Speciality of Lie-Jordan algebras

A. N. Grishkov Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brasil

Ivan P. Shestakov^{∗}

Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract

The class of so called Lie-Jordan algebras is introduced, which have one binary (Lie) operation [x, y] and one ternary (Jordan) operation {x, y, z}, that satisfy the identities (1)-(5) below. It is proved that any such an algebra is special, that is, isomorphic to a subalgebra of a Lie-Jordan algebra of the type A^{±}, obtained from an associative algebra A via the operations [x, y] = xy − yx, {x, y, z} = xyz + zyx. As an application, we prove the conjecture about associativity of a certain loop constructed in [2].

1 Introduction

It is well known that with every associative algebra A one can relate a Lie algebra A^{−} = ⟨A, +, [,]⟩, with the multiplication [x, y] = xy − yx, and a Jordan algebra A^{+} = ⟨A, +, ∘⟩, with the multiplication x∘y = \frac{1}{2} (xy+yx). Thus, we have the functors (.)^{−} and (.)^{+} from the category \underline{Ass} of associative algebras into the categories \underline{Lie} and \underline{Jord} of Lie and Jordan algebras, respectively. It is easy to see that both functors have left adjoint functors U : \underline{Lie} ⟶ \underline{Ass} and S : \underline{Jord} ⟶ \underline{Ass}, that is, for any L ∈ \underline{Lie}, J ∈ \underline{Jord}, A ∈ \underline{Ass} there are bijections

∗Supported by the FAPESP, Proc. 98/2162-6.

Hom\underline{Lie}(L, A^{−}) ⟶ Hom\underline{Ass}(U(L), A),
Hom\underline{Jord}(J, A^{+}) ⟶ Hom\underline{Ass}(S(J), A),

which are functorial in the variables L, J, A. The algebras U(L) and S(J) are called associative universal enveloping algebras of L and J, respectively. Furthermore, there are morphisms α_{L}^{−} : L ⟶ U(L)^{−} and α_{J}^{+} : J ⟶ S(J)^{+} with the universal properties (these morphisms correspond to the identical morphisms of the algebras A = U(L), A = S(J) in the bijections above).

But at this point the Lie and Jordan cases diverge: the mapping α_{L}^{−} is always injective when L is a free module over the ring of scalars, due to the celebrated Poincare—Birkhoff—Witt theorem, while the mapping α_{J}^{+} could have nonzero kernel even when J is a finite dimensional algebra over a field (see [4, 7]). A Jordan algebra J for which the universal mapping α_{J}^{+} is injective is called special. The condition of speciality plays an important role in the theory of Jordan algebras; there are many results and open problems concerning the speciality of particular Jordan algebras and classes of algebras (see [4, 7]).

One may consider other functors that conserve the additive structure of an algebra, and introduce for them a notion of speciality (see [6]). In this paper we consider the functor (.)^{±} that corresponds to an associative algebra A the algebra A^{±}, which has the same additive structure and two new multiplications: a Lie one [x, y] = xy − yx and a triple Jordan one {x, y, z} = xyz + zyx. It is easy to check that the operations [,] and {, , } satisfy the identities

[x, y] = −[y, x],  (1)
{x, y, z} = {z, y, x},  (2)
[[x, y], z] = {x, y, z} − {y, x, z},  (3)
[{x, y, z}, t] = {[x, t], y, z} + {x, [y, t], z} + {x, y, [z, t]},  (4)
{{x, y, z}, t, v} = {{x, t, v}, y, z} − {x, {y, v, t}, z} + {x, y, {z, t, v}}.(5)

Observe that the Jacobi identity follows from (2) and (3).

Definition 1 An algebra with two operations [,] and {, , } is called a Lie-Jordan algebra (an LJ-algebra for short) if identities (1)-(5) hold.

Note that an LJ-algebra is a Lie algebra relatively to the binary operation and a triple Jordan system relatively to the ternary operation.

Definition 2 An LJ-algebra L is called special if there exists an associative algebra A such that L is isomorphic to a subalgebra of A^{±}.

The main result of the present article is the following

Theorem 1 Every LJ-algebra over a field of characteristic ≠2 is special.

In the proof we follow the method developed in [5, 6] and construct the universal enveloping algebra U(L) for a Lie-Jordan algebra L as a deformation of its associated graded algebra gr U(L).

In the last section we apply this result to prove the conjecture from [2] that a certain loop constructed there is a group.

All the algebras considered in the paper are assumed to be over a field F of characteristic ≠ 2.

2 Universal enveloping algebra and associated graded algebra

Let L be an LJ-algebra; then an associative algebra U(L) is said to be a universal enveloping algebra for L if there exists a homomorphism α_{L} : L ⟶ U(L)^{±} such that for any associative algebra A and a homomorphism β : L ⟶ A^{±} there exists a homomorphism π of associative algebras π : U(L) ⟶ A such that β = α_{L} ∘ π. In other words, there is a bijection

Hom\underline{Lie−Jord}(L, A^{±}) ⟶ Hom\underline{Ass}(U(L), A),

which is functorial on the variables L and A.

The existence of a universal enveloping algebra U(L) for a given LJalgebra L is obvious. It is isomorphic to the quotient algebra of the tensor algebra T (L) by the ideal I generated by all the elements

a ⊗ b − b ⊗ a − [a, b], a ⊗ b ⊗ c + c ⊗ b ⊗ a − {a, b, c}, a, b, c ∈ L;

with the universal homomorphism α_{L} : a ⟶ a + I. Theorem 1 is equivalent to the fact that ker α = 0.

The subspace α(L) ⊆ U(L) generates U(L), and so it defines in U(L) an ascending filtration U_{1} = α(L) ⊆ U_{2} ⊆ ⋯, where U_{i} = U_{i−1} + U_{1}^{i}. Observe that for any a, b, c ∈ L we have

a ⊗ b ⊗ c − 1/2 ({a, b, c} + [b, c] ⊗ a + b ⊗ [a, c] + [a, b] ⊗ c) ∈ I,

which implies that U_{2} = U_{3} = U(L). Consider the associated graded algebra gr U = (gr U)_{1} ⊕ (gr U)_{2}, where (gr U)_{1} = U_{1}, (gr U)_{2} = U(L)/U_{1}; then it is easy to see that it is commutative and nilpotent of degree 3: (gr U)^{3} = 0. For more properties of this algebra we need the following

Lemma 1 In U(L)^{±} hold the inclusions

[U_{i}, U_{j}] ⊆ U_{i+j−1},
{U_{i}, U_{j}, U_{k}} ⊆ U_{i+j+k−2}.

Proof. Since U_{2} = U(L), both inclusions are evident if at least one of the indices is equal to 2. Furthermore, U_{1} = α(L), and for any a, b, c ∈ L we have

[α(a), α(b)] = α([a, b]), {α(a), α(b), α(c)} = α({a, b, c}),

which proves the inclusions for i = j = k = 1. □

Let \overline{c_{i}} = c_{i} + U_{i−1} ∈ (gr U)_{i}, for some c_{i} ∈ U_{i}. It follows from Lemma 1 that the following operations are correctly defined in gr U:

[\overline{c}_{i}, \overline{c}_{j}] = [c_{i}, c_{j}] + U_{i+j−2},
\{\overline{c}_{i}, \overline{c}_{j}, \overline{c}_{k}\} = \{c_{i}, c_{j}, c_{k}\} + U_{i+j+k−3}.

Since these operations are induced by those in the LJ-algebra U(L)^{±}, they satisfy identities (1) – (5); that is, the space gr U forms an LJ-algebra with respect to the operations [,], {, , }. Moreover, one can easily check that the following identities hold in U(L):

[xy, z] = x[y, z] + [x, z]y,
{x ∘ y, z, t} = {x, z, t} ∘ y + 1/2([[t, z], y] ∘ x + [[t ∘ z, y], x]),

where x ∘ y = xy + yx. The proof is straightforward. As a corollary, we obtain that the associated graded algebra gr U satisfies the identities

[xy, z] = x[y, z] + [x, z]y,  (6)
{xy, z, t} = {x, z, t}y + 1/2([[t, z], y]x + [[tz, y], x]).  (7)

Identity (6) shows that gr U is a Poisson algebra. By this reason, we will call a commutative associative algebra A with additional operations [,] and {, , } a Poisson LJ-algebra if ⟨A, +, [,], {, , }⟩ is an LJ-algebra, A^{3} = 0, and identities (6),(7) are satisfied in A. As we just have seen, for every LJ-algebra L, the algebra gr U(L) is a Poisson LJ-algebra.

Lemma 2 Every Poisson LJ-algebra A satisfies the identities

{x, z, t}y − {y, z, t}x − h(x, y, z, t) = 0,  (8)
[xy, zt] = {xy, z, tw} = {xy, zt, w} = 0,  (9)

where h(x, y, z, t) = 1/2([z, [x, y]]t + [t, [x, y]]z + [[t, z], x]y − [[t, z], y]x).

Proof. Interchanging x and y in (7) and subtracting the new identity from the former one, by the Jacobi identity and (6), we obtain (8). Furthermore, since A^{3} = 0, identities (9) follow from (7), (2), (3), and (6). □

Notice that α(L) = U_{1} = (gr U)_{1}, hence the problem of injectivity of α is reduced to the structure of the graded algebra gr U(L). In the next section we will try to construct this algebra starting with the algebra L.

3 The Poisson LJ-algebra \overline{V (L)}

Let L be an LJ-algebra. In this section we will construct a Poisson LJ-algebra \overline{V (L)} which eventually proves to be isomorphic to the algebra gr U(L).

As the first approximation, we consider the space V (L) = L ⊕ S^{2}(L). Here S^{2}(L) stands for the space of symmetric 2-degree tensors over L, which is spanned by the elements a • b = 1/2(a ⊗ b + b ⊗ a), a, b ∈ L. Define a multiplication · on V (L) by setting for a, b, c, d ∈ L

a · b = a • b, a · (b • c) = (b • c) · a = (a • b) · (c • d) = 0;

then V (L) becomes a commutative associative algebra with V (L)^{3} = 0. If no ambiguity occurs, we will write ab instead of a · b.

Extend now the operations [,] and {, , } from L to V (L) by setting

[x • x, y] = −[y, x • x] = 2x[x, y],
{x • x, y, z} = {z, y, x • x} = {x, y, z}x + 1/2([[z, y], x]x + [[zy, x], x]),
{x, y • y, z} = {y • y, x, z} + [[x, y^{2}], z],
{a, b, r} = {a, r, b} = {r, a, b} = [a, b] = 0,

for every x, y, z ∈ L, a, b, ∈ S^{2}(L), r ∈ V (L). Since the space S^{2}(L) is spanned by the elements x • x, x ∈ L, the operations are correctly defined.

Note that L is an LJ-subalgebra of V (L) though V (L) itself, in general, is not an LJ-algebra with respect to the defined operations.

By construction, we have a homomorphism of graded algebras

ρ : ⟨V (L), +, ·, [,], {, , }⟩ ⟶ gr U

such that ρ(a) = α(a), a ∈ L.

Obviously, the equality that we need, ker α = 0, is equivalent to the equality (ker ρ) ∩ L = 0. Since ker ρ is a graded ideal, this equality is equivalent to the inclusion

ker ρ ⊆ S^{2}(L).  (10)

Therefore, we need an information on the structure of ker ρ.

Notice that, by (8), for every elements x, y, z, t ∈ L we have

j(x, y, z, t) := {x, z, t}y − {y, z, t}x − h(x, y, z, t) ∈ ker ρ.  (11)

Let J = J(L) = vect {j(x, y, z, t)| x, y, z, t ∈ L} be the vector space generated by all the elements of this kind; then evidently J ⊆ S^{2}(L) ∩ ker ρ. We will eventually show that J = ker ρ; this will imply inclusion (10) and Theorem 1.

Our nearest objective is to prove that certain elements from ker ρ lie in J. Let L̃ = LJ[X] and A = A[X] be free Lie-Jordan and free associative

algebras with free generators X = {x_{1}, x_{2}, …}. Denote by S̃L = SLJ[X] a free special LJ-algebra, that is, the LJ-subalgebra of A^{±} generated by X. Evidently, there exists an epimorphism φ : L̃ ⟶ S̃L, identical on X. By definition of a universal enveloping algebra, φ extends uniquely to a homomorphism of associative algebras φ̃ : U(L̃) ⟶ A, φ̃(x) = x, x ∈ X. But A is a free algebra, hence φ̃ is an isomorphism. Thus, we have a homomorphism

ρ : ⟨V (L̃), +, ·, [,], {, , }⟩ ⟶ gr A = S̃L ⊕ A/S̃L.

such that ρ(a) = α(a), a ∈ L̃. Notice that the algebras V (L̃) and gr A have natural gradings by degrees with respect to X, and the homomorphism ρ respects this grading.

The following proposition is crucial for our construction.

Proposition 1 In the above notations, every multilinear element of degree ≤ 6 on X from ker ρ lies in J = J(L̃).

Proof. We will prove that J contains the multilinear elements of degree 6 from ker ρ. For the elements of smaller degree the proof is similar but more easy.

Denote by P the subspace of multilinear elements of degree 6 in V (L̃). A subset W of a vector space V we will call a pre-basis of V if vect {W } = V. We have a decomposition P = P_{1}⊕P_{2}, where P_{1} = P ∩L̃ and P_{2} = P ∩S^{2}(L̃). Let us find a pre-basis of the space (P_{2} +J)/J. Let L[X] be a Lie subalgebra in L̃ generated by X and B be a homogeneous basis of the space of multilinear elements of L[X]. We have B = ∪_{i=1}^{∞}B_{i}, where B_{i} = {b ∈ B | deg_{X} b = i}. Set

Q_{1} = {a_{i}b_{j} | a_{i} ∈ B_{i}, b_{j} ∈ B_{j}, i + j = 6},
Q_{2} = {a_{3}{x, y, z} | a_{3} ∈ B_{3}, x, y, z ∈ X},
Q_{3} = {a_{2}{b_{2}, y, z} | a_{2}, b_{2} ∈ B_{2}, y, z ∈ X},
Q_{4} = {{x, y, z}{u, v, w} | x, y, z, u, v, w ∈ X}.

Lemma 3 The set Q = ∪_{i=1}^{4}Q_{i} is a pre-basis of (P_{2} + J)/J.

Proof. Let v ∈ P_{2} be a monomial of degree 6. If v ∈ L[X] · L[X], then v ∈ vect {Q_{1}}.

If the operation {, , } appears only once in v, then we can assume, by (3) and (4), that v has one of the forms

v_{1} = z{a_{3}, x, y}, v_{2} = y{a_{2}, b_{2}, x}, v_{3} = a_{2}{b_{2}, x, y}, v_{4} = a_{3}{x, y, z},

where a_{i}, b_{i} ∈ B_{i}, x, y, z ∈ X. Now, for any a, b, c, d ∈ L̃ we have h(a, b, c, d) ∈ vect {Q_{1}}, hence, by (11),

{a, c, d}b − {b, c, d}a = j(a, b, c, d) + h(a, b, c, d) ∈ J + vect {Q_{1}}. (12)

Therefore, by (3), v_{1}, …, v_{4} ∈ vect {Q_{1} ∪ Q_{2} ∪ Q_{3}} + J.

Finally, if the operation {, , } appears twice in v, then evidently

v = ∑_{i}{{x_{i}, y_{i}, z_{i}}, t_{i}, u_{i}}w_{i} (mod vect {Q}),

for some x_{i}, …, w_{i} ∈ X. By (12), we have

{{x_{i}, y_{i}, z_{i}}, t_{i}, u_{i}}w_{i} ≡ {x_{i}, y_{i}, z_{i}}{w_{i}, t_{i}, u_{i}} (mod vect {Q} + J),

so v ∈ vect {Q} + J, and Lemma is proved. □

Set P_{2}^{j} = vect \{Q_{1} + ⋯ + Q_{j}\} + J, j = 1, 2, 3, 4, and \overline{P} _{2}^{j} = P_{2}^{j}/P_{2}^{j−1}, j = 2, 3, 4.

Lemma 4 We have

(i) dim(P_{2}^{1}/J) ≤ 274,  (ii) dim \overline{P} _{2}^{2} ≤ 40,
(iii) dim \overline{P} _{2}^{3} ≤ 45,  (iv) dim \overline{P} _{2}^{4} ≤ 1,
(v) dim(P_{2} + J/J) ≤ 360.

Proof. By [1, page 50], the dimension of the space of multilinear elements of degree m in a free n-generated Lie algebra is equal to B(n, m) = = (m − 1)!C_{m}^{n}. Therefore, we have

dim P_{2}^{1}/J ≤ |Q_{1}| ≤ B(6, 1)B(5, 5) + B(6, 2)B(4, 4) + B(6, 3)B(3, 3)
= 144 + 90 + 40 = 274.

Furthermore, by (2) and (3), we have for any a_{i} ∈ L[X], i = 1, 2, 3, and σ ∈ Sym{1, 2, 3}

{a_{σ(1)}, a_{σ(2)}, a_{σ(3)}} ≡ {a_{1}, a_{2}, a_{3}} (mod L[X]),  (13)

which implies that dim \overline{P} _{2}^{2} ≤ B(6, 3) = 40.

To prove (iii), notice that (12) implies

a_{2}{b_{2}, x, y} ≡ b_{2}{a_{2}, x, y} (mod L[X] + J),

hence dim \overline{P} _{2}^{3} ≤ 1/2 B(6, 2)B(4, 2) = 45.

For (iv), we need a certain corollary of identity (5). Adding (5) and the identity obtained from it by interchanging y and z, we get by (3)

{2{x, y, z} − [x, [y, z]], t, v} = 2{{x, t, v}, y, z} − [{x, t, v}, [y, z]]

−{x, {y, v, t}, z} + {x, y, {z, t, v}} − {x, {z, v, t}, y} + {x, z, {y, t, v}}

= 2{{x, t, v}, y, z} − [{x, t, v}, [y, z]] − {x, {y, t, v} + [y, [v.t]], z}
+{x, y, {z, v, t} + [z, [t, v]]} − {x, {z, v, t}, y} + {x, z, {y, t, v}}

= 2{{x, t, v}, y, z} − [{x, t, v}, [y, z]] − {x, [y, [v.t]], z} + {x, y, [z, [t, v]]}

−[x, [{y, t, v}, z]] + [x, [y, {z, v, t}]].

Since L̃ is an LJ-algebra, this implies that for any x_{i} ∈ X, i = 1, …, 6, we have

{{x_{1}, x_{2}, x_{3}}, x_{4}, x_{5}} = {{x_{1}, x_{4}, x_{5}}, x_{2}, x_{3}} + S,  (14)

where S is a sum of multilinear elements from L̃ on x_{1}, …, x_{5}, in which the operation {, , } appears only once. By the proof of Lemma 3, we have S · x_{6} ∈ P_{2}^{3}.

Now, for any x_{i} ∈ X, i = 1, …, 6 we have by (12)

{{x_{1}, x_{2}, x_{3}}, x_{4}, x_{5}}x_{6} ≡ {x_{1}, x_{2}, x_{3}}{x_{4}, x_{5}, x_{6}} (mod P_{2}^{3}),

which implies by (14) that

{x_{1}, x_{2}, x_{3}}{x_{4}, x_{5}, x_{6}} ≡ {x_{1}, x_{4}, x_{5}}{x_{2}, x_{3}, x_{6}} (mod P_{2}^{3}).  (15)

It follows easily from (13) and (15) that for any σ ∈ Sym{1, …, 6} holds

{x_{σ(1)}, x_{σ(2)}, x_{σ(3)}}{x_{σ(4)}, x_{σ(5)}, x_{σ(6)}} ≡ {x_{1}, x_{2}, x_{3}}{x_{4}, x_{5}, x_{6}} (mod P_{2}^{3}),

which implies (iv).

Finally, we have

dim(P_{2}+J/J) ≤ dim P_{2}^{1}/J+dim \overline{P} _{2}^{2}+dim \overline{P} _{2}^{3}+dim \overline{P} _{2}^{4} = 274+40+45+1 = 360,

proving the Lemma. □

Lemma 5 dim P_{1} ≤ 360.

Proof. Let us write P_{1} = P_{1}^{0} + P_{1}^{1} + P_{1}^{2}, where P_{1}^{i} means the space generated by monomials from P_{1} in which the operation {, , } appears i times. Furthermore, for i = 1, 2 we set \overline{P} _{1}^{i} = (P_{1}^{0} + ⋯ + P_{1}^{i})/(P_{1}^{0} + ⋯ + P_{1}^{i−1}). Clearly, dim P_{1}^{0} ≤ B(6, 6) = 5! = 120.

Let B and B_{i} be as in Lemma 3. By (13), the union of the sets {B_{4}, X, X}, \{B_{3}, B_{2}, X\}, and \{B_{2}, B_{2}, B_{2}\} forms a pre-basis of \overline{P} _{1}^{1}. Taking in account (13), we obtain

dim vect {{B_{4}, X, X}} ≤ B(6, 4) = 90,
dim vect {{B_{3}, B_{2}, X}} ≤ B(6, 3)B(3, 2) = 120,
dim vect {{B_{2}, B_{2}, B_{2}}} ≤ B(6, 2)B(4, 2)/3! = 15,

which implies that dim \overline{P} _{1}^{1} ≤ 90 + 120 + 15 = 225.

Furthermore, it follows from (14) and (13) that the set {{B_{2}, x_{i}, x_{j}}, x_{k}, x_{l}}, where x_{s} ∈ X, i < j < k < l, forms a pre-basis of \overline{P} _{1}^{2}; hence dim \overline{P} _{1}^{2} ≤ B(6, 2) = 15.

Thus, dim P_{1} = dim P_{1}^{0} + dim \overline{P} _{1}^{1} + dim \overline{P} _{1}^{2} ≤ 120 + 225 + 15 = 360. □ Return to the proof of Proposition 1. By Lemmas 4 and 5 we have that

dim P/P ∩ J = dim(P + J/J) ≤ dim P_{1} + dim(P_{2} + J/J) ≤ 360 + 360 = 720.

On the other hand, the image ρ(P) coincides with the space (gr A)_{⟨6⟩} of multilinear elements of degree 6 on X in gr A, which has the same dimension as the corresponding space A_{⟨6⟩} in A. Therefore, dim ρ(P) = dim A_{⟨6⟩} = 6! = 720, and hence P ∩ kerρ = P ∩ J. □

Corollary 1 J = J(L̃) is an ideal of the algebra ⟨V (L), +, ·, [,], {, , }⟩.

Proof. It suffices to prove that

[J, L̃] ⊆ J, {J, L̃, L̃} ⊆ J.

Moreover, the space J is evidently stable with respect to endomorphisms of the space V (L̃) induced by those of the LJ-algebra L̃; hence we need only to prove that for any x, y, z, t, u, v ∈ X hold the inclusions

[j(x, y, z, t), u] ∈ J, {j(x, y, z, t), u, v} ∈ J.

Observe that the elements under consideration belong to ker ρ. Since they are multilinear and of degrees 5 and 6, they belong to J as well. □

Corollary 2 The quotient algebra \overline{V (L̃)} = V (L̃)/J(L̃) is a Poisson LJalgebra.

Proof. We have to prove that identities (1)—(5) hold in V (L̃) modulo J(L̃). If an identity contains at least two arguments from S^{2}(L̃), it holds trivially. Consequently, we can assume, as before, that all but one arguments in identities under consideration belong to X, and one argument belongs to X ∪ X · X. The identities evidently hold modulo ker ρ. Since they are multilinear and of degree ≤ 6 on X, they hold modulo J(L̃) as well. □

Notice that the constructions of V (L) and J(L) are functorial; that is, any LJ-algebra homomorphism φ : L_{1} ⟶ L_{2} can be extended to a homomorphism

φ̃ : ⟨V (L_{1}), +, ·, [,], {, , }⟩ ⟶ ⟨V (L_{2}), +, ·, [,], {, , }⟩,

and φ̃(J(L_{1})) = J(φ(L_{1})) ⊆ J(L_{2}). Moreover, if φ is surjective then so is φ̃. Since every LJ-algebra can be represented as a quotient algebra of a free LJ-algebra, corollaries 1 and 2 imply

Proposition 2 For every LJ-algebra L the space J(L) is an ideal of the algebra V (L) with respect to the operations ·, [,], {, , }, and the quotient algebra \overline{V (L)} = V (L)/J(L) is a Poisson LJ-algebra.

Since J(L) ⊆ ker ρ, the homomorphism ρ : V (L) ⟶ gr U(L) induces a Poisson LJ-algebra homomorphism ρ̃: \overline{V (L)} ⟶ gr U(L), such that ρ̃(l) = α(l) for any l ∈ L. In the next section we will prove that ρ̃ is an isomorphism, which will imply that ker ρ = J(L), proving inclusion (10) and Theorem 1.

4 U(L) is a deformation of \overline{V (L)}

As we have seen, for the proving of theorem 1 it suffices to prove that \overline{V (L)} is isomorphic to gr U(L). We will prove it by constructing an associative multiplication on \overline{V (L)}, isomorphic to that in U(L).

Consider the space of polynomials \overline{V (L)}[t] over \overline{V (L)}. An associative multiplication ∗ on \overline{V (L)}[t] we will call a quantization deformation of \overline{V (L)}, if it satisfies the conditions:

\begin{matrix}(i) & a ∗ a = a^{2}, \\ (ii) & a ∗ b − b ∗ a = [a, b]t, \\ (iii) & a ∗ b ∗ c + c ∗ b ∗ a = \{a, b, c\}t^{2}, \\ (iv) & t ∗ r = r ∗ t, \end{matrix}

for any a, b, c ∈ L, r ∈ \overline{V (L)}.

The role of this definition is illustrated by the following Proposition 3 Let L be an LJ-algebra. If the Poisson LJ-algebra \overline{V (L)} admits a quantization deformation, then L is special. Moreover, in this case gr U(L) is isomorphic to \overline{V (L)} and U(L) is isomorphic to the quotient algebra ⟨\overline{V (L)}[t], +, ∗⟩/\overline{V (L)}[t](1 − t).

Proof. Assume that \overline{V (L)} = L ⊕ S^{2}(L)/J(L) admits a quantization deformation ∗. Define a new multiplication a ⋆ b = a ∗ b|_{t=1} on \overline{V (L)}; then the algebra B = ⟨\overline{V (L)}, +, ⋆⟩ is associative, and, by properties (ii) and (iii), the LJ-algebra L is a subalgebra of the LJ-algebra B^{±}. Therefore, L is special.

Consequently, there exists an algebra homomorphism π : U(L) ⟶ B such that πα(l) = l for any l ∈ L. The algebra B is evidently generated by L and has a filtration B_{1} = L ⊆ B_{2} = B, where B_{2} = L + L ⋆ L; and it is easily seen that π(U_{k}) ⊆ B_{k}. Therefore, π induces a homomorphism of associated graded algebras π̃ : gr U ⟶ gr B, with π̃α(l) = l for any l ∈ L. But gr B is easily seen to be isomorphic to \overline{V (L)}, and we have seen that there exists a graded algebra homomorphism ρ̃: \overline{V (L)} ⟶ gr U(L), such that ρ̃(l) = α(l) for any l ∈ L. This proves that gr U(L) is isomorphic to \overline{V (L)}. Observe that U(L) and gr U(L) are isomorphic as vector spaces. Choose in U(L) a base {u_{i}} such that the set {ū_{i}} is a graded base of gr U(L); then evidently {π(u_{i})} is a base of the space \overline{V (L)}, and so ker π = 0 and π is an isomorphism. □

Now, everything is finished by the following

Theorem 2 For any LJ-algebra L, the Poisson LJ-algebra \overline{V (L)} admits a quantization deformation.

Proof. We first define a multiplication ∗ on V (L)[t] by setting

a ∗ b = ab + 1/2 [a, b]t,
a^{2} ∗ b = 1/2 ([a^{2}, b]t + {a, a, b}t^{2}),
b ∗ a^{2} = 1/2 ([b, a^{2}]t + {a, a, b}t^{2}),
a^{2} ∗ b^{2} = 1/2 ({a, a, b^{2}}t^{2} + [{a, a, b}, b]t^{3}).

Let us check that J(L) ∗ V (L) + V (L) ∗ J(L) ⊆ J(L)[t]. Let x, y, z, t ∈ L; consider

j(x, y, z, t) = {x, z, t}y − {y, z, t}x − 1/2[z, [x, y]]t
−1/2 ([t, [x, y]]z + [[t, z], x]y − [[t, z], y]x)
= ∑_{i} j_{1}^{(i)}j_{2}^{(i)}.

We claim that for any a ∈ V (L) holds

∑_{i}{j_{1}^{(i)}, j_{2}^{(i)}, a} + {j_{2}^{(i)}, j_{1}^{(i)}, a} ∈ J.  (16)

In fact, we have

∑_{i}{j_{1}^{(i)}, j_{2}^{(i)}, a} + {j_{2}^{(i)}, j_{1}^{(i)}, a} =

= {{x, z, t}, y, a} + {y, {x, z, t}, a} − {{y, z, t}, x, a} −{x, {y, z, t}, a} − 1/2{[z, [x, y]], t, a} − 1/2{t, [z, [x, y]], a} −1/2{[t, [x, y]], z, a} − 1/2{z, [t, [x, y]], a} − 1/2{[[t, z], x], y, a} −1/2{y, [[t, z], x], a} + 1/2{[[t, z], y], x, a} + 1/2{x, [[t, z], y], a}.

One can straightforwardly check that this sum is identically zero in the algebra S̃L. Since it is of degree 5, by Proposition 1 it is an identity in the free LJ-algebra L̃. Consequently, this identity holds in the LJ-algebra \overline{V (L)} = V (L)/J(L), which implies inclusion (16).

Now, by definition of the multiplication ∗, we have

j(x, y, z, t) ∗ a = 1/2 [j, a]t + 1/4 ∑_{i}({j_{1}^{(i)}, j_{2}^{(i)}, a} + {j_{2}^{(i)}, j_{1}^{(i)}, a})t^{2},

which lies in J[t] by Corollary 1 and (16). Furthermore, if a ∈ S^{2}(L), a = b^{2}, then

j(x, y, z, t) ∗ b^{2} = 1/2 ∑_{i}({j_{1}^{(i)}, j_{2}^{(i)}, b^{2}} + {j_{2}^{(i)}, j_{1}^{(i)}, b^{2}})t^{2}

+1/2 ∑_{i}[{j_{1}^{(i)}, j_{2}^{(i)}, b} + {j_{2}^{(i)}, j_{1}^{(i)}, b}, b]t^{3},

which lies in J[t] by the same reasons. Therefore, V (L) ∗ J(L) ⊆ J(L)[t]. Similarly, J(L) ∗ V (L) ⊆ J(L)[t].

The inclusions just proved show that ∗ can be defined naturally on the quotient algebra \overline{V (L)}[t]. Conditions (i)–(iv) are evidently satisfied by ∗. We have only to prove that ∗ is associative.

We will consider only the most complicated case; the remaining cases are proved similarly.

Compute

(a^{2} ∗ b^{2}) ∗ c^{2} = 1/2 ({a, a, b^{2}} ∗ c^{2})t^{2} + 1/2([{a, a, b}, b] ∗ c^{2})t^{3}.

One can easily check that in any Poisson LJ-algebra hold the identities

{a, a, b^{2}} = {a, a, b}b + [a, b]^{2} + a[[a, b], b].

Hence,

{a, a, b^{2}} ∗ c^{2} = 1/4 ({{{a, a, b}, b, c^{2}} + {b, {a, a, b}, c^{2}})t^{2}
+1/4 ({a, [[a, b], b], c^{2}} + {[[a, b], b], a, c^{2}})t^{2}
+1/2 {[a, b], [a, b], c^{2}}t^{2} + 1/2[{[a, b], [a, b], c}, c]t^{3}
+1/4 ([{{a, a, b}, b, c}, c] + [{b, {a, a, b}, c}, c])t^{3}
+1/4 ([{{a, [[a, b], b], c}, c] + [{[[a, b], b], a, c}, c])t^{3}.

Therefore, (a^{2} ∗ b^{2}) ∗ c^{2} = At^{4} + Bt^{5}, where

A = 1/8 (({{a, a, b}, b, c^{2}} + {b, {a, a, b}, c^{2}}
+ {a, [[a, b], b], c^{2}} + {[[a, b], b], a, c^{2}},
+2 [[{a, a, b}, b], c^{2}]) + 2{[a, b], [a, b], c^{2}},

and

B = 1/8 ([{{a, a, b}, b, c}, c] + [{b, {a, a, b}, c}, c]
+ [{{a, [[a, b], b], c}, c] + [{[[a, b], b], a, c}, c]
+2 [{[a, b], [a, b], c}, c] + 2{c, c, [{a, a, b}, b]}.

Similarly, a^{2} ∗ (b^{2} ∗ c^{2}) = A_{1}t^{4} + B_{1}t^{5}, where

A_{1} = 1/4 ({a, a, {b, b, c^{2}}} + [a^{2}, [{b, b, c}, c]]),

and

B_{1} = 1/8 ([{a, a, b}, {b, b, c}] + [{a, a, {b, b, c}}, c]
+ [{a, a, b}, [[b, c], c]] + [{a, a, [[b, c], c]}, b]
+2 [{a, a, [b, c]}, [b, c]] + 2 {a, a, [{b, b, c}, c]}).

One can straightforwardly check that A = A_{1} and B = B_{1} in the algebra gr A[X] for any a, b, c ∈ X. Since the elements A−A_{1}, B−B_{1} are of degree 6, by Proposition 1 we have A − A_{1}, B − B_{1} ∈ J(L̃), and hence A = A_{1}, B = B_{1} in \overline{V (L)}.

This proves the theorem. □

5 Application

In order to formulate an application of our result, we recall some definitions from [2].

Let A = A[x, y] be the free associative algebra over a field F of characteristic p > 0 on free generators x, y. For any f = f(x, y) ∈ A we define a space A_{f} as a minimal subspace of A with the properties:

(i) x, y ∈ A_{f},

(ii) If a, b ∈ A_{f}, then f(a, b) ∈ A_{f}.

A series E(x) = ∑_{i=0}^{∞} a_{i}x^{i} is called an f-exponent if a_{0} = 1 and in the natural completion \overline{A} of A we have

E(x)E(y) = E(z),  (17)

where z = ∑_{i=0}^{∞} z_{i}, deg z_{i} = i, z_{i} ∈ A_{f}.

A series E(x) we will call a p-exponent if E(x) is an f-exponent for some polynomial f, and for any other polynomial g we have A_{f} ⊆ A_{g}.

It was proved in [2] that in the case p = 0 a 0-exponent exists and coincides with the classical exponent exp x = ∑_{i=0}^{∞} x^{i}/i!. It was also conjectured there that a p-exponent exists (certainly not unique!) and coincides with an f_{p}- exponent, where the element f_{p} ∈ A is defined as follows.

It is well known (see for instance [3, section 5.7]) that for every simple p > 0 an equality of the following type holds:

(x + y)^{p} = x^{p} + y^{p} + s(x, y) + pf̃_{p}(x, y),

where s(x, y) and f̃_{p}(x, y) are a Lie and associative polynomials on x, y with integer coefficients. Of course, the polynomial f̃_{p}(x, y) is not uniquely defined by this equation; but it is unique modulo Lie polynomials.

Now, we fix f̃_{p}, reduce its coefficients modulo p, and set f_{p}(x, y) = [x, y]+ f̃_{p}(x, y).

For p = 3, it is easy to see that we can take f̃_{3} = xyx + yxy; hence f_{3} = xy − yx + xyx + yxy, and the subspace A_{f_{3}} coincides with the free special LJ-algebra SLJ[x, y]. Consequently, the elements z_{i} in equality (17) belong to SLJ[x, y] and can be written as concrete polynomials on x, y in terms of operations [,] and {, , }. More specifically, it was proved in [2] that the series

E(x) = x + \sqrt{1 + x^{2}} = 1 + x − x^{2} + x^{4} + ⋯

is a 3-exponent and an f_{3}-exponent; and the element z from (17) is given by

z = E^{−1}(E(x)E(y)) = x ∗ y = x + y − [x, y] + {x, x, y} + {x, y, y} + · · · . (18)

The following Theorem was conjectured in [2].

Theorem 3 Let L = LJ[x, y] be the free LJ-algebra over a field F of characteristic 3 on generators x, y, and \overline{L} be the completion of L; then the operation u∗v = w, defined by the right part of (18), introduces on \overline{L} a group structure.

Proof. By Theorem 1, the LJ-algebra L is special; hence it is isomorphic to SL = SLJ[x, y], and by (17) we have u ∗ v = E^{−1}(E(u)E(v)) for any u, v ∈ L = SL ⊆ A. This implies easily the statement of the Theorem. □

References

[1] Yu. A. Bahturin, Identities of Lie algebras, Moscow, Nauka, 1985 (in Russian); English translation by VNU Science Press: Utrecht, 1987.

[2] A. Grishkov, An analog of exponential map and Campbell–Hausdorff series in characteristic 3, to appear.

[3] N. Jacobson, Lie algebras, Interscience Publishers, 1962.

[4] N. Jacobson, Structure and representations of Jordan algebras, AMS, Providence, RI, 1968.

[5] I. P. Shestakov, Every Akivis algebra is linear, Geometriae dedicata, to appear.

[6] I. P. Shestakov, Speciality and Deformations of Algebras, Proceedings of the International Algebraic Conference dedicated to memory of A.G.Kurosh, May 1998, Moscow, to appear.

[7] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, A. I. Shirshov, Rings that are nearly associative, Moscow, Nauka, 1978 (in Russian); English translation by Academic Press: N.Y., 1982.