄_{(2)}⟨x̄_{(3)}; d′⟩; d⟩
= − ∑ x̄_{(1)}⟨x̄_{(2)}; ⟨x̄_{(3)}; d⟩, ⟨x_{(4)}; d′⟩⟩ + ∑ x̄_{(1)}⟨x̄_{(2)}⟨x̄_{(3)}; d⟩; d′⟩
− ∑ x̄_{(1)}⟨x̄_{(2)}⟨x̄_{(3)}; d′⟩; d⟩
= − ∑ x̄_{(1)}⟨x̄_{(2)}; [d, d′]⟩
= x̄[d, d′],

then the map

ρ: E → End_{F} (S̃(V))^{(−)}
d ↦ d: x̄ ↦ − ∑ x̄_{(1)}⟨x̄_{(2)}; d⟩

is a homomorphism of Lie algebras. Moreover, given a map d ∈ E such that ∑ x̄_{(1)}⟨x̄_{(2)}; d⟩ = 0 for any x̄ ∈ S̃(V) then it follows by induction on |x|, the degree of x, that ⟨x̄; d⟩ = 0. Hence ρ is injective and the result follows. □

In fact, Hom(S̃(V), V) is a Lie envelope of (V, ⟨; ⟩). Consider H = {d ∈ Hom(S̃(V), V) | ¯1d = 0} and, for any a ∈ V,

τ_{a}: S̃(V) → V
¯1 ↦ −a
x̄ ↦ 0 |x̄| ≥ 1.

The decomposition

E = H ⊕ V̄

with V̄ = {τ_{a} | a ∈ V } induces on V̄ the structure of Sabinin algebra.

Proposition 25 The map

τ: V → V̄
a ↦ τ_{a}

is an isomorphism of Sabinin algebras.

PROOF. Clearly τ is bijective. The structure of Sabinin algebra on V̄ is determined by the relation

\overline{wd} = − ∑ \overline{w_{(1)}⟨w_{(2)}; d⟩} ∀w ∈ U(E), d ∈ E (23)

on U(E)/H U(E), and by definition given τ_{x} = τ_{a_{1}} ⋯ τ_{a_{n}} ∈ T(V̄),

⟨τ_{x}; τ_{a}, τ_{b}⟩ = ⟨τ_{x}; [τ_{a}, τ_{b}]⟩.

Since ⟨τ_{x}; τ_{a}, τ_{b}⟩ belongs to V̄ then there exists ⟨x; a, b⟩′ ∈ V such that

⟨τ_{x}; τ_{a}, τ_{b}⟩ = τ_{⟨x;a,b⟩′}.

Using the action of E on S̃(V) and the fact that H kills ¯1 then by (23) with w = τ_{x} and d = [τ_{a}, τ_{b}] we obtain

¯1τ_{x}[τ_{a}, τ_{b}] = \{ \begin{matrix}\overline{xab} − \overline{xba} = − ∑ \overline{x_{(1)}}⟨x_{(2)}; a, b⟩ \\ − ∑ \overline{1}τ_{x_{(1)}}⟨τ_{x_{(2)}}; [τ_{a}, τ_{b}]⟩ = − ∑ \overline{x_{(1)}}⟨x_{(2)}; a, b⟩′ \end{matrix}

from which ⟨x; a, b⟩′ = ⟨x; a, b⟩. Therefore, ⟨τ_{a_{1}} ⋯ τ_{a_{n}}; τ_{a}, τ_{b}⟩ = τ_{⟨a_{1}···a_{n};a,b⟩}. □

Corollary 26 The map

ψ: U(E)/H U(E) → S̃(V)
D̄ ↦ ¯1D

is an isomorphism of E-modules.

PROOF. Since ¯1τ_{a_{1}} ⋯ τ_{a_{n}} = \overline{a_{1} ⋯ a}_{n} then by the Poincaré-Birkhoff-Witt Theorem ψ sends a basis of U(E)/H U(E) to a basis of S̃(V), so ψ is an linear isomorphism. Furthermore,

(\overline{τ}_{x}d)ψ = \overline{1}τ_{x}d = \overline{x}d = (\overline{τ}_{x})ψd

for any d ∈ E implies that ψ is an isomorphism as modules. □

Now we proceed to construct the algebras (S̃(V), l). This is done using a fixed Poincaré–Birkhoff–Witt basis \{\overline{a}_{I}\}_{I} of S̃(V). The independence of this basis is a consequence of Proposition 30. The definition is based on certain map l which existence at this point might be questionable. After Theorem 31 we will present a natural source for these maps.

Given l: S̃(V) ⊗ E → H satisfying

l(¯1, d) = d + τ_{⟨1;d⟩}
l(ȳ, [d, d′]) = l(ȳd, d′) − l(ȳd′, d) + ∑[l(ȳ_{(1)}, d), l(ȳ_{(2)}, d′)]  (24)

we define a product on S̃(V) by

x̄¯1 = x̄
x̄ā = \overline{xa}
x̄ā_{I} = ∑(x̄ā_{I′})ā_{i_{n}} − ∑ x̄l(ā_{I′(1)}, τ_{a_{in}})ā_{I′(2)}

where ā_{I} = \overline{a_{i_{1}} ⋯ a}_{i_{n}} with i_{1} ≤ ⋯ ≤ i_{n} and ā_{I′} = \overline{a_{i_{1}} ⋯ a}_{n−1}. We will denote this algebra by (S̃(V), l).

Proposition 27 The element ¯1 is the unit element of (S̃(V), l).

PROOF. By definition, ā¯1 = ¯1 and ¯1ā = ā. Using induction we have

¯1ā_{I} = (¯1ā_{I′})ā_{i_{n}} − ∑ ¯1l(ā_{I′(1)}, τ_{a_{in}})ā_{I′(2)} = ā_{I}

as desired. □

Formally we denote by u: F → S̃(V) the unit map 1 ↦ ¯1. Recall that in (22) we saw that S̃(V) is a coassociative, cocommutative connected coalgebra.

Proposition 28 (S̃(V), Δ, ϵ, ·, u) is a coassociative, cocommutative connected unital bialgebra and V ⊆ Prim(S̃(V)).

PROOF. We will show that Δ(x̄ā_{I}) = Δ(x̄)Δ(ā_{I}) by induction on the degree |I| of I (the case |I| = 0 is trivial). By the very definition of the product on (S̃(V), l) and the action of E on S̃(V) we have

Δ(x̄ā_{I}) = Δ(x̄)Δ(ā_{I′}) · Δ(ā_{i_{n}}) − ∑ Δ(x̄l(ā_{I′(1)}, τ_{a_{in}}))Δ(ā_{I′(2)})
= ∑ x̄_{(1)}ā_{I′(1)} · ā_{i_{n}} ⊗ x̄_{(2)}ā_{I′(2)} + ∑ x̄_{(1)}ā_{I′(1)} ⊗ x̄_{(2)}ā_{I′(2)} · ā_{i_{n}}
− ∑ x̄_{(1)}⟨x̄_{(2)}; l(ā_{I′(1)}, ā_{i_{n}})⟩ · ā_{I′(2)} ⊗ x̄_{(3)}ā_{I′(3)}
− ∑ x̄_{(1)}ā_{I′(1)} ⊗ x̄_{(2)}⟨x̄_{(3)}; l(ā_{I′(2)}, ā_{i_{n}})⟩ · ā_{I′(3)}
= ∑ x̄_{(1)}(ā_{I′(1)}ā_{i_{n}}) ⊗ x̄_{(2)}ā_{I′(2)} + ∑ x̄_{(1)}ā_{I′(1)} ⊗ x̄_{(2)}(ā_{I′(2)}ā_{i_{n}})
= Δ(x̄)Δ(ā_{I}).

Similarly one obtains that ϵ(x̄ā_{I}) = ϵ(x̄)ϵ(ā_{I}). □

The following corollary follows from Proposition 5.

Corollary 29 There exit unique \ and / such that (S̃(V), Δ, ϵ, ·, u, \, /) is an H–bialgebra.

Notice that the left and right divisions are uniquely determined by (1) and (2) and the coalgebra structure indeed. For instance, ¯1\x̄ = x̄ = x̄/¯1, ā\x̄ = −āx̄ and x̄/ā = −x̄ā for any a ∈ V and x̄ ∈ S̃(V).

Proposition 30 In (S̃(V), l) we have that

(x̄ȳ)d = x̄(ȳd) + ∑ x̄l(ȳ_{(1)}, d)ȳ_{(2)} ∀d ∈ E.

PROOF. We may assume that ȳ = ā_{I} and use induction on |I|. If |I| = 0 then ȳ = ¯1 and

x̄(¯1d) + x̄l(¯1, d)¯1 = −x̄τ_{⟨¯1;d⟩} + x̄(d + τ_{⟨¯1;d⟩}) = x̄d

which proves the statement in this case. In general, we have that

(x̄ā_{I})d =^{⟨1⟩} (x̄ā_{I′})τ_{a_{in}}d − ∑ (x̄l(ā_{I′(1)}, τ_{a_{in}})ā_{I′(2)}) d
=^{⟨2⟩} (x̄ā_{I′})dτ_{a_{in}} + (x̄ā_{I′})[τ_{a_{in}}, d] − ∑ x̄l(ā_{I′(1)}, τ_{a_{in}})(ā_{I′(2)}d)
− ∑ x̄l(ā_{I′(1)}, τ_{a_{in}})l(ā_{I′(2)}, d)ā_{I′(3)}
=^{⟨3⟩} (x̄ā_{I′})dτ_{a_{in}} + x̄(ā_{I}d) − x̄(ā_{I′}dτ_{a_{in}})
+ ∑ x̄l(ā_{I′(1)}, [τ_{a_{in}}, d])ā_{I′(2)} − ∑ x̄l(ā_{I′(1)}, τ_{a_{in}})(ā_{I′(2)}d)

− ∑ x̄l(ā_{I′(1)}, τ_{a_{in}})l(ā_{I′(2)}, d)ā_{I′(3)}
=^{⟨4⟩} (x̄ā_{I′})dτ_{a_{in}} + x̄(ā_{I}d) − x̄(ā_{I′}dτ_{a_{in}})
+ ∑ x̄l(\overline{a_{I′(1)}a_{i_{n}}}, d)ā_{I′(2)} − ∑ x̄l(ā_{I′(1)}d, τ_{a_{in}})ā_{I′(2)}

− ∑ x̄l(ā_{I′(1)}, d)l(ā_{I′(2)}, τ_{a_{in}})ā_{I′(3)} − ∑ x̄l(ā_{I′(1)}, τ_{a_{in}})(ā_{I′(2)}d)

=^{⟨5⟩} x̄(ā_{I}d) + ∑ x̄l(ā_{I(1)}, d)ā_{I(2)}
+(x̄(ā_{I′}d))τ_{a_{in}} − x̄(ā_{I′}dτ_{a_{in}}) − ∑ x̄l((ā_{I′}d)_{(1)}, τ_{a_{in}})(ā_{I′}d)_{(2)}

where ⟨1⟩ follows by definition, ⟨2⟩ and ⟨3⟩ by induction, ⟨4⟩ by the properties imposed to l and in ⟨5⟩ we have used that

(x̄ā_{I′})dτ_{a_{in}} = (x̄(ā_{I′}d) + ∑ x̄l(ā_{I′(1)}, d)ā_{I′(2)})τ_{a_{in}}
= (x̄(ā_{I′}d))τ_{a_{in}} + ∑ x̄l(ā_{I′(1)}, d) · ā_{I′(2)}ā_{i_{n}}
+ ∑ x̄l(a_{I′(1)}, d)l(ā_{I′(2)}, τ_{a_{in}})ā_{I′(3)}
= (x̄(ā_{I′}d))τ_{a_{in}} + ∑ x̄l(ā_{I(1)}, d) · ā_{I(2)}

− ∑ x̄l(ā_{I′(1)}τ_{a_{in}}, d)ā_{I′(2)} + ∑ x̄l(a_{I′(1)}, d)l(ā_{I′(2)}, τ_{a_{in}})ā_{I′(3)}.

In case that d = τ_{a} with a ≥ a_{i_{n}} then the statement follows from the definition of the product, and the previous computation is superfluous. In case that d ∈ H then by the hypothesis of induction (x̄(ā_{I′}d))τ_{a_{in}} − x̄(ā_{I′}dτ_{a_{in}}) − ∑ x̄l((ā_{I′}d)_{(1)}, τ_{a_{in}})(ā_{I′}d)_{(2)} = 0 and the result follows. Finally, in case that d = τ_{a} with a < a_{i_{n}} then up to terms of lower degree we may order the monomial ā_{I′}d falling in one of the previous cases. □

Theorem 31 The map l extends to l: S̃(V) ⊗ U(E) → U(H) with

(x̄ȳ)D = ∑ x̄l(ȳ_{(1)}, D_{(1)})(ȳ_{(2)}D_{(2)})

for any D ∈ U(E) and x̄, ȳ ∈ S̃(V).

PROOF. If we define l(ȳ, 1) = ϵ(ȳ)1 then Proposition 30 says that

(x̄ȳ)d = ∑ x̄l(ȳ_{(1)}, d_{(1)}) · ȳ_{(2)}d_{(2)}

and

(x̄ȳ)1 = ∑ x̄l(ȳ_{(1)}, 1) · ȳ_{(2)}1.

We may extend these formulas to l: S̃(V) ⊗ T(E) → U(H) by

l(ȳ, dD) = ∑ l(ȳ_{(1)}, d)l(ȳ_{(2)}, D) + l(ȳd, D)

for any d ∈ E and D ∈ T(E). First we claim that

l(ȳ, DD′) = ∑ l(ȳ_{(1)}, D_{(1)})l(ȳ_{(2)}D_{(2)}, D′). (25)

To show this equation we proceed by induction on |D|:

l(ȳ, (dD)D′) = ∑ l(ȳ_{(1)}, d)l(ȳ_{(2)}, DD′) + l(ȳd, DD′)
= ∑ l(ȳ_{(1)}, d)l(ȳ_{(2)}, D_{(1)})l(ȳ_{(3)}D_{(2)}, D′)
+ ∑ l(ȳ_{(1)}d, D_{(1)})l(ȳ_{(2)}D_{(2)}, D′)
∑ l(ȳ_{(1)}, D_{(1)})l(ȳ_{(2)}dD_{(2)}, D′)
= ∑ l(ȳ_{(1)}, dD_{(1)})l(ȳ_{(2)}D_{(2)}, D′)
+ ∑ l(ȳ_{(1)}, D_{(1)})l(ȳ_{(2)}dD_{(2)}, D′)
= ∑ l(ȳ_{(1)}(dD)_{(1)})l(ȳ_{(2)}(dD)_{(2)}, D′).

We also claim that

(x̄ȳ)D = ∑ x̄l(ȳ_{(1)}, D_{(1)}) · ȳ_{(2)}D_{(2)}.

Again by induction, with D = dD′,

(x̄ȳ)D = (x̄ȳ)dD′ = ∑(x̄l(ȳ_{(1)}, d_{(1)}) · ȳ_{(2)}d_{(2)})D′
= ∑ x̄l(ȳ_{(1)}, d_{(1)})l(ȳ_{(2)}d_{(2)}, D′_{(1)}) · ȳ_{(3)}d_{(3)}D′_{(2)}
= ∑ x̄l(ȳ_{(1)}, d_{(1)}D′_{(1)}) · ȳ_{(2)}d_{(2)}D′_{(2)}
= ∑ x̄l(ȳ_{(1)}, D_{(1)}) · ȳ_{(2)}D_{(2)}

So, to prove the theorem it suffices to check that l induces a corresponding map l: S̃(V) ⊗ U(E) → U(H), i.e.,

l(ȳ, Ddd′D′) − l(ȳ, Dd′dD′) = l(ȳ, D[d, d′]D′).

By (25) it its enough to show that

l(ȳ, dd′D′) − l(ȳ, d′dD′) − l(ȳ, [d, d′]D′) = 0.

Since

l(ȳ, dd′D′) = ∑ l(ȳ_{(1)}, d)l(ȳ_{(2)}, d′D′) + l(ȳd, d′D′)
= ∑ l(ȳ_{(1)}, d)l(ȳ_{(2)}, d′)l(ȳ_{(3)}, D′) + ∑ l(ȳ_{(1)}, d)l(ȳ_{(2)}d′, D′)
+ ∑ l(ȳ_{(1)}d, d′)l(ȳ_{(2)}, D′) + ∑ l(ȳ_{(1)}, d′)l(ȳ_{(2)}d, D′)
+l(ȳdd′, D′)

then

l(ȳ, dd′D′) − l(ȳ, d′dD′) = ∑[l(ȳ_{(1)}, d), l(ȳ_{(2)}, d′)]l(ȳ_{(3)}, D′)
+ ∑ l(ȳ_{(1)}d, d′)l(ȳ_{(2)}, D′) − ∑ l(ȳ_{(1)}d′, d)l(ȳ_{(2)}, D′)
+l(ȳ[d, d′], D′)
= ∑ l(ȳ_{(1)}, [d, d′])l(ȳ_{(2)}, D′) + l(ȳ[d, d′], D′)
= l(ȳ, [d, d′]D)

as desired. □

Finally, let us show how to obtain adequate maps l to carry out the previous construction. Let Φ̂: S̃(V) ⊗ S̃(V) → V be an arbitrary map that verifies

Φ̂(¯1, ȳ) = 0 ∀ȳ
Φ̂(x̄, ā) = Φ̂(x̄, ¯1) = 0 ∀a ∈ V, x̄ ∈ S̃(V),

and B = {a_{i}}_{i∈Λ} an ordered basis of V. While the first requirement about Φ̂ will be necessary to ensure that the image of l lays in H, the second is superfluous and does not play any special role in the following.

From Φ̂ and the ordered basis B we define another map

l: S̃(V) ⊗ E → H

as follows:

1. l(¯1, d) = d + τ_{⟨1;d⟩}
2. l(ā_{I}, τ_{a}) = Φ̂(−, \overline{a_{I}a}) if a_{i_{n}} ≤ a  (26)
3. l(ā_{I}, d) = l(ā_{I′}d, τ_{a_{in}}) + l(ā_{I′}, [τ_{a_{in}}, d]) − ∑[l(ā_{I′(1)}, τ_{a_{in}}), l(ā_{I′(2)}, d)]
if d ∈ H or d = τ_{a} with a < a_{i_{n}}.

We should notice that in this later case,

l(ā_{I′}, [τ_{a_{in}}, d]) = l(ā_{I′}τ_{a_{in}}, d) − l(ā_{I′}d, τ_{a_{in}})

+ ∑[l(ā_{I′(1)}, τ_{a_{in}}), l(ā_{I′(2)}, d)]  (27)

In case that d = τ_{a} with a ≥ a_{i_{n}} then it also follows by skew–symmetry. Proposition 32 For any d, d′ ∈ E and ȳ ∈ S̃(V) we have

l(ȳ, [d, d′]) = l(ȳd, d′) − l(ȳd′, d) + ∑[l(ȳ_{(1)}, d), l(ȳ_{(2)}, d′)]

PROOF. We may assume that ȳ = ā_{I} and use induction on |I|. If |I| = 0 then ȳ = ¯1 and

l(¯1, [d, d′]) = l(¯1d, d′) − l(¯1d′, d) + [l(¯1, d), l(¯1, d′)]

follows trivially if d, d′ ∈ H, and by the remark otherwise. Now we will deal with the general case. We have by (27) and the Jacobi identity

l(ā_{I}, [d, d′]) = l(ā_{I′}τ_{a_{in}}, [d, d′]) = l(ā_{I′}, [τ_{a_{in}}, [d, d′]]) + l(ā_{I′}[d, d′], τ_{a_{in}})

− ∑[l(ā_{I′(1)}, τ_{a_{in}}), l(ā_{I′(2)}, [d, d′])]

= l(ā_{I′}, [[τ_{a_{in}}, d], d′]) + l(ā_{I′}, [d, [τ_{a_{in}}, d′]]) + l(ā_{I′}dd′, τ_{a_{in}})

−l(ā_{I′}d′d, τ_{a_{in}}) − ∑[l(ā_{I′(1)}, τ_{a_{in}}), l(ā_{I′(2)}, [d, d′])].

Now, applying the hypothesis of induction to this last expression we get

l(ā_{I}, [d, d′]) = l(ā_{I′}[τ_{a_{in}}, d], d′) − l(ā_{I′}d′, [τ_{a_{in}}, d])

+ ∑[l(ā_{I′(1)}, [τ_{a_{in}}, d]), l(ā_{I′(2)}, d′)] + l(ā_{I′}d, [τ_{a_{in}}, d′]) −l(ā_{I′}[τ_{a_{in}}, d′], d) + ∑[l(ā_{I′(1)}, d), l(ā_{I′(2)}, [τ_{a_{in}}, d′])] +l(ā′_{I}dd′, τ_{a_{in}}) − l(ā_{I′}d′d, τ_{a_{in}}) − ∑[l(ā_{I′(1)}, τ_{a_{in}}), l(ā_{I′(2)}, [d, d′])].

Again, by (27) and the hypothesis of induction,

l(ā_{I}, [d, d′]) = l(ā_{I}d, d′) − l(ā_{I′}dτ_{a_{in}}, d′) + l(ā_{I′}d, [τ_{a_{in}}, d′]) + l(ā_{I′}dd′, τ_{a_{in}})

−l(ā_{I}d′, d) + l(ā_{I′}d′τ_{a_{in}}, d) − l(ā_{I′}d′, [τ_{a_{in}}, d]) − l(ā_{I′}d′d, τ_{a_{in}}) +( ∑[l(ā_{I′(1)}ā_{i_{n}}, d), l(ā_{I′(2)}, d′)] − ∑[l(ā_{I′(1)}d, τ_{a_{in}}), l(ā_{I′(2)}, d′)] + ∑[[l(ā_{I′(1)}, τ_{a_{in}}), l(ā_{I′(2)}, d)], l(ā_{I′(3)}, d′)]) +( ∑[l(ā_{I′(1)}, d), l(ā_{I′(2)}τ_{a_{in}}, d′)]

− ∑[l(ā_{I′(1)}, d), l(ā_{I′(2)}d′, τ_{a_{in}})]
+ ∑[l(ā_{I′(1)}, d), [l(ā_{I′(2)}, τ_{a_{in}}), l(ā_{I′(3)}, d′)]])
+( − ∑[l(ā_{I′(1)}, τ_{a_{in}}), l(ā_{I′(2)}d, d′)]
+ ∑[l(ā_{I′(1)}, τ_{a_{in}}), l(ā_{I′(2)}d′, d)]
− ∑[l(ā_{I′(1)}, τ_{a_{in}}), [l(ā_{I′(2)}, d), l(ā_{I′(3)}, d′)]]).

The cyclic sum on d, d′, τ_{a_{in}} of ∑[[l(ā_{I′(1)}, τ_{a_{in}}), l(ā_{I′(2)}, d)], l(ā_{I′(3)}, d′)] in this last term vanishes by the Jacobi identity. So, collecting together some terms we obtain

l(ā_{I}, [d, d′]) = l(ā_{I}d, d′) − l(ā_{I}d′, d) + ∑[l(ā_{I(1)}, d), l(ā_{I(2)}, d′)]
+l(ā_{I′}d, [τ_{a_{in}}, d′]) − l(ā_{I′}dτ_{a_{in}}, d′) + l(ā_{I′}dd′, τ_{a_{in}})
− ∑[l((ā_{I′}d)_{(1)}, τ_{a_{in}}), l((ā_{I′}d)_{(2)}, d′)]
−l(ā_{I′}d′, [τ_{a_{in}}, d]) + l(ā_{I′}d′τ_{a_{in}}, d) − l(ā_{I′}d′d, τ_{a_{in}})
+ ∑[l((ā_{I′}d′)_{(1)}, τ_{a_{in}}), l((ā_{I′}d′))_{(2)}, d].

If d ∈ H or d = τ_{a} with a ≤ a_{i_{n}} and also d′ ∈ H or d′ = τ_{b} with b ≤ a_{i_{n}}, then we may use the hypothesis of induction and (27) to conclude that

l(ā_{I′}d, [τ_{a_{in}}, d′]) − l(ā_{I′}dτ_{a_{in}}, d′) + l(ā_{I′}dd′, τ_{a_{in}})
− ∑[l((ā_{I′}d)_{(1)}, τ_{a_{in}}), l((ā_{I′}d)_{(2)}, d′)] = 0

and

l(ā_{I′}d′, [τ_{a_{in}}, d]) + l(ā_{I′}d′τ_{a_{in}}, d) − l(ā_{I′}d′d, τ_{a_{in}})
+ ∑[l((ā_{I′}d′)_{(1)}, τ_{a_{in}}), l((ā_{I′}d′)_{(2)}, d)] = 0,

therefore, the statement follows. Otherwise, i.e. d or d′ is of the form τ_{a} with a ≥ a_{i_{n}} then the statement is a direct consequence of (27). □

6.2 Sabinin algebras from primitive elements (II): A universal enveloping algebra for Sabinin algebras.

Recall that given a nonassociative algebra C over a field of characteristic zero one may consider a structure of Sabinin algebra on C, denoted by G(C), where the operations are given by

⟨y, z⟩ = −[y, z]
⟨x_{1}, …, x_{m}; y, z⟩ = −q_{m,1}(x_{1}, …, x_{m}, y, z) + q_{m,1}(x_{1}, …, x_{m}, z, y)

Φ(x_{1}, …, x_{m}; y_{1}, …, y_{n}) =

\frac{1}{m!} \frac{1}{n!} ∑_{τ∈S_{m},δ∈S_{n}} q_{m,n−1}(x_{τ(1)}, …, x_{τ(m)}, y_{δ(1)}, …, y_{δ(n)}).

In case that C is a bialgebra then Prim(C) is a Sabinin subalgebra of G(C). The main problem posed in [19] is stated as follows: Does any Sabinin algebra appears as a Sabinin subalgebra of G(C) for some algebra C? In this section we will provide an affirmative answer to this problem. For simplicity we will omit the indices on q.

Proposition 33 In (S̃(V), l) we have that

q(ū, v̄, ā) = −⟨ū; l(v̄, τ_{a})⟩.

PROOF. Since

(ū, v̄, ā) = \{ \begin{matrix}∑ ū_{(1)}q(ū_{(2)}, v̄_{(1)}, ā) · v̄_{(2)} (by definition of q) \\ (ūv̄)τ_{a} − ū(v̄τ_{a}) = ∑ ūl(v̄_{(1)}, τ_{a})v̄_{(2)} \end{matrix}

then the statement follows trivially for v̄ = ¯1. Now we may proceed by induction on |v̄| to prove that − ∑ ū_{(1)}⟨ū_{(2)}; l(v̄, ā)⟩ = ∑ ū_{(1)}q(ū_{(2)}, v̄, ā):

∑ ūl(v̄_{(1)}, τ_{a})v̄_{(2)} = (ū, v̄, ā) = ∑ ū_{(1)}q(ū_{(2)}, v̄_{(1)}, ā) · v̄_{(2)}
= ∑ ū_{(1)}q(ū_{(2)}, v̄, ā) + ∑_{|v̄_{(2)}|≥1} ū_{(1)}q(ū_{(2)}, v̄_{(1)}, ā) · v̄_{(2)}
= ∑ ū_{(1)}q(ū_{(2)}, v̄, ā) + ∑_{|v̄_{(2)}|≥1} ūl(v̄_{(1)}, τ_{a})v̄_{(2)}

implies that ūl(v̄, ā) = ∑ ū_{(1)}q(ū_{(2)}, v̄, ā). Therefore, − ∑ ū_{(1)}⟨ū_{(2)}; l(v̄, ā)⟩ = ∑ ū_{(1)}q(ū_{(2)}, v̄, ā). From this relation one easily obtains the result by induction on the degree of ū. □

Note 2 It is worthwhile to notice that although in the previous proof we used induction, we might avoid it just by computing ∑ ū_{(1)}\((ū_{2}, v̄_{(1)}, ā)/v̄_{(2)}) in two different ways as in the proof. One leads to q(ū, v̄, ā) while the other leads to −⟨ū; l(ū, τ_{a})⟩.

Proposition 34 In (S̃(V), l) we have that if |u| ≥ 1 then

⟨ū; ā, b̄⟩ = q(ū, b̄, ā) − q(ū, ā, b̄) and ⟨ā, b̄⟩ = −[ā, b̄].

PROOF. We first note that since l(¯1, τ_{a}) = 0 then

l(¯1, [τ_{a}, τ_{b}]) = l(ā, τ_{b}) − l(b̄, τ_{a}).

So,

q(ū, b̄, ā) − q(ū, ā, b̄) = −⟨ū; l(b̄, τ_{a})⟩ + ⟨ū; l(ā, τ_{b})⟩
= ⟨ū; l(¯1, [τ_{a}, τ_{b}])⟩ = ⟨ū; [τ_{a}, τ_{b}] + τ_{⟨¯1;[τ_{a},τ_{b}]⟩}⟩
=^{⟨1⟩} ⟨ū; [τ_{a}, τ_{b}]⟩ = ⟨ū; ā, b̄⟩

where ⟨1⟩ follows because |ū| ≥ 1 and the definition of τ_{a}. □

Theorem 35 Let (V, ⟨; ⟩, Φ) be a Sabinin algebra over a field of characteristic zero. There exist a unital algebra U(V, ⟨; ⟩, Φ) and a monomorphism of Sabinin algebras ι: V → G(U(V, ⟨; ⟩, Φ)) such that for any unital algebra C and any homomorphism of Sabinin algebras φ: V → G(C) there exists a unique homomorphism of unital algebras φ̄: U(V, ⟨; ⟩, Φ) → C with φ = φ̄ ∘ ι.

PROOF. We will construct an algebra (S̃(V), l) with the property that ι: V → G((S̃(V), l)) is a monomorphism of Sabinin algebras. The universal property will follow easily.

By the Poincaré–Birkhoff–Witt theorem we may look at V as contained in S̃(V). We will fix an ordered basis {a_{i}}_{i∈Λ} of V.

By Proposition 33, in order to obtain that

Φ(x̄_{1}, …, x̄_{m}; ȳ_{1}, …, ȳ_{n}) =

\frac{1}{m!} \frac{1}{n!} ∑_{τ∈S_{m},δ∈S_{n}} q_{m,n−1}(x̄_{τ(1)}, …, x̄_{τ(m)}, ȳ_{δ(1)}, …, ȳ_{δ(n)})

the map l should satisfy

−\frac{1}{m!} \frac{1}{n!} ∑_{τ∈S_{m},δ∈S_{n}}⟨\overline{x_{τ(1)} ⋯ x_{τ(m)}}; l(\overline{y_{δ(1)} ⋯ y_{δ(n−1)}}, τ_{y_{δ(n)}})⟩ =
Φ(x̄_{1}, …, x̄_{m}; ȳ_{1}, …, ȳ_{n}). (28)

The map l is defined through an auxiliary map Φ̂ as in (26). We set Φ̂(−, ¯1) = Φ̂(−, ȳ_{1}) = 0 and assume that we have defined Φ̂(−, \overline{a_{i_{1}} ⋯ a_{i_{n}}}) (n ≥ 2) so that the corresponding l in (26) satisfies (28). Modulo the subalgebra of H generated by l(S̃(V)_{n−1}, E) (which by (27) at this point is already constructed) we have that

−\frac{1}{(n + 1)!} ∑_{δ∈S_{n+1}} l(\overline{a_{i_{δ(1)}} ⋯ a_{i_{δ(n)}}}, τ_{a_{i_{δ(n+1)}}}) ≡ −l(\overline{a_{i_{1}} ⋯ a_{i_{n}}}, τ_{a_{i_{n+1}}})
= −Φ̂(−, \overline{a_{i_{1}} ⋯ a_{i_{n+1}}})

therefore, we may choose Φ̂(−, \overline{a_{i_{1}} ⋯ a_{i_{n+1}}}) so that Φ̂(¯1, \overline{a_{i_{1}} ⋯ a_{i_{n+1}}}) = 0 and the corresponding l in (26) satisfies

−\frac{1}{m!} \frac{1}{(n + 1)!} ∑_{τ∈S_{m},δ∈S_{n+1}}⟨\overline{x_{τ(1)} ⋯ x_{τ(m)}}; l(\overline{a_{i_{δ(1)}} ⋯ a_{i_{δ(n)}}}, τ_{a_{i_{δ(n+1)}}})⟩ =
Φ(x̄_{1}, …, x̄_{m}; ā_{i_{1}}, …, ā_{i_{n+1}})

as desired (the argument also shows how to set the initial step n = 2 of the induction). This together with Proposition 34 proves that ι: V → G((S̃(V), l)) is a monomorphism of Sabinin algebras.

Now we proceed to prove the universal property. Consider a unital algebra C and a homomorphism of Sabinin algebras φ: V → G(C). Since in C we have

(ua)b − (ub)a + u[a, b] = (u, a, b) − (u, b, a)
= − ∑_{|u_{(2)}|≥1} u_{(1)} (q(u_{(2)}, b, a) − q(u_{(2)}, a, b))
= − ∑_{|u_{(2)}|≥1} u_{(1)}⟨u_{(2)}; a, b⟩

then (ub)a − (ua)b = − ∑ u_{(1)}⟨u_{(2)}; a, b⟩. Therefore, there exists a well defined linear map

φ̄: S̃(V) → C
\overline{x_{1} ⋯ x_{n}} ↦ ((φ(x_{1})φ(x_{2})) ⋯)φ(x_{n}).

We have to show that this map is a homomorphism of algebras. Since S̃(V) is spanned by {ā^{m} | a ∈ V, m ∈ ℕ} we only have to check that φ̄(x̄b̄^{n}) = φ̄(x̄)φ̄(b̄^{n}). We use induction on n. If n = 0 the result is obvious. In general we may assume that x̄ = ā^{m} so

φ̄(ā^{m}b̄^{n}) = φ̄ ((ā^{m}b̄^{n−1})b̄ − ∑ ā_{(1)}^{m}q(ā_{(2)}^{m}, b̄_{(2)}^{n−1}, b̄) · b̄_{(1)}^{n−1})
= φ̄ ((ā^{m}b̄^{n−1})b̄ − ∑_{i,j} \binom{m}{i}\binom{n − 1}{j}ā^{m−i}q(ā^{i}, b̄^{j}, b̄) · b̄^{n−1−j})

= φ̄ ((ā^{m}b̄^{n−1})b̄ − ∑_{i,j} \binom{m}{i}\binom{n − 1}{j}ā^{m−i}Φ(a, …

…, a; b, …, b, b) · b̄^{n−1−j})
= (φ̄(ā)^{m}φ̄(b̄)^{n−1})φ̄(b̄) − ∑_{i,j} \binom{m}{i}\binom{n − 1}{j}φ̄(ā)^{m−i}φ̄(Φ(a, …
…, a; b, …, b, b)) · φ̄(b̄)^{n−1−j}
= φ̄(ā)^{m}φ̄(b̄)^{n} = φ̄(ā^{m})φ̄(b̄^{n})

and the result follows. □

Corollary 36 Any cocommutative connected unital H–bialgebra H over a field of characteristic zero is isomorphic to the universal enveloping algebra U(Prim(H)) of the Sabinin algebra Prim(H).

PROOF. Since Prim(H) is a Sabinin subalgebra of G(H) then by the universal property of U(Prim(H)) there exists a homomorphism of unital algebras φ̄: U(Prim(H)) → H extending the inclusion φ: Prim(H) ↪ H. By results in [19], for instance, this map is injective. Moreover, the arguments in the proof of Theorem 5.6.5 in [8] work verbatim in this case to obtain the bijectivity of φ̄. □

6.3 Envelopes of Malcev and Bol algebras (II)

Let (V, [ , , ], [ , ]) be a right Bol algebra over a field of characteristic ≠ 2, and U = U(V) its universal enveloping algebra as defined in [10]. We have that V ⊆ RN_{alt}(U) and the ternary and binary products are recovered as [a, b, c] = cb · a − ca · b − [b, a]c and [a, b] = ab − ba. We define on V the operations

⟨1; a, b⟩ = −[a, b],
⟨c; a, b⟩ = [a, b, c] − [[a, b], c] and
⟨xc; a, b⟩ = ∑⟨x_{(1)}; c, ⟨x_{(2)}; a, b⟩⟩ if |x| ≥ 1.

Proposition 37 (V, ⟨; ⟩) is a Sabinin subalgebra of G(U(V)).

PROOF. By (19) it is enough to prove that x[R_{a}, R_{b}] = − ∑ x_{(1)}⟨x_{(2)}; a, b⟩. We will use induction on the degree of x. If x = 1 then the result is obvious.

If x = c for some c ∈ V then c[R_{a}, R_{b}] = ca · b − cb · a = −[a, b, c] − [b, a]c = −c[b, a] − [a, b, c] − [[b, a], c] = c[a, b] − ⟨c; a, b⟩. In the general case we have

(xc)[R_{a}, R_{b}] = −(x)[[R_{a}, R_{b}], R_{c}] + (x)[R_{a}, R_{b}]R_{c}
= −x[a, b, c] − ∑ x_{(1)}⟨x_{(2)}; a, b⟩ · c

where the last equality follows from the relation [[R_{a}, R_{b}], R_{c}] = R_{[a,b,c]} valid for any a, b, c ∈ RN_{alt}(U) (see [10]) and the hypothesis of induction. Thus,

(xc)[R_{a}, R_{b}] = −x[a, b, c] − ∑ x_{(1)}c · ⟨x_{(2)}; a, b⟩ + ∑ x_{(1)}[R_{c}, R_{⟨x_{(2)};a,b⟩}]
= −x[a, b, c] − ∑ x_{(1)}c · ⟨x_{(2)}; a, b⟩
− ∑ x_{(1)}⟨x_{(2)}; c, ⟨x_{(3)}; a, b⟩⟩.

Since by definition we have that

∑ x_{(1)}⟨x_{(2)}c; a, b⟩ = ⟨c; a, b⟩ + ∑_{|x_{(2)}|≥1} x_{(1)}⟨x_{(2)}c; a, b⟩
= ⟨c; a, b⟩ − ∑_{|x_{(1)}|<|x|} x_{(1)}⟨x_{(2)}; c, ⟨x_{(3)}; a, b⟩⟩
= ⟨c; a, b⟩ + x[c, [a, b]] − ∑ x_{(1)}⟨x_{(2)}; c, ⟨x_{(3)}; a, b⟩⟩

then

(xc)[R_{a}, R_{b}] = −x[a, b, c] − ∑ x_{(1)}c · ⟨x_{(2)}; a, b⟩
− ∑ x_{(1)}⟨x_{(2)}c; a, b⟩ + x⟨c; a, b⟩ − x[c, [a, b]]
= − ∑(xc)_{(1)}⟨(xc)_{(2)}; a, b⟩

as desired. □

Proposition 38 The multioperator inherited by V from G(U(V)) is zero.

PROOF. By the definition of Φ it is enough to prove that (x, a^{n}, a) = 0 ∀a ∈ V. Recall that if a ∈ RN_{alt}(U) then R_{ya+ay} = R_{y}R_{a} + R_{a}R_{y} (see [10]). So if we assume proved that R_{a^{n}} = R_{a}^{n} then a^{n}a = aa^{n−1} · a = aR_{a}^{n−1}R_{a} = aR_{a}^{n} = aR_{a^{n}} = aa^{n}, so 2R_{a^{n+1}} = R_{aa^{n}+a^{n}a} = R_{a^{n}}R_{a} + R_{a}R_{a^{n}} = 2R_{a}^{n+1}. Therefore, R_{a}^{n} = R_{a^{n}} for any n. In particular (x, a^{n}, a) = xR_{a^{n}}R_{a} − xR_{a^{n+1}} = 0. □

A Sabinin algebra (V, ( )) is called homogeneous if there exists a bilinear operation ⋄: V × V → V such that for any l ≥ 2

(x_{0}x_{1} ⋯ x_{l}) = (x_{0} ⋄ x_{1} ⋯ x_{l}) + ⋯ + (x_{1} ⋯ x_{0} ⋄ x_{l}) − x_{0} ⋄ (x_{1} ⋯ x_{l}).

(29)

The structure of homogeneous Sabinin algebras is determined by the bilinear operations (xy) and x⋄y. From Differential Geometry [7] it is very well known that Malcev algebras (over fields of characteristic ≠ 2, 3) constitute a natural example of homogeneous Sabinin algebras where the maps ( ) and ⋄ are proportional. Any Malcev algebra (M, [ , ]) is seen as an homogeneous Sabinin algebra by setting (xy) = [x, y] and x ⋄ y = −\frac{1}{3}[x, y].

Since any Malcev algebra is a particular example of Bol algebra where [a, b, c] = [[a, b], c] − \frac{1}{3}J(a, b, c) and J(a, b, c) denotes the Jacobian of a, b and c, then we may wonder about the relationship between the structure induced by the universal enveloping algebra U(M) and the previously known (29). Surprisingly the are essentially the same.

For any Malcev algebra (M, [ , ]), the structure of Sabinin algebra on M induced by its universal enveloping algebra is given by

⟨1; a, b⟩ = −[a, b]
⟨c; a, b⟩ = −\frac{1}{3}J(a, b, c)
⟨xc; a, b⟩ = ∑⟨x_{(1)}; c, ⟨x_{(2)}; a, b⟩.

The structure of M^{opp} = (M, [ , ]′) (the opposite of (M, [ , ]), which is isomorphic to M) for being an homogeneous Sabinin algebra is given by

a ⋄ b = −\frac{1}{3}[a, b]′ = \frac{1}{3}[a, b]
(ab) = (1ab) = [a, b]′ = −[a, b]
(x_{0}x_{1} ⋯ x_{l}) = (x_{0} ⋄ x_{1} ⋯ x_{l}) + ⋯ + (x_{1} ⋯ x_{0} ⋄ x_{l}) − x_{0} ⋄ (x_{1} ⋯ x_{l}).

For instance,

(cab) = (c ⋄ ab) + (ac ⋄ b) − c ⋄ (ab)
= −\frac{1}{3}[[c, a], b] − \frac{1}{3}[a, [c, b]] + \frac{1}{3}[c, [a, b]]
= −\frac{1}{3}J(a, b, c) = ⟨c; a, b⟩.

Proposition 39 We have that ⟨x_{0} ⋯ x_{l−2}; x_{l−1}, x_{l}⟩ = (x_{0} ⋯ x_{l}).

PROOF. We only have to check that (xcab) = ∑(x_{(1)}c(x_{(2)}ab)) if |x| ≥ 1. If x = d ∈ M then the relation becomes

0 = (dcab) − (dc(ab)) − (c(dab))
= (dcab) − \frac{1}{3}J(d, c, [a, b]) − \frac{1}{3}[c, J(d, a, b)]
= (dcab) − 2(d, c, [a, b]) − 2[c, (d, a, b)]
= −\frac{2}{3}([d, c], a, b) − \frac{2}{3}(c, [d, a], b) − \frac{2}{3}(c, a, [d, b]) + \frac{2}{3}[d, (c, a, b)]
−2(d, c, [a, b]) − 2[c, (d, a, b)].

Since J(a, b, c) = 6(a, b, c) in U(M), we should check that −3J(d, c, [a, b]) − 3[c, J(d, a, b)] − J([d, c], a, b) − J(c, [d, a], b) − J(c, a, [d, b]) + [d, J(c, a, b)] = 0 on any Malcev algebra. With the notation J(a, b, c) = aΔ(b, c) as in [13] this equality is equivalent to 2[− ad_{d}, Δ(a, b)] = 2Δ([d, a], b) + 2Δ(a, [d, b]) − 6Δ(d, [a, b]) + 6(− ad_{J(d,a,b)}). Using that 2[− ad_{d}, Δ(a, b)] = −3Δ(d, [a, b]) + Δ(a, [b, d]) + Δ(b, [d, a]), formula (2.35) in [13], then we only need to prove that 0 = Δ([d, a], b)+Δ(a, [d, b])−Δ(d, [a, b])+2(− ad_{J(d,a,b)}) which is exactly the identity (2.32) in [13].

Assume that we have proved that (xab) = ∑(x′_{(1)}x_{l}(x′_{(2)}ab)) with x = x_{1} ⋯ x_{l} and x′ = x_{1} ⋯ x_{l−1}. Then using a dot to separate arguments and using the notation (x_{0} ⋄ x · ab) = (x_{0} ⋄ x_{1} · x_{2} ⋯ x_{l}ab) + ⋯ + (x_{1}x_{2} ⋯ x_{0} ⋄ x_{l} · ab) we have

(x_{0}xab) = (x_{0} ⋄ x · ab) + (x · x_{0} ⋄ a · b) + (xa · x_{0} ⋄ b) − x_{0} ⋄ (xab)
= ∑(x_{0} ⋄ x′_{(1)} · x_{l}(x′_{(2)}ab)) + ∑(x′_{(1)} · x_{0} ⋄ x_{l} · (x′_{(2)}ab))
+ ∑(x′_{(1)}x_{l}(x_{0} ⋄ x′_{(2)} · ab)) + ∑(x′_{(1)}x_{l}(x′_{(2)} · x_{0} ⋄ a · b))
+ ∑(x′_{(1)}x_{l}(x′_{(2)}a · x_{0} ⋄ b)) − ∑ x_{0} ⋄ (x′_{(1)}x_{l}(x′_{(2)}ab))
= ∑(x_{0}x′_{(1)}x_{l}(x′_{(2)}ab)) + ∑(x′_{(1)}x_{l}(x_{0}x′_{(2)}ab))
= ∑((x_{0}x′)_{(1)}x_{l}((x_{0}x′)_{(2)}ab))

as desired. □

Corollary 40 Let (M, [ , ]) be a Malcev algebra and (M, ( )) the homogeneous Sabinin algebra defined by (29) with (xy) = [x, y], x ⋄ y = −\frac{1}{3}[x, y] and trivial multioperator. Then the universal enveloping algebra U((M, ( ))) of the Sabinin algebra (M, ( )) is isomorphic (as H–bialgebra) to the universal enveloping algebra U(M) of the Malcev algebra (M, [ , ]).

PROOF. Consider M^{opp} = (M, [ , ]′) the opposite algebra of (M, [ , ]). By Proposition 39 and Proposition 38 we have a monomorphism of Sabinin algebras (M, ( )) → G(U(M^{opp})). By the universal property of U((M, ( ))) this map extends to a homomorphism U((M, ( ))) → U(M^{opp}). Since this map obviously sends the Poincaré–Birkhoff–Witt basis of U((M, ( ))) to the Poincaré– Birkhoff–Witt basis of U(M^{opp}) then it is an isomorphism as bialgebras. Since the left and right division on U((M, ( ))) and U(M^{opp}) are determined by the bialgebra structure (they both are connected coalgebras) then this map is an isomorphism as H–bialgebras indeed. Finally, it is easy to prove that the isomorphism x ↦ −x form M^{opp} to M extends to an isomorphism of H–bialgebras between U(M^{opp}) and U(M). □

References

[1] S. Burris and H. Sankappanaver, A Course in Universal Algebra. The Millennium Edition (http://thoralf.uwaterloo.ca/htdocs/UALG/univalgebra.pdf).

[2] W.A. de Graaf, Lie algebras: theory and algorithms (North-Holland, Amsterdam, 2000).

[3] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer–Verlag, New York, 1972).

[4] N. Jacobson, Lie Algebras (Dover, New York, 1979).

[5] K. Kunen, Moufang Quasigroups, J. Algebra 183 (1996), no. 1, 231–234.

[6] A.I. Malcev, Analytic loops, Mat. Sb. (N. S.) 36 (78), no.3 (1955), 569-575.

[7] P.O. Miheev, Pseudoautomorphisms of smooth loops, Algebras, Groups and Geometries 8 (1991) no. 2, 131–144.

[8] S. Montgomery, Hopf Algebras and Their Actions on Rings (CBMS Regional Conference Series in Mathematics, 82, A.M.S., 1993).

[9] P.J. Morandi, J.M. Pérez–Izquierdo and S. Pumplün, On The Tensor Product of Composition Algebras, J. Algebra 243 (2001), 41-68.

[10] J.M. Pérez–Izquierdo, An Envelope for Bol Algebras, to appear in J. Algebra.

[11] J.M. Pérez–Izquierdo and I.P. Shestakov, An Envelope for Malcev Algebras, J. Algebra 272 (2004), 379–393.

[12] L.V. Sabinin, Smooth Quasigroups and Loops (Mathematics and Its Applications, 492, Kluwer Academic Publishers, 1999).

[13] A.A. Sagle, Malcev algebras, Trans. Amer. Math. Soc. 101 (1961), 426–458.

[14] L.V. Sabinin and P.O. Miheev, Analytic Bol loops, Webs and quasigroups (Kalinin. Gos. Univ., Kalinin, 1982) 102–109.

[15] L.V. Sabinin and P.O. Miheev, On local analytic loops and their corresponding hyperalgebras, Proceedings of 9th conference of young researchers (VINITI Press, Moscow, 1986) 34–54.

[16] L.V. Sabinin and P.O. Miheev, Infinitesimal theory of local analytic loops, Soviet Math. Dokl. 36 (1988), no. 3, 545-548.

[17] L.V. Sabinin and P.O. Miheev, Quasigroups and differential geometry, Quasigroups and loops: theory and applications (Sigma Ser. Pure Math., 8, Heldermann, Berlin, 1990) 357–430.

[18] V. Shcherbacov and V. Izbash, On quasigroups with Moufang identity, Buletinul A. S. a R. M. Matematica 27 (1998), no. 2, 109–116.

[19] I.P. Shestakov and U.U. Umirbaev, Free Akivis algebras, primitive elements, and hyperalgebras, J. Algebra 250 (2002), no. 2, 533–548.

[20] M. Takeuchi, Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), 561–582.

[21] K.A. Zhevlakov, A.M. Slinko, I.P. Shestakov and A.I. Shirshov, Rings that are nearly associative (Mowcow, Nauka, 1978; English transl.: Academic Press, N.Y. 1982).