ROOT-GRADED LIE ALGEBRAS WITH COMPATIBLE GRADING

Yoji Yoshii

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, K1N 6N5 Canada

Abstract

Lie algebras graded by a finite irreducible reduced root system Δ will be generalized to predivision (Δ, G)-graded Lie algebras for an abelian group G. In this paper such algebras are classified, up to central extensions, when Δ = A_{l} for l ≥ 3, D or E, and G = ℤ^{n}.

Introduction

The concept of a Lie algebra over a field F of characteristic 0 graded by a finite irreducible reduced root system Δ or a Δ-graded Lie algebra was introduced by Berman and Moody [3]. It is a Lie algebra L together with a finite dimensional split simple Lie algebra 𝔤, a split Cartan subalgebra 𝔥 of 𝔤 and the root system Δ, so that 𝔤 has the root space decomposition 𝔤 = 𝔥 ⊕ ( ⊕_{μ∈Δ} 𝔤_{μ}) with 𝔥 = 𝔤_{0}, satisfying the following three conditions:

(i) L contains 𝔤 as a subalgebra; (ii) L = ⊕_{μ∈Δ∪{0}} L_{μ}, where L_{μ} = {x ∈ L | [h, x] = μ(h)x for all h ∈ 𝔥}; and (iii) L_{0} = ∑_{μ∈Δ} [L_{μ}, L_{−μ}].

The subalgebra 𝔤 is called the grading subalgebra of L.

Berman and Moody classified Δ-graded Lie algebras, up to central extensions, when Δ has type A_{l}, l ≥ 2, D or E in [3], and then Benkart and Zelmanov completed the classification for the other types in [5]. (In [7], using the connection to Jordan pairs, Δ-graded Lie algebras were classified, where Δ ≠ E_{8}, F_{4} or G_{2}. The results in [7] hold for root systems Δ of infinite rank, as well as for Lie algebras over rings.)

Typeset by AMS-TEX

Let L = ⊕_{μ∈Δ∪{0}} L_{μ} be a Δ-graded Lie algebra over F and let G be an abelian group. We say that L admits a compatible G-grading or simply L is a (Δ, G)-graded Lie algebra if L = ⊕_{g∈G} L^{g} is a G-graded Lie algebra such that 𝔤 ⊂ L^{0}. Then we have

L = ⨁_{μ∈Δ∪{0}} ⨁_{g∈G} L_{μ}^{g},

where L_{μ}^{g} = L_{μ} ∩ L^{g} (see Definition 2.5). Let Z(L) be the centre of L and let

{h_{μ} ∈ 𝔥 | μ ∈ Δ}

be the set of coroots. Then L is called a division (Δ, G)-graded Lie algebra if for any μ ∈ Δ and any 0 ≠ x ∈ L_{μ}^{g}, there exists y ∈ L_{−μ}^{−g} such that [x, y] ≡ h_{μ} modulo Z(L).

Let us explain the case Δ = A_{l} for l ≥ 3 in order to describe our motivation of this paper. By [3], an A_{l}-graded Lie algebra covers psl_{l+1}(A) for some unital associative algebra A (see Definition 2.9). Then Berman, Gao and Krylyuk showed in [4] that the core of an extended affine Lie algebra of type A_{l} for l ≥ 3 is an A_{l}-graded Lie algebra and covers sl_{l+1}(ℂ_{q}) where ℂ_{q} = ℂ_{q}[t_{1}^{±}, …, t_{n}^{±}] is a certain ℤ^{n}-graded associative algebra, called a quantum torus over ℂ (see §2 below). We will see that L = sl_{l+1}(ℂ_{q}) is a division (A_{l}, ℤ^{n})-graded Lie algebra over ℂ so that L = ⊕_{μ∈Δ∪{0}} ⊕_{α∈ℤ^{n}} L_{μ}^{α}. Moreover, L satisfies

(∗)  dim_{ℂ} L_{μ}^{α} = 1 for all μ ∈ Δ and α ∈ ℤ^{n}.

Our goal is to describe division (A_{l}, ℤ^{n})-graded Lie algebras without assuming (∗). This generalizes the core of an extended affine Lie algebra of type A_{l} (see Example 2.8(c)). One of the main results of the paper, which is contained in Proposition 2.13 is the following: Result 1. Let l ≥ 3. Then any division (A_{l}, G)-graded Lie algebra covers psl_{l+1}(P) where P is a division G-graded associative algebra.

For a group G, a division G-graded algebra is defined as a G-graded algebra whose nonzero homogeneous elements are all invertible. A division G-graded associative algebra over a field F can be considered as a crossed product algebra D ∗ G for an associative division algebra D over F (see §1). Our next goal is to describe D ∗ ℤ^{n}. For this purpose, we introduce the following definition: A triple (D, φ, q) is called a division ℤ^{n}-grading triple over F if

(1) D is an associative division algebra over F;

(2) φ = (φ_{1}, …, φ_{n}) is an n-tuple of F -automorphisms φ_{i} of D; and

(3) q = (q_{ij}) is an n × n matrix over D satisfying, for all 1 ≤ i, j, k ≤ n,

q_{ii} = 1 and q_{ji}^{−1} = q_{ij},
φ_{j}φ_{i} = I(q_{ij})φ_{i}φ_{j},
φ_{k}(q_{ij}) = q_{jk}φ_{j}(q_{ik})q_{ij}φ_{i}(q_{kj})q_{ki},

where I(q_{ij}) is the inner automorphism of D determined by q_{ij}, i.e.,

I(q_{ij})(d) = q_{ij}dq_{ij}^{−1} for d ∈ D.

We will show that any D∗ℤ^{n} can be constructed from a division ℤ^{n}-grading triple (D, φ, q).

Let us briefly explain how this works. First we consider the simplest example of D ∗ ℤ^{n}, namely, the ring D[t_{1}^{±1}, …, t_{n}^{±1}] of Laurent polynomials over D in n-variables. Note that D[t_{1}^{±1}, …, t_{n}^{±1}] = ⊕_{α∈ℤ^{n}} Dt_{α} is a ℤ^{n}-graded algebra, where t_{α} = t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}} for α = (α_{1}, …, α_{n}) ∈ ℤ^{n}, and the multiplication rule is determined by

t_{i}d = dt_{i} and t_{j}t_{i} = t_{i}t_{j} for all d ∈ D and all i, j.

Then one sees that D ∗ ℤ^{n} has the same ℤ^{n}-grading as in D[t_{1}^{±1}, …, t_{n}^{±1}], i.e., D ∗ ℤ^{n} = ⊕_{α∈ℤ^{n}} Dt_{α} as a D-vector space. It is easily seen that the multiplication rule in D ∗ ℤ^{n} determines a division ℤ^{n}-grading triple (D, φ, q) as follows:

(∗∗)  t_{i}d = φ_{i}(d)t_{i} and t_{j}t_{i} = q_{ij}t_{i}t_{j}, for all 1 ≤ i, j ≤ n,

as the defining relations in the quantum torus F_{q} = F_{q}[t_{1}^{±1}, …, t_{n}^{±1}] for q = (q_{ij}).

Conversely, for a division ℤ^{n}-grading triple (D, φ, q), let D_{φ,q} = D_{φ,q}[t_{1}^{±1}, …, t_{n}^{±1}] be the same ℤ^{n}-graded D-vector space as D[t_{1}^{±1}, …, t_{n}^{±1}] above. We will show that the relations (∗∗) determine an associative multiplication on D_{φ,q}. Thus we will get the following:

Result 2. For any division ℤ^{n}-grading triple (D, φ, q), there exists a crossed product D_{φ,q} = D_{φ,q}[t_{1}^{±1}, …, t_{n}^{±1}] such that D_{φ,q} = ⊕_{α∈ℤ^{n}} Dt_{α} has the same ℤ^{n}-grading as D[t_{1}^{±1}, …, t_{n}^{±1}] above, and the multiplication rule is determined by (∗∗). Conversely, any crossed product D ∗ ℤ^{n} is isomorphic to D_{φ,q} for some φ and q (see Theorem 3.3 for more precise statements).

Note that if D = F, then φ = 1 = (id, …, id) and F_{1,q} = F_{q} is the quantum torus.

Consequently, one gets that any division (A_{l}, ℤ^{n})-graded Lie algebra for l ≥ 3 covers psl_{l+1}(D_{φ,q}). We will also classify division (Δ, ℤ^{n})-graded Lie algebras when Δ = D or E, which is simpler than the case A. Moreover, our concept of “division” can be generalized to “predivision” (see Definition 2.7). Results 1 and 2 above will be proved in this more general set-up.

The organization of the paper is as follows. In §1 we review basic concepts of graded algebras and crossed product algebras. In §2 we prove some properties of (Δ, G)-graded Lie algebras. Then predivision or division (Δ, G)-graded Lie algebras are defined. After describing some examples of them, we classify predivision (Δ, G)-graded Lie algebras for Δ = A_{l} (l ≥ 3), D and E types. In §3 we classify crossed product algebras R ∗ ℤ^{n}. Finally in §4 we give a summary of our results.

§1 Basic Concepts

For any group G and any G-graded algebra L = ⊕_{g∈G} L_{g}, we denote

supp L := {g ∈ G | L_{g} ≠ (0)}.

Then we have L = ⊕_{g∈G′} L_{g} where G′ = ⟨supp L⟩ is the subgroup of G generated by supp L. Because of this, we will in the following always assume

(1.1)  G = ⟨supp L⟩.

Whenever a class of algebras has a notion of invertibility, one can make the following definition:

Definition 1.2. Let G be a group. A G-graded algebra P = ⊕_{g∈G} P_{g} is called a predivision G-graded algebra if P_{g} contains an invertible element for all g ∈ supp P. Also, P is called a division G-graded algebra if all nonzero homogeneous elements are invertible.

One can easily check that if P is a predivision G-graded associative algebra, then supp P = G and P is strongly graded, i.e., P_{g}P_{h} = P_{gh} for all g, h ∈ G. This is not true if P is a Jordan algebra (see [9]). Predivision G-graded associative algebras are realized as crossed product algebras, which we recall here:

Definition 1.3. Let R be a unital associative algebra over a field F and G a group. Let R ∗ G be the free left R-module with basis \overline{G} = \{\overline{g} | g ∈ G\}, a copy of G. Define a multiplication on R ∗ G by linear extension of

(r\overline{g})(s\overline{h}) = rσ_{g}(s)τ(g, h)\overline{gh},

for r, s ∈ R and g, h ∈ G, where

(action) σ : G ⟶ Aut_{F} (R), the group of F -automorphisms of R,
(twisting) τ : G × G ⟶ U(R), the group of units of R,

are arbitrary maps and σ_{g} := σ(g). It is easily seen that R ∗ G is an algebra over F. R ∗ G = (R, G, σ, τ) is called a crossed product algebra over F if the multiplication is associative. If there is no action or twisting, that is, if σ_{g} = id and τ(g, h) = 1 for all g, h ∈ G, then R ∗ G = R[G] is the ordinary group algebra. If the action is trivial, then R ∗ G =: R^{t}[G] is called a twisted group algebra. Finally, if the twisting is trivial, then R ∗ G =: RG is called a skew group algebra.

Remark 1.4. If a crossed product algebra R ∗ G is commutative, then the action is clearly trivial, and so R ∗ G = R^{t}[G].

The following lemma characterizes σ and τ (see [8], Lemma 1.1 p.2). We denote by I(d) the inner automorphism determined by d ∈ U(R), i.e., I(d)(r) = drd^{−1} for r ∈ R.

1.5. The associativity of R ∗ G is equivalent to the following two conditions: for all g, h, k ∈ G,

(i) σ_{g}σ_{h} = I(τ(g, h))σ_{gh},

(ii) σ_{g}(τ(h, k))τ(g, hk) = τ(g, h)τ(gh, k).

Remark 1.6. If R is commutative, then the action σ : G ⟶ Aut_{F} (R) becomes a group homomorphism by condition (i) in 1.5. So the action is really a “group action” in usual sense. Also, for a skew group algebra RG, the action becomes a group homomorphism for the same reason. Conversely, any group action G ⟶ Aut_{F} (R) defines a skew group algebra RG.

If d : G ⟶ U(R) assigns to each element g ∈ G a unit d_{g}, then G̃ = \{d_{g}\overline{g} | g ∈ G\} yields another R-basis for R ∗ G so that R ∗ G is a crossed product algebra for the new basis. One calls this a diagonal change of basis ([8], p.3). Any crossed product algebra has an identity element. It is of the form 1 = u\overline{e} for some unit u in R where e is the identity element of G ([8], Exercise 2 p.9). We can and will assume that 1 = \overline{e}, via a diagonal change of basis, and so τ(g, e) = τ(e, g) = 1 for all g ∈ G. The embedding of R into R ∗ G is then given by r ↦ r\overline{e}. Also, we have ([8], p.3)

(1.7)  r\overline{g} is invertible if and only if r ∈ U(R).

Now, it is clear that a crossed product algebra R ∗ G = ⊕_{g∈G} R\overline{g} is a predivision Ggraded associative algebra. Conversely, suppose that A = ⊕_{g∈G} A_{g} is a predivision G-graded associative algebra over F. Then we have A = ⊕_{g∈G} Rx_{g} where R = A_{e} and an invertible element x_{g} ∈ A_{g}, which exists since A is predivision graded and supp A = G. Moreover, for h ∈ G, we have x_{g}x_{h} = x_{g}x_{h}(x_{gh})^{−1}x_{gh}. So we can put τ(g, h) := x_{g}x_{h}(x_{gh})^{−1} ∈ U(R). Then we have x_{g}x_{h} = τ(g, h)x_{gh}. Also, let I(x_{g}) be the inner automorphism determined by x_{g} and let σ_{g} := I(x_{g}) |_{R}. Then, σ_{g} is clearly an F -automorphism of R and for r, r′ ∈ R,

(rx_{g})(r′x_{h}) = r(x_{g}r′x_{g}^{−1})x_{g}x_{h} = rσ_{g}(r′)x_{g}x_{h} = rσ_{g}(r′)τ(g, h)x_{gh}.

Hence A is a crossed product algebra R ∗ G determined by these σ and τ. So the two concepts, a crossed product algebra R ∗ G and a predivision G-graded associative algebra, coincide (see [8], Exercise 2 p.18). In particular, a division G-graded associative algebra is a crossed product algebra R ∗ G where R is a division algebra.

By Remark 1.4, a predivision G-graded commutative associative algebra Z = ⊕_{g∈G} Z_{g} (G is necessarily abelian) is a twisted group algebra K^{t}[G] where K := Z_{e}. Moreover (see [8], Exercise 6 p.10):

1.8. If the abelian group G is free, then Z is a group algebra K[G]. In particular, when G = ℤ^{n}, Z is the algebra K[z_{1}^{±1}, …, z_{n}^{±1}] of Laurent polynomials for invertible elements z_{i} ∈ Z_{ε_{i}}, i = 1, …, n, where {ε_{1}, …, ε_{n}} is a basis of ℤ^{n}.

§2 Predivision (Δ, G)-graded Lie algebras

In this section F is a field of characteristic 0 and Δ is a finite irreducible reduced root system. The concept of a Δ-graded Lie algebra L = ⊕_{μ∈Δ∪{0}} L_{μ} over F as a triple (L, 𝔤, 𝔥) has been defined in the introduction. When no confusion is likely to arise we will use the abbreviation L for (L, 𝔤, 𝔥). Also, we note that the centre Z(L) of L is contained in L_{0}.

A homomorphism (resp. an isomorphism) φ : L ⟶ L′ of Δ-graded Lie algebras L = (L, 𝔤, 𝔥) and L′ = (L′, 𝔤′, 𝔥′), which have the same type Δ, is called a Δ-homomorphism (resp. a Δ-isomorphism) if φ(𝔤) = 𝔤′ and φ(𝔥) = 𝔥′ (cf. Definition 1.20 in [3]). Then one can check that φ(L_{α}) ⊂ L′_{α} for all α ∈ Δ, and so φ(L_{0}) ⊂ L′_{0}. In other words, a Δ-homomorphism is graded.

Recall that a cover L̃ = (L̃, π) of a Lie algebra L is an epimorphism π : L̃ ⟶ L of Lie algebras so that L̃ is perfect, i.e., L̃ = [L̃, L̃], and ker π is contained in the centre of L̃.

Definition 2.1. Let L̃ and L be Δ-graded Lie algebras. If π : L̃ ⟶ L is a cover and a Δ-homomorphism, L̃ = (L̃, π) is called a Δ-cover of L. Also, for Δ-graded Lie algebras L and L′, if there exist a Δ-graded Lie algebra L̃ and maps π : L̃ ⟶ L and π′ : L̃ ⟶ L′ such that (L̃, π) and (L̃, π′) are both Δ-covers, we say that L and L′ are Δ-isogeneous.

Example 2.2. Let L = (L, 𝔤, 𝔥) be a Δ-graded Lie algebra with centre Z(L). Then, for any subspace V of Z(L), L/V = (L/V, 𝔤+V, 𝔥+V) is a Δ-graded Lie algebra, and the canonical epimorphism L ⟶ L/V is a Δ-cover. In particular, L and L/V are Δ-isogeneous.

We will show that if L and L′ are Δ-isogeneous, then L/Z(L) and L′/Z(L′) are Δisomorphic, i.e., there exists a Δ-isomorphism between them.

Lemma 2.3. Let π : L̃ ⟶ L be a cover. Then Z(L̃) = π^{−1}(Z(L)). Hence, if ω : L ⟶ L/Z(L) is the canonical epimorphism, we have ker(ω ∘ π) = Z(L̃).

Proof. For x̃ ∈ L̃ we have x̃ ∈ π^{−1}(Z(L)) ⇔ π([x̃, L̃]) = 0 ⇔ [x, L̃] ⊂ ker π. Since ker π ⊂ Z(L̃) and L̃ is perfect, it follows that x̃ ∈ Z(L̃), whence π^{−1}(Z(L)) ⊂ Z(L̃). The other inclusion is clear. The map ω ∘ π : L̃ ⟶ L/Z(L) is a cover. Perfectness of L implies that L/Z(L) is centreless, whence ker(ω ∘ π) = Z(L̃). □

Corollary 2.4. Suppose that L and L′ are Δ-isogeneous. Then L/Z(L) and L′/Z(L′) are Δ-isomorphic.

Proof. By assumption, there exists a Δ-graded Lie algebra L̃ = (L̃, 𝔤̃, 𝔥̃) such that π : L̃ = (L, 𝔤, 𝔥) ⟶ L and π′ : L̃ ⟶ L′ = (L′, 𝔤′, 𝔥′) are both Δ-covers. Let ω : L ⟶ L/Z(L) and ω′ : L′ ⟶ L′/Z(L′) be the canonical epimorphisms. Then, by Lemma 2.3, we have ker(ω ∘ π) = Z(L̃) = ker(ω′ ∘ π′). Hence there exists the induced isomorphism

φ : L/Z(L) =(L/Z(L), 𝔤 + Z(L), 𝔥 + Z(L))
⟶L′/Z(L′) = (L′/Z(L′), 𝔤′ + Z(L′), 𝔥′ + Z(L′))

such that φ∘ω∘π = ω′ ∘π′. In particular, φ(𝔤+Z(L)) = φ∘ω∘π(𝔤̃) = ω′ ∘π′(𝔤̃) = 𝔤′ +Z(L′) and similarly φ(𝔥 + Z(L)) = 𝔥′ + Z(L′). Therefore, φ is a Δ-isomorphism. □

Now we define new concepts.

Definition 2.5. Let L = (L, 𝔤, 𝔥) = ⊕_{μ∈Δ∪{0}} L_{μ} be a Δ-graded Lie algebra over F. Let G be an abelian group. We say that L admits a compatible G-grading or simply L is a (Δ, G)- graded Lie algebra if L = ⊕_{g∈G} L^{g} is a G-graded Lie algebra such that 𝔤 ⊂ L^{0}. In this case, L^{g} is a 𝔥-module for all g ∈ G via the adjoint action. Hence we have L^{g} = ⊕_{μ∈Δ∪{0}} L_{μ}^{g} where L_{μ}^{g} = L_{μ} ∩ L^{g} (see [6] Proposition 1, p.92). Therefore, L_{μ} = ⊕_{g∈G} L_{μ}^{g} and

L = ⨁_{μ∈Δ∪{0}} ⨁_{g∈G} L_{μ}^{g}.

Remark 2.6. (i) The compatible G-grading is completely determined by L_{μ}^{g} for all μ ∈ Δ and g ∈ G since L_{0}^{g} = ∑_{μ∈Δ} ∑_{g=h+k} [L_{μ}^{h}, L_{−μ}^{k}].

(ii) Let supp L_{μ} := {g ∈ G | L_{μ}^{g} ≠ (0)}. Recall supp L = {g ∈ G | L^{g} ≠ (0)} as defined in the beginning of §1. If g ∈ supp L, then L_{0}^{g} ≠ (0) or there exists some μ ∈ Δ such that L_{μ}^{g} ≠ (0). If L_{μ}^{g} ≠ (0), we have g ∈ supp L_{μ}. If L_{0}^{g} ≠ (0), then g = h+k ∈ supp L_{μ}+supp L_{−μ} for some μ ∈ Δ and h, k ∈ G by (i). Thus since 0 ∈ supp L_{−μ}, we obtain

supp L ⊂ ⋃_{μ∈Δ} (supp L_{μ} + supp L_{−μ}).

Definition 2.7. Let L = (L, 𝔤, 𝔥) be a (Δ, G)-graded Lie algebra with centre Z(L) and let

{h_{μ} ∈ 𝔥 | μ ∈ Δ}

be the set of coroots. Then L is called predivision if

(pd) for any μ ∈ Δ and any L_{μ}^{g} ≠ (0), there exist x ∈ L_{μ}^{g} and y ∈ L_{−μ}^{−g} such that

[x, y] ≡ h_{μ} modulo Z(L); and division if

(d) for any μ ∈ Δ and any 0 ≠ x ∈ L_{μ}^{g}, there exists y ∈ L_{−μ}^{−g} such that [x, y] ≡ h_{μ} modulo Z(L).

Note that (d) implies (pd), i.e., ‘division’ ⟹ ‘predivision’. If dim_{F} L_{μ}^{g} ≤ 1 for all μ ∈ Δ and g ∈ G, then two concepts, ‘predivision’ and ‘division’, coincide.

Example 2.8. (a) A Δ-graded Lie algebra is a predivision (Δ, G_{0})-graded algebra for the trivial group G_{0} = {0}.

(b) The core of an extended affine Lie algebra of reduced type Δ with nullity n is a division (Δ, Λ)-graded Lie algebra over ℂ, where Λ is a free abelian group of rank n. Indeed, it is known that such a core is a Δ-graded Lie algebra over ℂ, say L = ⊕_{μ∈Δ∪{0}} L_{μ}, and each L_{μ} has a decomposition L_{μ} = ⊕_{δ∈Λ} L_{μ}^{δ}, where Λ is defined as the group generated by the isotropic roots δ (we use the notation L_{μ}^{δ} instead of L_{μ+δ} which is normally used in the theory of extended affine Lie algebras). It turns out that Λ is a lattice of rank n with ⟨supp L⟩ = Λ (for details see [2]). Let L^{δ} := ⊕_{μ∈Δ∪{0}} L_{μ}^{δ}. Then the grading subalgebra 𝔤 is contained in L^{0} so that L = ⊕_{δ∈Λ} L^{δ} gives a compatible Λ-grading. Thus L is a (Δ, Λ)-graded Lie algebra.

We recall one of the basic properties of extended affine Lie algebras (see [1]): For any μ ∈ Δ, δ ∈ Λ and any 0 ≠ e_{μ}^{δ} ∈ L_{μ}^{δ}, there exist some f_{μ}^{δ} ∈ L_{−μ}^{−δ} and h_{μ}^{δ} ∈ L_{0}^{0} such that ⟨e_{μ}^{δ}, f_{μ}^{δ}, h_{μ}^{δ}⟩ is an sl_{2}-triplet, and in particular [e_{μ}^{δ}, f_{μ}^{δ}] = h_{μ}^{δ}.

One can check that h_{μ} − h_{μ}^{δ} ∈ Z(L) for all coroots h_{μ} = h_{μ}^{0} of 𝔤. Therefore L is a division (Δ, Λ)-graded Lie algebra. We note that dim_{ℂ} L_{μ}^{δ} ≤ 1 for all μ ∈ Δ and δ ∈ Λ, which is also one of the basic properties of extended affine Lie algebras.

(c) Let Z = ⊕_{g∈G} Z_{g} be a G-graded commutative associative algebra over F and let 𝔤 = 𝔥 ⊕ ( ⊕_{μ∈Δ} 𝔤_{μ}) be a finite dimensional split simple Lie algebra over F of type Δ with the set {h_{μ} ∈ 𝔥 | μ ∈ Δ} of coroots. Then L := 𝔤 ⊗_{F} Z is a (Δ, G)-graded Lie algebra. In fact, L = ⊕_{μ∈Δ∪{0}} (𝔤_{μ} ⊗_{F} Z) for 𝔤_{0} = 𝔥 is a Δ-graded Lie algebra with grading subalgebra 𝔤 = 𝔤 ⊗ 1. We put L^{g} := 𝔤 ⊗_{F} Z_{g} for all g ∈ G. Then supp L = supp Z and L = ⊕_{g∈G} L^{g} is a G-graded Lie algebra with 𝔤 ⊂ L^{0}, i.e, the G-grading is compatible. Hence L is a (Δ, G)-graded Lie algebra. We call the compatible G-grading of L = 𝔤 ⊗_{F} Z the natural compatible G-grading obtained from the G-grading of Z.

Suppose that Z = ⊕_{g∈G} K\overline{g} is a crossed product commutative algebra over F. Let e ∈ 𝔤_{μ} and f ∈ 𝔤_{−μ} such that [e, f] = h_{μ}. Then e ⊗ \overline{g} ∈ L_{μ}^{g}, f ⊗ \overline{g}^{−1} ∈ L_{−μ}^{−g} and

[e ⊗ \overline{g}, f ⊗ \overline{g}^{−1}] = [e, f] ⊗ \overline{g} \overline{g}^{−1} = h_{μ} ⊗ 1 = h_{μ}

for all g ∈ G, and so L is a predivision (Δ, G)-graded Lie algebra over F. Note that Z(L) = (0). Also, if K is a field, then L is a division (Δ, G)-graded Lie algebra.

Suppose that L̃ = (L̃, 𝔤̃, 𝔥̃) = ⊕_{g∈G} L̃^{g} is a (Δ, G)-graded Lie algebra and that π : L̃ ⟶ L is a cover of a Lie algebra L. Then L = (L, π(𝔤̃), π(𝔥̃)) becomes a Δ-graded Lie algebra so that (L̃, π) is a Δ-cover of L. Moreover, if ker π is G-graded, then L admits the induced compatible G-grading L = ⊕_{g∈G} π(L̃^{g}). In particular, since the centre Z(L̃) is always G-graded, L̃/Z(L̃) is a (Δ, G)-graded Lie algebra. Definition 2.9. Let P be a unital associative algebra over F and let 𝔤𝔩_{l+1}(P) be the Lie algebra consisting of all (l + 1) × (l + 1) matrices over P under the commutator product (l ≥ 1). Let e_{ij}(a) ∈ 𝔤𝔩_{l+1}(P) whose (i, j)-entry is a and the other entries are all 0. We define sl_{l+1}(P) as the subalgebra of 𝔤𝔩_{l+1}(P) generated by e_{ij}(a) for all a ∈ P and 1 ≤ i ≠ j ≤ l+1. The centre Z(sl_{l+1}(P)) of sl_{l+1}(P) consists of ∑_{i=1}^{l+1} e_{ii}(a) for a ∈ [P, P]∩Z(P) where [P, P] is the span of all commutators in P and Z(P) is the centre of P. We define psl_{l+1}(P) as sl_{l+1}(P)/Z(sl_{l+1}(P)).

It is well-known that sl_{l+1}(P) is an A_{l}-graded Lie algebra (see [3]): Denote {e_{ij}(b) | b ∈ B} by e_{ij}(B) for any subset B ⊂ P. Let

sl_{l+1}(F) = 𝔥 ⊕ ⨁_{1≤i≠j≤l+1} e_{ij}(F 1) ⊂ sl_{l+1}(P),

be the split simple Lie algebra over F of type A_{l} where 𝔥 is the Cartan subalgebra consisting of diagonal matrices of sl_{l+1}(F). Let ε_{i} : 𝔥 ⟶ F be the projection onto the (i, j)-entry for i = 1, …, l + 1, and Δ := {ε_{i} − ε_{j} | i ≠ j}, which is a root system of type A_{l}. Then

sl_{l+1}(P) = L_{0} ⊕ ( ⨁_{ε_{i}−ε_{j}∈Δ} e_{ij}(P)),

where L_{0} = ∑_{ε_{i}−ε_{j}∈Δ} [e_{ij}(P), e_{ji}(P)], is an A_{l}-graded Lie algebra with grading subalgebra sl_{l+1}(F). Let Z := Z(sl_{l+1}(P)). We can and will identify sl_{l+1}(F) + Z with sl_{l+1}(F) and e_{ij}(P) + Z with e_{ij}(P), and so

psl_{l+1}(P) = (L_{0}/Z) ⊕ ( ⨁_{ε_{i}−ε_{j}∈Δ} e_{ij}(P))

is also an A_{l}-graded Lie algebra with the same grading subalgebra sl_{l+1}(F).

Example 2.10. Let L = sl_{l+1}(P) be the A_{l}-graded Lie algebra over F with grading subalgebra sl_{l+1}(F) described above. If P = ⊕_{g∈G} P_{g} is a G-graded algebra, then L admits a compatible G-grading. Indeed, let

L^{g} := { ∑_{i,j} e_{ij}(P_{g}) | ∑_{i,j} e_{ij}(P_{g}) ⊂ L}.

Then L = ⊕_{g∈G} L^{g}, and it is a G-graded Lie algebra with sl_{l+1}(F) ⊂ L^{0}. Note that supp L ⊃ supp P, and so ⟨supp L⟩ = G. Also, psl_{l+1}(P) admits the induced compatible G-grading. We call the compatible G-grading of L or psl_{l+1}(P) the natural compatible G-grading obtained from the G-grading of P. This grading is the unique G-grading so that

L_{ε_{i}−ε_{j}}^{g} = e_{ij}(P_{g}) = psl_{l+1}(P)_{ε_{i}−ε_{j}}^{g} for all ε_{i} − ε_{j} ∈ Δ and g ∈ G.

If P = ⊕_{g∈G} R\overline{g} is a crossed product algebra, then

[e_{ij}(\overline{g}), e_{ji}(\overline{g}^{−1})] = e_{ii}(1) − e_{jj}(1) = [e_{ij}(1), e_{ji}(1)] = h_{ε_{i}−ε_{j}}

for all g ∈ G. Thus L and psl_{l+1}(P) with the natural compatible G-gradings from the G-grading of P are predivision (A_{l}, G)-graded Lie algebras over F. Also, if R is a division algebra, then the (A_{l}, G)-graded Lie algebras L and psl_{l+1}(P) are division.

Lemma 2.11. (i) Let P be a unital associative algebra. Suppose that l ≥ 2 and that the A_{l}- graded Lie algebra psl_{l+1}(P) described above admits a predivision (resp. division) compatible G-grading. Then P is a predivision (resp. division) G-graded algebra, and the G-grading of psl_{l+1}(P) is the natural compatible G-grading obtained from the G-grading of P.

(ii) Let Z be a unital commutative associative algebra. Suppose that the Δ-graded Lie algebra 𝔤 ⊗_{F} Z described in Example 2.8(c) admits a predivision (resp. division) compatible G-grading. Then Z is a predivision (resp. division) G-graded algebra, and the G-grading of 𝔤 ⊗_{F} Z is the natural compatible G-grading obtained from the G-grading of Z.

Proof. (i): By assumption, psl_{l+1}(P) = psl_{l+1}(P)_{0}⊕(⊕_{ε_{i}−ε_{j}∈Δ} e_{ij}(P)) admits a predivision (resp. division) compatible G-grading, say

psl_{l+1}(P) = psl_{l+1}(P)_{0} ⊕ ( ⊕_{ε_{i}−ε_{j}∈Δ} ⊕_{g∈G} e_{ij}(P)^{g}).

Let

P_{g}^{ij} := {p ∈ P | e_{ij}(p) ∈ e_{ij}(P)^{g}} for i ≠ j.

We claim that P_{g}^{ij} = P_{g}^{rs} for all ε_{r} − ε_{s} ∈ Δ.

In general, it is well-known that for any distinct α, β ∈ Δ = A_{l}, l ≥ 2, D or E, there exists a sequence α_{1}, …, α_{t} ∈ Δ so that α_{1} = α, α_{t} = β and α_{i+1} − α_{i} ∈ Δ for i = 1, …, t − 1.

Now, it is enough to show that P_{g}^{ij} ⊂ P_{g}^{rs}. Let p ∈ P_{g}^{ij}. We apply the above for α = ε_{i}−ε_{j} and β = ε_{r} − ε_{s}. For p ∈ P_{g}^{ij},

[··[[e_{ij}(p), e_{α_{2}−α_{1}}(1)], e_{α_{3}−α_{2}}(1)], …, e_{α_{t}−α_{t−1}}(1)] = ±e_{α_{t}}(p) = ±e_{rs}(p) ∈ e_{rs}(P)^{g}

since [e_{ij}(p), e_{kl}(1)] = δ_{jk}e_{il}(p) − δ_{li}e_{kj}(p) and e_{α_{i+1}−α_{i}}(1) ∈ L_{α_{i+1}−α_{i}}^{0}. Hence p ∈ P_{g}^{rs} and our claim is settled.

Thus one can write P_{g} = P_{g}^{ij} and P = ⊕_{g∈G} P_{g}. Since, for p ∈ P_{g} and q ∈ P_{h} (g, h ∈ G),

[e_{ij}(p), e_{jk}(q)] = e_{ik}(pq) ∈ e_{ik}(P)^{g+h} for i ≠ k,

we have pq ∈ P_{g+h}. Also, one can see that supp L ⊂ supp P + supp P (see Remark 2.6(ii)), and so ⟨supp P ⟩ ⊃ ⟨supp L⟩ = G, whence ⟨supp P ⟩ = G. Therefore, P is a G-graded algebra. Note that e_{ij}(P)^{g} = e_{ij}(P_{g}) for all ε_{i}−ε_{j} ∈ Δ and g ∈ G, and hence the G-grading is natural (see Remark 2.6(i)).

By (pd), for any ε_{i} − ε_{j} ∈ Δ and any g ∈ supp P, there exist e_{ij}(p) ∈ e_{ij}(P_{g}) and e_{ji}(q) ∈ e_{ji}(P_{−g}) such that

[e_{ij}(p), e_{ji}(q)] = [e_{ij}(1), e_{ji}(1)] + z for some z ∈ Z(sl_{l+1}(P)).

Hence e_{ii}(pq) − e_{jj}(qp) = e_{ii}(1) − e_{jj}(1) + ∑_{k=1}^{l+1} e_{kk}(a) for some a ∈ P, and so a = 0 and pq = qp = 1, i.e., p is invertible. Also, p is invertible in P ⇔ p is invertible in P^{+}. Therefore, P = ⊕_{g∈G} P_{g} is a predivision G-graded associative algebra. The statement for ‘division’ can be shown in the same manner.

(ii): Let Z_{g} := {z ∈ Z | 𝔤 ⊗ z ⊂ (𝔤 ⊗_{F} Z)^{g}}. Then Z = ⊕_{g∈G} Z_{g} becomes a G-graded algebra. The rest can be shown in the same manner as in (i). □

Definition 2.12. Let L̃ = ⊕_{g∈G} L̃^{g} and L = ⊕_{g∈G} L^{g} be (Δ, G)-graded Lie algebras and suppose that π : L̃ ⟶ L is a Δ-cover. If L^{g} = π(L̃^{g}) for all g ∈ G, then L̃ = (L̃, π) is called a (Δ, G)-cover of L. Also, for (Δ, G)-graded Lie algebras L and L′, if there exist a (Δ, G)-graded Lie algebra L̃ and maps π : L̃ ⟶ L and π′ : L̃ ⟶ L′ such that (L̃, π) and (L̃, π′) are both (Δ, G)-covers, we say that L and L′ are (Δ, G)-isogeneous.

It is clear using Lemma 2.3 that if L̃ is a (Δ, G)-cover of L, then

L̃ is is predivision (resp. division) ⟺ L is predivision (resp. division).

Also, by the proof of Corollary 2.4, if L and L′ are (Δ, G)-isogeneous, then L/Z(L) and L′/Z(L′) are (Δ, G)-isomorphic, i.e., there exists a Δ-isomorphism which is also G-graded between them. In particular, L̃/Z(L̃) and L/Z(L) above are (Δ, G)-isomorphic.

Proposition 2.13. (i) Let l ≥ 3. Then a predivision (resp. division) (A_{l}, G)-graded Lie algebra L over F is an (A_{l}, G)-cover of psl_{l+1}(P) admitting the natural compatible G-grading obtained from the G-grading of a predivision (resp. division) G-graded associative algebra P over F. Hence L/Z(L) and psl_{l+1}(P) are (Δ, G)-isomorphic.

(ii) Let Δ = D or E and let 𝔤 be a finite dimensional split simple Lie algebra L over F of type Δ. Then a predivision (resp. division) (Δ, G)-graded Lie algebra over F is a (Δ, G)- cover of 𝔤 ⊗_{F} Z admitting the natural compatible G-grading obtained from the G-grading of a predivision (resp. division) G-graded commutative associative algebra Z over F. Hence L/Z(L) and 𝔤 ⊗_{F} Z are (Δ, G)-isomorphic.

Proof. For (i), let L be a predivision (A_{l}, G)-graded Lie algebra over F. Berman and Moody showed in [3] that L is A_{l}-isogeneous to (sl_{l+1}(P), sl_{l+1}(F)) (the Steinberg Lie algebra st_{l+1}(P) serves as an A_{l}-cover of L and sl_{l+1}(P)). Hence, by Corollary 2.4, L/Z(L) is A_{l}-isomorphic to psl_{l+1}(P). Thus (psl_{l+1}(P), sl_{l+1}(F)) admits a compatible G-grading via the A_{l}-isomorphism from the compatible G-grading of L/Z(L) induced by the compatible G-grading of L. Therefore, the statement follows from Lemma 2.11.

(ii): Let L be a predivision (Δ, G)-graded Lie algebra over F. Berman and Moody showed in [3] that L is a Δ-cover of 𝔤 ⊗_{F} Z. Thus the statement follows from Lemma 2.11. □

In this paper we will classify predivision (Δ, ℤ^{n})-graded Lie algebras for Δ = A_{l}, l ≥ 3, D or E, up to central extensions. By Proposition 2.13, it remains to classify crossed product algebras R ∗ ℤ^{n}. We determine such algebras as a generalization of quantum tori. Namely, let q = (q_{ij}) be an n × n matrix over F such that

q_{ii} = 1 and q_{ji} = q_{ij}^{−1}.

The quantum torus F_{q} = F_{q}[t_{1}^{±1}, …, t_{n}^{±1}] determined by q is defined as the associative algebra over F with 2n generators t_{1}^{±1}, …, t_{n}^{±1}, and relations

t_{i}t_{i}^{−1} = t_{i}^{−1}t_{i} = 1 and t_{j}t_{i} = q_{ij}t_{i}t_{j}

for all 1 ≤ i, j ≤ n. Quantum tori are characterized as predivision ℤ^{n}-graded associative algebras whose homogeneous spaces are all 1-dimensional (see [4]). Note that F_{q} is commutative ⟺ q = 1 whose entries are all 1, i.e., F_{1} = F [t_{1}^{±1}, …, t_{n}^{±1}] is the algebra of Laurent polynomials. Also, a quantum torus is a twisted group algebra F^{t}[ℤ^{n}].

§3 Classification of R ∗ ℤ^{n}

Throughout this section F is an arbitrary field and G is an arbitrary group. For a Ggraded algebra S = ⊕_{g∈G} S_{g} over F in general, we denote by GrAut_{F} (S) the group of graded automorphisms of S, i.e.,

GrAut_{F} (S) := {σ ∈ Aut_{F} (S) | σ(S_{g}) = S_{g} for all g ∈ G}.

Lemma 3.1. Let R∗G = (R, G, σ, τ) be a crossed product algebra over F and (R∗G)∗M = (R ∗ G, M, η, ξ) a crossed product algebra over F for a group M, an action η and a twisting ξ. Suppose that η(M) ⊂ GrAut_{F} (R ∗ G) and that ξ(m, l) ∈ U(R) for all m, l ∈ M. Then, (R ∗ G) ∗ M is a crossed product algebra R ∗ (G × M) = (R, (G × M), σ′, τ′) over F for some action σ′ and twisting τ′.

Proof. We have

(R ∗ G) ∗ M = ⊕_{m∈M} (R ∗ G)\overline{m} = ⊕_{m∈M} (⊕_{g∈G} R\overline{g})\overline{m} = ⊕_{(g,m)∈G×M} R\overline{g}\overline{m}

as free R-modules, where \overline{g}\overline{m} = \overline{g} \overline{m}. We define η_{m} = η(m) |_{R1} an F -automorphism of R for every m ∈ M. Also for h ∈ G, \overline{h} is a unit in R ∗ G (see 1.6). Since η_{m} is a graded automorphism of R ∗ G by our first assumption, η(m)(\overline{h}) = d_{m,h}\overline{h} for some d_{m,h} ∈ U(R). Therefore, for r\overline{g}\overline{m} ∈ R\overline{g}\overline{m} and s\overline{h}\overline{l} ∈ R\overline{h}\overline{l}, we have

(r\overline{g}\overline{m})(s\overline{h}\overline{l}) = r\overline{g}η(m)(s\overline{h})\overline{m}\overline{l}
= r\overline{g}η_{m}(s)η(m)(\overline{h})ξ(m, l)\overline{ml}
= r\overline{g}η_{m}(s)d_{m,h}\overline{h}ξ(m, l)\overline{ml}
= r\overline{g}η_{m}(s)d_{m,h}σ_{h}(ξ(m, l))\overline{h}\overline{ml} (by our second assumption)
= rσ_{g}η_{m}(s)σ_{g}(d_{m,h})σ_{gh}(ξ(m, l))\overline{g}\overline{h}\overline{ml}
= rσ_{g}η_{m}(s)σ_{g}(d_{m,h})σ_{gh}(ξ(m, l))τ(g, h)\overline{gh} \overline{ml}.

Thus we have the action

σ′ : G × M ⟶ Aut_{F} R by σ′_{(g,m)} = σ_{g}η_{m},

and the twisting τ′ : (G × M) × (G × M) ⟶ U(R) by

τ′((g, m), (h, l))= σ_{g}(d_{m,h})σ_{gh}(ξ(m, l))τ(g, h).

Since the crossed product algebra (R ∗ G) ∗ M is associative, we get

(R ∗ G) ∗ M = R ∗ (G × M) = (R, G × M, σ′, τ′). □

A triple (R, φ, q) where R is a unital associative algebra over F,

φ = (φ_{1}, …, φ_{n})

is an n-tuple of F -automorphisms φ_{i} of R, and q = (q_{ij}) is an n×n matrix over R satisfying (G1) q_{ii} = 1 for 1 ≤ i ≤ n and q_{ji}^{−1} = q_{ij} for 1 ≤ i < j ≤ n, φ_{j}φ_{i} = I(q_{ij})φ_{i}φ_{j} for 1 ≤ i < j ≤ n, (G2) φ_{k}(q_{ij}) = q_{jk}φ_{j}(q_{ik})q_{ij}φ_{i}(q_{kj})q_{ki} for 1 ≤ i < j < k ≤ n, (G3) is called a ℤ^{n}-grading triple over F, and a division ℤ^{n}-grading triple over F if R is a division algebra. It follows easily from (G1)-(G3) that

these equations hold for all i, j, k satisfying 1 ≤ i, j, k ≤ n.

For a ℤ^{n}-grading triple, we introduce several notations and prove some identities. Notations.

(N1)  ε_{i} = (0, …, 0, 1, 0, …, 0) ∈ ℤ^{n},

i.e., the i-th coordinate is 1 and the others are 0.

(N2)  q_{ij}^{(m)} := \{ \begin{matrix}q_{ij}φ_{i}(q_{ij})φ_{i}^{2}(q_{ij}) ⋯ φ_{i}^{m−1}(q_{ij}) = ∏_{l=0}^{m−1} φ_{i}^{l}(q_{ij}), & if m > 0 \\ 1, & if m = 0 \\ φ_{i}^{−1}(q_{ji})φ_{i}^{−2}(q_{ji}) ⋯ φ_{i}^{m}(q_{ji}) = ∏_{l=−1}^{m} φ_{i}^{l}(q_{ji}), & if m < 0, \end{matrix}
and q_{ij}^{−(m)} := (q_{ij}^{(m)})^{−1}.

For α = (α_{1}, …, α_{n}), β = (β_{1}, …, β_{n}) ∈ ℤ^{n} and k = 0, 1, 2, …, n,

(N3)  φ^{(α)_{k}} := \{ \begin{matrix}id, & if k = 0, 1 \\ φ_{1}^{α_{1}} ⋯ φ_{k−1}^{α_{k−1}}, & if k > 1, \end{matrix}
and φ^{α} := φ_{1}^{α_{1}} ⋯ φ_{n}^{α_{n}}.
(N4)  q_{ε_{1},α} := 1 and q_{ε_{j},α} := ∏_{i=1}^{j−1} φ^{(α)_{i}}(q_{ij}^{(α_{i})}) for j > 1.
(N5)  q_{ε_{j},α}^{(m)} := \{ \begin{matrix}∏_{l=m−1}^{0} φ_{j}^{l}(q_{ε_{j},α}), & if m > 0 \\ 1, & if m = 0 \\ ∏_{l=m}^{−1} φ_{j}^{l}(q_{ε_{j},α}^{−1}), & if m < 0. \end{matrix}
(N6)  q_{α,β} := ∏_{j=n}^{1} φ^{(α)_{j}}(q_{ε_{j},β}^{(α_{j})}).

Lemma 3.2. For m ∈ ℤ and α = (α_{1}, …, α_{n}) ∈ ℤ^{n}, we have

φ_{i}^{−m}(q_{ij}^{−(m)}) = q_{ij}^{(−m)}, (1) φ_{j}φ_{i}^{m} = I(q_{ij}^{(m)})φ_{i}^{m}φ_{j}, (2)

(3) φ_{j}φ^{(α)_{i}} = \{ \begin{matrix}I(∏_{l=1}^{i−1} φ^{(α)_{l}}(q_{lj}^{(α_{l})}))φ^{(α)_{i}}φ_{j} for j ≥ i, \\ I( ∏_{l=1}^{j−1} φ^{(α)_{l}}(q_{lj}^{(α_{l})}))φ^{(α+ε_{j})_{i}} for j < i, \end{matrix}

q_{ij}^{(m+1)} = q_{ij}φ_{i}(q_{ij}^{(m)}) and q_{ij}^{−(m+1)} = φ_{i}(q_{ij}^{−(m)})q_{ji}, (4) φ_{k}(q_{ij}^{(m)}) = q_{jk}φ_{j}(q_{ik}^{(m)})q_{ij}^{(m)}φ_{i}^{m}(q_{kj})q_{ik}^{−(m)}. (5)

Proof. For (1), we have from (N2),

q_{ij}^{−(m)} = \{ \begin{matrix}φ_{i}^{m−1}(q_{ji}) ⋯ φ_{i}(q_{ji})q_{ji} = ∏_{l=m−1}^{1} φ_{i}^{l}(q_{ji}), & if m > 0 \\ 1, & if m = 0 \\ φ_{i}^{m}(q_{ij}) ⋯ φ_{i}^{−2}(q_{ij})φ_{i}^{−1}(q_{ij}) = ∏_{l=m}^{−1} φ_{i}^{l}(q_{ij}), & if m < 0. \end{matrix}

So we get

φ_{i}^{−m}(q_{ij}^{−(m)}) = \{ \begin{matrix}φ_{i}^{−1}(q_{ji}) ⋯ φ_{i}^{−m}(q_{ji}) = ∏_{l=−1}^{−m} φ_{i}^{l}(q_{ji}), & if m > 0 \\ 1, & if m = 0 \\ q_{ij}φ_{i}(q_{ij}) ⋯ φ_{i}^{−m−1}(q_{ij}) = ∏_{l=1}^{−m−1} φ_{i}^{l}(q_{ij}), & if m < 0, \end{matrix}

which is exactly q_{ij}^{(−m)}.

For (2), the case m = 0 is clear. Assume that m > 0. Put q := q_{ij} for simplicity. Then we have

φ_{j}φ_{i}^{m} = φ_{j}φ_{i}^{m−1}φ_{i}
= I(q^{(m−1)})φ_{i}^{m−1}φ_{j}φ_{i} by induction on m
= I(q^{(m−1)})φ_{i}^{m−1}I(q)φ_{i}φ_{j} by (G2)
= I(q^{(m−1)})I(φ_{i}^{m−1}(q))φ_{i}^{m}φ_{j}
= I(q^{(m)})φ_{i}^{m}φ_{j}.

Also, (φ_{j}φ_{i}^{m})^{−1} = (I(q_{ij}^{(m)})φ_{i}^{m}φ_{j})^{−1} for m > 0, and so

φ_{i}^{−m}φ_{j}^{−1} = φ_{j}^{−1}φ_{i}^{−m}(I(q_{ij}^{−(m)}) = φ_{j}^{−1}I(φ_{i}^{−m}(q_{ij}^{−(m)}))φ_{i}^{−m} = φ_{j}^{−1}I(q_{ij}^{(−m)})φ_{i}^{−m},

by (1). Hence we get φ_{j}φ_{i}^{−m} = I(q_{ij}^{(−m)})φ_{i}^{−m}φ_{j}, and (2) holds for all m ∈ ℤ.

For (3), when j ≥ i, using (2), we have

φ_{j}φ_{i}^{(α)_{i}} = φ_{j}φ_{1}^{α_{1}} ⋯ φ_{i−1}^{α_{i−1}}

= I(q_{1j}^{(α_{1})})φ_{1}^{α_{1}}φ_{j}φ_{2}^{α_{2}} ⋯ φ_{i−1}^{α_{i−1}}

= I(q_{1j}^{(α_{1})})φ_{1}^{α_{1}}I(q_{2j}^{(α_{2})})φ_{2}^{α_{2}}φ_{j}φ_{3}^{α_{3}} ⋯ φ_{i−1}^{α_{i−1}}

· · · · · ·

= I(q_{1j}^{(α_{1})})φ_{1}^{α_{1}}I(q_{2j}^{(α_{2})})φ_{2}^{α_{2}}I(q_{3j}^{(α_{3})})φ_{3}^{α_{3}} ⋯ I(q_{i−1,j}^{(α_{i−1})})φ_{i−1}^{α_{i−1}}φ_{j}

= I(∏_{l=1}^{i−1} φ^{(α)_{l}}(q_{lj}^{(α_{l})}))φ^{(α)_{i}}φ_{j}. (Note φ^{(α)_{0}} = id when i = 1)

When j < i, we have

φ_{j}φ_{i}^{(α)_{i}} = φ_{j}φ_{1}^{α_{1}} ⋯ φ_{i−1}^{α_{i−1}}

= I(q_{1j}^{(α_{1})})φ_{1}^{α_{1}}φ_{j}φ_{2}^{α_{2}} ⋯ φ_{j}^{α_{j}} ⋯ φ_{i−1}^{α_{i−1}}

· · · · · ·

= I(q_{1j}^{(α_{1})})φ_{1}^{α_{1}} ⋯ I(q_{j−1,j}^{(α_{j−1})})φ_{j−1}^{α_{j−1}}I(q_{jj}^{(α_{j})})φ_{j}^{α_{j}}φ_{j} ⋯ φ_{i−1}^{α_{i−1}} = I(q_{1j}^{(α_{1})})φ_{1}^{α_{1}} ⋯ I(q_{j−1,j}^{(α_{j−1})})φ_{j−1}^{α_{j−1}}φ^{α_{j}+1} ⋯ φ_{i−1}^{α_{i−1}}

= I(∏_{l=1}^{j−1} φ^{(α)_{l}}(q_{lj}^{(α_{l})}))φ^{(α+ε_{j})_{i}}. (Note φ^{(α)_{0}} = id when j = 1)

For the first formula of (4), the case m = 0 is clear. We put q := q_{ij}, p := q^{−1} and φ := φ_{i} for simplicity. For m > 0, we have

q^{(m+1)} = qφ(q)φ^{2}(q) ⋯ φ^{m}(q)
= qφ(qφ(q) ⋯ φ^{m−1}(q))= qφ(q^{(m)}).

For m = −1, we have q^{(−1+1)} = 1, while qφ(q^{(−1)}) = qφφ^{−1}(p) = 1. For m < −1, we have

q^{(m+1)} = φ^{−1}(p)φ^{−2}(p) ⋯ φ^{m+1}(p)
= qpφ^{−1}(p)φ^{−2}(p) ⋯ φ^{m+1}(p)
= qφ(φ^{−1}(p)φ^{−2}(p) ⋯ φ^{m}(p))= qφ(q^{(m)}).

The second formula follows from the first since q_{ij}^{−(m+1)} = (q_{ij}^{(m+1)})^{−1}.

For (5), the case m = 0 is clear. Assume that m > 0. Then we have φ_{k}(q_{ij}^{(m)})

= φ_{k}(q_{ij})φ_{k}φ_{i}(q_{ij}^{(m−1)}) by (4) = φ_{k}(q_{ij})q_{ik}φ_{i}φ_{k}(q_{ij}^{(m−1)})q_{ki} by (G2) = q_{jk}φ_{j}(q_{ik})q_{ij}φ_{i}(q_{kj})q_{ki}q_{ik}φ_{i}(q_{jk}φ_{j}(q_{ik}^{(m−1)})q_{ij}^{(m−1)}φ_{i}^{m−1}(q_{kj})(q_{ik}^{−(m−1)}))q_{ki}

by (G3) and induction on m = q_{jk}φ_{j}(q_{ik})q_{ij}φ_{i}φ_{j}(q_{ik}^{(m−1)})φ_{i}(q_{ij}^{(m−1)})φ_{i}^{m}(q_{kj})φ_{i}(q_{ik}^{−(m−1)})q_{ki} = q_{jk}φ_{j}(q_{ik})q_{ij}q_{ji}φ_{j}φ_{i}(q_{ik}^{(m−1)})q_{ij}φ_{i}(q_{ij}^{(m−1)})φ_{i}^{m}(q_{kj})q_{ik}^{−(m)}

by (G2) and (3)

= q_{jk}φ_{j}(q_{ik})φ_{j}φ_{i}(q_{ik}^{(m−1)})q_{ij}^{(m)}φ_{i}^{m}(q_{kj})q_{ik}^{−(m)} by (4) = q_{jk}φ_{j}(q_{ik}^{(m)})q_{ij}^{(m)}φ_{i}^{m}(q_{kj})q_{ik}^{−(m)} by (4).

Also, one has (φ_{k}(q_{ij}^{(m)}))^{−1}=(q_{jk}φ_{j}(q_{ik}^{(m)})q_{ij}^{(m)}φ_{i}^{m}(q_{kj})q_{ik}^{−(m)})^{−1} for m > 0, and so φ_{k}(q_{ji}^{−(m)}) = q_{ik}^{(m)}φ_{i}^{m}(q_{jk})q_{ij}^{−(m)}φ_{j}(q_{ik}^{−(m)})q_{kj}. Applying φ_{i}^{−m} in both hands, we get

φ_{i}^{−m}φ_{k}(q_{ij}^{−(m)}) = φ_{i}^{−m}(q_{ik}^{(m)}φ_{i}^{m}(q_{jk})q_{ij}^{−(m)}φ_{j}(q_{ik}^{−(m)})q_{kj})

= φ_{i}^{−m}(q_{ik}^{(m)})q_{jk}q_{ij}^{(−m)}φ_{i}^{−m}φ_{j}(q_{ik}^{−(m)})φ_{i}^{−m}(q_{kj}) by (1).

Then, by (1) and (2), we have

I(q_{ik}^{−(−m)})φ_{k}(q_{ij}^{(−m)}) = q_{ik}^{−(−m)}q_{jk}q_{ij}^{(−m)}I(q_{ij}^{−(−m)})φ_{j}(q_{ik}^{(−m)})φ_{i}^{−m}(q_{kj}),

and we obtain

φ_{k}(q_{ij}^{(−m)}) = q_{jk}φ_{j}(q_{ik}^{(−m)})q_{ij}^{(−m)}φ_{i}^{−m}(q_{kj})q_{ik}^{−(−m)} for m > 0.

Hence, (5) holds for all m ∈ ℤ. □

Now we are ready to state our theorem.

Theorem 3.3. Let (R, φ, q) be a ℤ^{n}-grading triple and let R_{φ,q} := ⊕_{α∈ℤ^{n}} Rt_{α} be a free left R-module with basis {t_{α} | α ∈ ℤ^{n}}. Then there exists a unique associative multiplication on R_{φ,q} such that, for t_{i} := t_{ε_{i}}, i = 1, …, n, α = (α_{1}, …, α_{n}) and r ∈ R,

(3.4)  t_{α} = t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}}, t_{i}t_{i}^{−1} = t_{i}^{−1}t_{i} = 1, t_{i}r = φ_{i}(r)t_{i} and t_{j}t_{i} = q_{ij}t_{i}t_{j}.

Moreover, for rt_{α}, r′t_{β} ∈ R_{φ,q}, we have

rt_{α}r′t_{β} = rφ^{α}(r′)q_{α,β}t_{α+β},

where φ^{α} and q_{α,β} are defined in (N3) and (N6). In particular, R_{φ,q} is a crossed product algebra R ∗ ℤ^{n} with

(action) σ : ℤ^{n} ⟶ Aut_{F} (R) by σ(α) = φ^{α}
(twisting) τ : ℤ^{n} × ℤ^{n} ⟶ U(R) by τ(α, β) = q_{α,β}.

Conversely, for any crossed product algebra R ∗ ℤ^{n}, there exists a ℤ^{n}-grading triple (R, φ, q) such that R ∗ ℤ^{n} = R_{φ,q}.

Proof. We first consider a crossed product algebra R ∗ ℤ. Let t := \overline{1} ∈ R ∗ ℤ. Then, t^{m} is a unit in R\overline{m} for all m ∈ ℤ. Using the diagonal basis change, one can obtain an R-basis {t^{m} | m ∈ ℤ}. So we have t^{m}t^{l} = t^{m+l} for all m, l ∈ ℤ. Hence, R ∗ ℤ = Rℤ is a skew group algebra. Let ψ be the action of 1, i.e., t(r1) = ψ(r)t for r ∈ R. (Note that 1 = \overline{0}.) Then the action of m is ψ^{m}, i.e.,

t^{m}(r1) = ψ^{m}(r)t^{m}.

Conversely, it is clear that any F -automorphism ψ of R determines a skew group algebra Rℤ by the action m ↦ ψ^{m} (see Remark 1.3). We denote this Rℤ by R[t; ψ].

Let R^{(1)} := R[t_{1}; ψ_{1}] where ψ_{1} = φ_{1}. Let ψ_{2} be a graded F -automorphism ψ_{2} of R^{(1)} and R^{(2)} := R^{(1)}[t_{2}; ψ_{2}]. Then, by Lemma 3.1, we get R^{(2)} = (Rℤ)ℤ = R ∗ ℤ^{2}. Repeating this process n times, one can construct R ∗ ℤ^{n} inductively. Namely, for a crossed product algebra R^{(k−1)} = R ∗ ℤ^{k−1}, if we specify an F -graded automorphism ψ_{k} of R^{(k−1)}, then

R^{(k)} := R^{(k−1)}[t_{k}; ψ_{k}] = R ∗ ℤ^{k},

and we obtain R^{(n)} = R ∗ ℤ^{n}. Thus, our task is to specify ψ_{k} on R^{(k−1)} and to show that ψ_{k} is a graded F -automorphism where k ≥ 2. We note that

{t_{1}^{α_{1}} ⋯ t_{k−1}^{α_{k−1}} | (α_{1}, …, α_{k−1}) ∈ ℤ^{k−1}}

is a basis of the free R-module R^{(k−1)}. For convenience, we put

t^{(α)_{k}} = t_{1}^{α_{1}} ⋯ t_{k−1}^{α_{k−1}},

and define an F -linear transformation ψ_{k} on R^{(k−1)} by

ψ_{k}(rt^{(α)_{k}}) = φ_{k}(r)[ ∏_{i=1}^{k−1} φ^{(α)_{i}}(q_{ik}^{(α_{i})})]t^{(α)_{k}} for r ∈ R,

which is clearly graded. If ψ_{k}(rt^{(α)_{k}}) = 0, then φ_{k}(r) = 0, and hence r = 0, and so ψ_{k} is injective. Since

ψ_{k}(φ_{k}^{−1}(r[ ∏_{i=1}^{k−1} φ^{(α)_{i}}(q_{ik}^{(α_{i})})]^{−1})t^{(α)_{k}}) = rt^{(α)_{k}},

ψ_{k} is surjective. Therefore, ψ_{k} is an F -linear graded isomorphism on R^{(k−1)}. So it remains to prove that ψ_{k} is a homomorphism. For this purpose, we use a well-known fact.

3.5. Let A and B be unital associative algebras over F and f a F -linear map from A into B. Let {t_{i}}_{i∈I} be a generating set of the F -algebra A. Then, f is a homomorphism if and only if f(t_{i}y) = f(t_{i})f(y) for all i ∈ I and y ∈ A. Moreover, if {t_{i}^{±1}}_{i∈I} is a generating set of A, then f is a homomorphism if and only if f(t_{i}y) = f(t_{i})f(y) and f(t_{i}^{−1}) = f(t_{i})^{−1} for all i ∈ I and y ∈ A.

We have a generating set R ∪ {t_{1}^{±1}, …, t_{k−1}^{±1}} of R^{(k−1)} over F, and

ψ_{k}(t_{j}^{−1}) = q_{jk}^{(−1)}t_{j}^{−1} = φ_{j}^{−1}(q_{kj})t_{j}^{−1}
= (t_{j}φ_{j}^{−1}(q_{jk}))^{−1} = (q_{jk}t_{j})^{−1} = ψ_{k}(t_{j})^{−1}.

So, by 3.5, we only need to show that, for all r, r′ ∈ R and 1 ≤ j ≤ k − 1,

(A)  ψ_{k}(rr′t^{(α)_{k}}) = ψ_{k}(r)ψ_{k}(r′t^{(α)_{k}}),
(B)  ψ_{k}(t_{j}rt^{(α)_{k}}) = ψ_{k}(t_{j})ψ_{k}(rt^{(α)_{k}}).

For (A), we have

ψ_{k}(rr′t^{(α)_{k}}) = φ_{k}(rr′) ∏_{i=1}^{k−1} φ^{(α)_{i}}(q_{ik}^{(α_{i})})t^{(α)_{k}}

= φ_{k}(r)φ_{k}(r′) ∏_{i=1}^{k−1} φ^{(α)_{i}}(d_{ik}^{(α_{i})})t^{(α)_{k}}

= ψ_{k}(r)ψ_{k}(r′t^{(α)_{k}}).

For (B), we first note that there is the embedding of R^{(j)} into R^{(k−1)} for 1 ≤ j ≤ k − 1, and so

t_{j}t^{(α)_{j}} = ψ_{j}(t^{(α)_{j}})t_{j} = [ ∏_{i=1}^{j−1} φ^{(α)_{i}}(q_{ij}^{(α_{i})})]t^{(α)_{j}}t_{j}.

Thus we have

ψ_{k}(t_{j}rt^{(α)_{k}}) = ψ_{k}(φ_{j}(r)t_{j}t^{(α)_{k}})

= ψ_{k}(φ_{j}(r)ψ_{j}(t^{(α)_{j}})t_{j}^{α_{j}+1} ⋯ t_{k−1}^{α_{k−1}})

= ψ_{k}(φ_{j}(r)[ ∏_{i=1}^{j−1} φ^{(α)_{i}}(q_{ij}^{(α_{i})})]t^{(α+ε_{j})_{k}})

= φ_{k}φ_{j}(r)[ ∏_{i=1}^{j−1} φ_{k}φ^{(α)_{i}}(q_{ij}^{(α_{i})})][ ∏_{i=1}^{k−1} φ^{(α+ε_{j})_{i}}(q_{ik}^{(α_{i}+δ_{ij})})]t^{(α+ε_{j})_{k}}

: = ABCt^{(α+ε_{j})_{k}},

where A = φ_{k}φ_{j}(r), B = ∏_{i=1}^{j−1} φ_{k}φ^{(α)_{i}}(q_{ij}^{(α_{i})}) and C = ∏_{i=1}^{k−1} φ^{(α+ε_{j})_{i}}(q_{ik}^{(α_{i}+δ_{ij})}). First of all, we have

A = φ_{k}φ_{j}(r) = q_{jk}φ_{j}φ_{k}(r)q_{kj} by (G2).

Secondly, by Lemma 3.2(3) and (5), we have

φ_{k}φ^{(α)_{i}}(q_{ij}^{(α_{i})})

= [ ∏_{l=1}^{i−1} φ^{(α)_{l}}(q_{lk}^{(α_{l})})]φ^{(α)_{i}}φ_{k}(q_{ij}^{(α_{i})})[ ∏_{l=1}^{i−1} φ^{(α)_{l}}(d_{lk}^{(α_{l})})]^{−1}

= [ ∏_{l=1}^{i−1} φ^{(α)_{l}}(q_{lk}^{(α_{l})})]φ^{(α)_{i}}(q_{jk}φ_{j}(q_{ik}^{(α_{i})})q_{ij}^{(α_{i})}φ_{i}^{α_{i}}(q_{kj})q_{ik}^{−(α_{i})})[ ∏_{l=1}^{i−1} φ^{(α)_{l}}(q_{lk}^{(α_{l})})]^{−1}.

Note that

φ^{(α)_{i}}(q_{ki}^{−(α_{i})})[ ∏_{l=1}^{i−1} φ^{(α)_{l}}(q_{lk}^{(α_{l})})]^{−1} = [ ∏_{l=1}^{i} φ^{(α)_{l}}(q_{lk}^{(α_{l})})]^{−1}
and φ^{(α)_{i}}φ_{i}^{α_{i}}(q_{kj}) = φ^{(α)_{i+1}}(q_{kj}).

So we have

(φ_{k}φ^{(α)_{i}}(q_{ij}^{(α_{i})}))

= [ ∏_{l=1}^{i−1} φ^{(α)_{l}}(q_{lk}^{(α_{l})})]φ^{(α)_{i}}(q_{jk})φ^{(α)_{i}}(φ_{j}(q_{ik}^{(α_{i})})q_{ij}^{(α_{i})})φ^{(α)_{i+1}}(q_{kj})[ ∏_{l=1}^{i} φ^{(α)_{l}}(q_{lk}^{(α_{l})})]^{−1}.

Thus, after cancellations, we get

B = ∏_{i=1}^{j−1} φ_{k}φ^{(α)_{i}}(q_{ij}^{(α_{i})})

= q_{jk}[ ∏_{i=1}^{j−1} φ^{(α)_{i}}(φ_{j}(q_{ik}^{(α_{i})})q_{ij}^{(α_{i})})]φ^{(α)_{j}}(q_{kj})[ ∏_{i=1}^{j−1} φ^{(α)_{i}}(q_{ik}^{(α_{i})})]^{−1}.

Thirdly, we have

C = ∏_{i=1}^{k−1} φ^{(α+ε_{j})_{i}}(q_{ik}^{(α_{i}+δ_{ij})})
=[ ∏_{i=1}^{j−1} φ^{(α)_{i}}(q_{ik}^{(α_{i})})]φ^{(α)_{j}}(q_{jk}^{(α_{j}+1)}) ∏_{i=j+1}^{k−1} φ^{(α+ε_{j})_{i}}(q_{ik}^{(α_{i})})
=[ ∏_{i=1}^{j−1} φ^{(α)_{i}}(q_{ik}^{(α_{i})})]φ^{(α)_{j}}(q_{jk}φ_{j}(q_{jk}^{(α_{j})})) ∏_{i=j+1}^{k−1} φ^{(α+ε_{j})_{i}}(q_{ik}^{(α_{i})}),

by Lemma 3.2(4). Consequently, after cancellations and noting that q_{ii} = 1, we obtain

ψ_{k}(t_{j}rt^{(α)_{k}}) = ABCt^{(α+ε_{j})_{k}}

= q_{jk}φ_{j}φ_{k}(r)[ ∏_{i=1}^{j} φ^{(α)_{i}}(φ_{j}(q_{ik}^{(α_{i})})q_{ij}^{(α_{i})})][ ∏_{i=j+1}^{k−1} φ^{(α+ε_{j})_{i}}(q_{ik}^{(α_{i})})]t^{(α+ε_{j})_{k}}. (∗)

On the other hand, we have

ψ_{k}(t_{j})ψ_{k}(rt^{(α)_{k}}) = q_{jk}t_{j}φ_{k}(r)[ ∏_{i=1}^{k−1} φ^{(α)_{i}}(q_{ik}^{(α_{i})})]t^{(α)_{k}}

= q_{jk}φ_{j}[φ_{k}(r) ∏_{i=1}^{k−1} φ^{(α)_{i}}(q_{ik}^{(α_{i})})]t_{j}t^{(α)_{k}}

= q_{jk}φ_{j}φ_{k}(r)[ ∏_{i=1}^{k−1} φ_{j}φ^{(α)_{i}}(q_{ik}^{(α_{i})})][ ∏_{l=1}^{j−1} φ^{(α)_{l}}(q_{lj}^{(α_{l})})]t^{(α+ε_{j})_{k}}.

We rewrite D := ∏_{i=1}^{k−1} φ_{j}φ^{(α)_{i}}(q_{ik}^{(α_{i})}). To find an expression for D, we use the following lemma:

Lemma 3.6. Let A be a unital associative algebra, a_{0} = 1, a_{1}, …, a_{k} ∈ A units and b_{1}, …, b_{k} ∈ A. Then we have

(1)  ∏_{i=1}^{k} (I( ∏_{l=1}^{i−1} a_{l})(b_{i})) = ( ∏_{i=1}^{k} b_{i}a_{i})b_{k}( ∏_{l=1}^{k−1} a_{l})^{−1}.
(2)  ∏_{i=j+1}^{k} (I( ∏_{l=1}^{j−1} a_{l})(b_{i})) = I( ∏_{l=1}^{j−1} a_{l})( ∏_{i=j+1}^{k} b_{i}).

Proof. (1) is straightforward and (2) is obvious. □

By Lemma 3.2(3), we have, for i ≤ j,

φ_{j}φ^{(α)_{i}}(q_{ik}^{(α_{i})}) = I( ∏_{l=1}^{i−1} φ^{(α)_{l}}(q_{lj}^{(α_{l})}))(φ^{(α)_{i}}φ_{j}(q_{ik}^{(α_{i})})).

So, by Lemma 3.6(1), we get using q_{jj} = 1 that

∏_{i=1}^{j} φ_{j}φ^{(α)_{i}}(q_{ik}^{(α_{i})}) = [ ∏_{i=1}^{j} φ^{(α)_{i}}(φ_{j}(q_{ik}^{(α_{i})})(q_{ij}^{(α_{i})}))][ ∏_{l=1}^{j−1} φ^{(α)_{l}}(q_{lj}^{(α_{l})})]^{−1}.

By Lemma 3.2(3), we have, for j < i,

φ_{j}φ^{(α)_{i}}(q_{ik}^{(α_{i})}) = I( ∏_{l=1}^{j−1} φ^{(α)_{l}}(q_{lj}^{(α_{l})}))(φ^{(α+ε_{j})_{i}}(q_{ik}^{(α_{i})})).

So, by Lemma 3.6(2), we get

∏_{i=j+1}^{k−1} φ_{j}φ^{(α)_{i}}(q_{ik}^{(α_{i})}) = I( ∏_{l=1}^{j−1} φ^{(α)_{l}}(q_{lj}^{(α_{l})}))( ∏_{i=j+1}^{k−1} φ^{(α+ε_{j})_{i}}(q_{ik}^{(α_{i})})).

Hence we get

D = ∏_{i=1}^{k−1} φ_{j}φ^{(α)_{i}}(q_{ik}^{(α_{i})})

= [ ∏_{i=1}^{j} φ^{(α)_{i}}(φ_{j}(q_{ik}^{(α_{i})})q_{ij}^{(α_{i})})][ ∏_{i=j+1}^{k−1} φ^{(α+ε_{j})_{i}}(q_{ik}^{(α_{i})})][ ∏_{l=1}^{j−1} φ^{(α)_{l}}(q_{lj}^{(α_{l})})]^{−1}.

Consequently, we obtain

ψ_{k}(t_{j})ψ_{k}(rt^{(α)_{k}})

= q_{jk}φ_{j}φ_{k}(r)[ ∏_{i=1}^{j} φ^{(α)_{i}}(φ_{j}(q_{ik}^{(α_{i})})q_{ij}^{(α_{i})})][ ∏_{i=j+1}^{k−1} φ^{(α+ε_{j})_{i}}(q_{ik}^{(α_{i})})]t^{(α+ε_{j})_{k}},

which is exactly (∗). Hence we have shown (B) and constructed a crossed product algebra R ∗ ℤ^{k} = R^{(k)} for k = 1, …, n from (R, φ, q).

Let us put R_{φ,q} := R^{(n)} = ⊕_{α∈ℤ^{n}} Rt_{α} where α = (α_{1}, …, α_{n}) ∈ ℤ^{n} and t_{α} = t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}}. Since ψ_{k} |_{R}= φ_{k} for k = 1, …, n, we have t_{i}r = φ_{i}(r)t_{i}. Also, we have t_{j}t_{i} = ψ_{j}(t_{i})t_{j} = q_{ij}t_{i}t_{j} for 1 ≤ i < j ≤ n, and so t_{j}t_{i} = q_{ij}t_{i}t_{j} for all 1 ≤ i, j ≤ n. Hence, our R_{φ,q} satisfies (3.4). The uniqueness of the multiplication on R_{φ,q} is clear since R ∪ {t_{1}^{±1}, …, t_{n}^{±1}} is a generating set of R_{φ,q}.

Now, one can easily check that ψ_{j}^{α_{j}}(t^{(β)_{j}}) = q_{ε_{j},β}^{(α_{j})}t^{(β)_{j}}. So for rt_{α}, r′t_{β} ∈ R_{φ,q}, we get

rt_{α}r′t_{β} = rφ^{α}(r′)t_{α}t_{β}
= rφ^{α}(r′)t^{(α)_{n}}t_{n}^{α_{n}}t^{(β)_{n}}t_{n}^{β_{n}}
= rφ^{α}(r′)t^{(α)_{n}}ψ_{n}^{α_{n}}(t^{(β)_{n}})t_{n}^{α_{n}+β_{n}}
= rφ^{α}(r′)t^{(α)_{n}}q_{ε_{n},β}^{(α_{n})}t^{(β)_{n}}t_{n}^{α_{n}+β_{n}}
= rφ^{α}(r′)φ^{(α)_{n}}(q_{ε_{n},β}^{(α_{n})})t^{(α)_{n}}t^{(β)_{n}}t_{n}^{α_{n}+β_{n}}
· · · · · ·
= rφ^{α}(r′)φ^{(α)_{n}}(q_{ε_{n},β}^{(α_{n})}) ⋯ φ^{(α)_{2}}(q_{ε_{2},β}^{(α_{2})})t_{1}^{α_{1}+β_{1}} ⋯ t_{n}^{α_{n}+β_{n}}
= rφ^{α}(r′)q_{α,β}t_{α+β}.

Conversely, for any crossed product algebra R ∗ ℤ^{n} = (R, ℤ^{n}, τ, σ) = ⊕_{α∈ℤ^{n}} R\overline{α}, we take a new R-basis \{t_{α} | α ∈ ℤ^{n}\} of R ∗ ℤ^{n} where t_{α} = \overline{ε}_{1}^{α_{1}} ⋯ \overline{ε}_{n}^{α_{n}}. We set q_{ij} := τ(ε_{j}, ε_{i}) for 1 ≤ i ≤ j ≤ n, q_{ji} := q_{ij}^{−1} and φ_{i} := σ_{ε_{i}}. Note that τ(ε_{i}, ε_{j}) = 1 for i ≤ j. Then one can check that the triple (R, φ, q) is a ℤ^{n}-grading triple:

(G1) is clear. Let t_{i} := \overline{ε}_{i} for i = 1, …, n. Then, for i ≤ j and r ∈ R, we have t_{j}t_{i}r = φ_{j}φ_{i}(r)t_{j}t_{i} = φ_{j}φ_{i}(r)q_{ij}t_{i}t_{j} and t_{j}t_{i}r = q_{ij}t_{i}t_{j}r = q_{ij}φ_{i}φ_{j}(r)t_{i}t_{j}. Hence, φ_{j}φ_{i}(r)q_{ij} = q_{ij}φ_{i}φ_{j}(r), i.e., (G2) holds. For i ≤ j ≤ k, we have t_{k}t_{j}t_{i} = t_{k}q_{ij}t_{i}t_{j} = φ_{k}(q_{ij})q_{ik}t_{i}t_{k}t_{j} = φ_{k}(q_{ij})q_{ik}φ_{i}(q_{jk})t_{i}t_{j}t_{k} and t_{k}t_{j}t_{i} = q_{jk}t_{j}t_{k}t_{i} = q_{jk}φ_{j}(q_{ik})t_{j}t_{i}t_{k} = q_{jk}φ_{j}(q_{ik})q_{ij}t_{i}t_{j}t_{k}. Hence, φ_{k}(q_{ij})q_{ik}φ_{i}(q_{jk}) = q_{jk}φ_{j}(q_{ik})q_{ij}, i.e., (G3) holds.

Finally, it is clear that R∗ℤ^{n} = ⊕_{α∈ℤ^{n}} Rt_{α} satisfies (3.4). Therefore, we obtain R∗ℤ^{n} = R_{φ,q}. □

Thus the following is clear:

Corollary 3.7. Let (D, φ, q) be a division ℤ^{n}-grading triple. Then, D_{φ,q} is a division ℤ^{n}-graded algebra. Conversely, for any division ℤ^{n}-graded algebra A, there exists a division ℤ^{n}-grading triple (D, φ, q) such that A = D_{φ,q}.

Remark. What we have shown in Theorem 3.3 can be written in the following way: Let B := {ε_{1}, …, ε_{n}} and C := {(ε_{j}, ε_{i}) | 1 ≤ i < j ≤ n}. Suppose that maps

σ : B ⟶ Aut_{F} (R) and τ : C ⟶ U(R)

satisfy

σ_{ε_{j}}σ_{ε_{i}} = I(τ(ε_{j}, ε_{i}))σ_{ε_{i}}σ_{ε_{j}} and (a) σ_{ε_{k}}(τ(ε_{j}, ε_{i}))τ(ε_{k}, ε_{i})σ_{ε_{i}}(τ(ε_{k}, ε_{j})) = τ(ε_{k}, ε_{j})σ_{ε_{j}}(τ(ε_{k}, ε_{i}))τ(ε_{j}, ε_{i}) (b)

for all 1 ≤ i < j < k ≤ n. Then there exist unique action σ̃ : ℤ^{n} ⟶ Aut_{F} (R) and twisting τ̃ : ℤ^{n} × ℤ^{n} ⟶ U(R) such that σ̃ |_{B}= σ, τ̃ |_{C}= τ and

(c)  τ̃(α_{1}ε_{1} + ⋯ + α_{i}ε_{i}, α_{j}ε_{j} + ⋯ + α_{n}ε_{n}) = 1 for all 1 ≤ i ≤ j ≤ n.

Conversely, for any crossed product algebra R ∗ ℤ^{n}, we can use the diagonal basis change so that the action and twisting satisfy (a), (b) and (c).

In a certain situation, the condition (G3) for a ℤ^{n}-grading triple is not needed.

Lemma 3.8. Let R be a unital associative algebra over F, φ =(I(d_{1}), …, I(d_{n})) an ntuple of inner automorphisms φ_{i} of R for some d_{1}, …, d_{n} ∈ U(R) and q = (q_{ij}) an n × n matrix over F. Suppose that a triple (R, φ, q) satisfies (G1) and (G2). Then, (R, φ, q) is a ℤ^{n}-grading triple.

Proof. We only need to check (G3). By (G1) and (G2), we have, for all 1 ≤ i, j ≤ n, I(d_{j})I(d_{i}) = I(q_{ij})I(d_{i})I(d_{j}). So for all r ∈ R, d_{j}d_{i}rd_{i}^{−1}d_{j}^{−1} = q_{ij}d_{i}d_{j}rd_{j}^{−1}d_{i}^{−1}q_{ji} and hence rd_{i}^{−1}d_{j}^{−1}q_{ij}d_{i}d_{j} = d_{i}^{−1}d_{j}^{−1}q_{ij}d_{i}d_{j}r, i.e., d_{i}^{−1}d_{j}^{−1}q_{ij}d_{i}d_{j} =: c_{ij} is in the centre of R. Note that c_{ji}^{−1} = c_{ij}. Thus we have

q_{ij} = c_{ij}[d_{j}, d_{i}]

for all i, j, where [d_{j}, d_{i}] = d_{j}^{−1}d_{i}^{−1}d_{j}d_{i}. Using this identity, we get (G3): for all 1 ≤ i < j < k ≤ n,

q_{jk}φ_{j}(q_{ik})q_{ij}φ_{i}(q_{kj})q_{ki}
=c_{jk}[d_{k}, d_{j}]d_{j}c_{ik}[d_{k}, d_{i}]d_{j}^{−1}c_{ij}[d_{j}, d_{i}]d_{i}c_{kj}[d_{j}, d_{k}]d_{i}^{−1}c_{ki}[d_{i}, d_{k}]
=d_{k}c_{ij}[d_{j}, d_{i}]d_{k}^{−1} = φ_{k}(q_{ij}). □

By this lemma, if R is a finite dimensional central simple associative algebra, the defining identities of a ℤ^{n}-grading triple are just (G1) and (G2).

Remark 3.9. (1) For a ℤ^{n}-grading triple (R, φ, q), if φ = 1 := (id, …, id), then the crossed product algebra R_{1,q} has the trivial action by Theorem 3.3. So, R_{1,q} = R^{t}[ℤ^{n}] is a twisted group algebra.

(2) For a ℤ^{n}-grading triple (R, φ, q), if q = 1_{n} = 1 := ( \begin{matrix}1 & ⋯ & 1 \\ ⋮ & ... & ⋮ \\ 1 & ⋯ & 1 \end{matrix} ), then a crossed

product algebra R_{φ,1} has the trivial twisting by Theorem 3.3. So, R_{φ,1} = Rℤ^{n} is a skew group algebra.

(3) By (G2), (R, φ, 1) is a ℤ^{n}-grading triple if and only if

(∗)  φ_{j}φ_{i} = φ_{i}φ_{j} for all i, j.

Finally, we give some examples.

Example. (1) Let F_{q} be an arbitrary quantum torus and R an arbitrary associative algebra. Then it is easy to see that R ⊗_{F} F_{q} is a predivision ℤ^{n}-graded associative algebra (division ℤ^{n}-graded if R is a division algebra) and is isomorphic to R_{1,q}. Note also if R is a field, then this example becomes a quantum torus over R. Conversely, for a division ℤ^{n}-grading triple (D, φ, q), if φ = 1, then I(q_{ij}) = id for all q_{ij}, by (G2). Hence q_{ij} is in the centre of D, say K, and we can show that D_{1,q} ≅ D ⊗_{K} K_{q}. Therefore, D_{φ,q} is a tensor product with D and some quantum torus if and only if φ = 1.

(2) Let Q = ⟨i, j⟩ be a quaternion algebra over a field, where i and j are the standard generators, φ = φ_{3} = (I(i), I(j), I(ij)) and 1 = 1_{3}. Then one can easily check (∗) in Remark 3.9(3), and hence Q_{φ,1} is a predivision ℤ^{3}-graded associative algebra.

(3) Let K = ℚ(ζ_{5}) be a cyclotomic extension of ℚ (the field of rational numbers) where ζ := ζ_{5} is a primitive 5th root of unity, and φ the automorphism of K defined by φ(ζ) = ζ^{2}. Let φ = (φ, φ^{2}, φ^{3}) and

q = ( \begin{matrix}1 & ζ & ζ^{2} \\ ζ^{−1} & 1 & ζ^{−1} \\ ζ^{3} & ζ & 1 \end{matrix} ) .

Then one can easily check that (K, φ, q) is a division ℤ^{3}-grading triple, and hence K_{φ,q} is a division ℤ^{3}-graded associative algebra over ℚ.

(4) Let ℍ = ⟨i, j⟩ be Hamilton’s quaternion over ℝ (the field of real numbers), i.e., the unique quaternion division algebra over ℝ. Put k := ij. Let φ = (I(d_{1}), I(d_{2}), I(d_{3})) where d_{1} = 1 + i, d_{2} = 1 + j and d_{3} = 1 + k. We put q_{ij} = 2[d_{j}, d_{i}] for 1 ≤ i < j ≤ 3, q_{ji} = q_{ij}^{−1} and q_{ii} = 1. Then, (ℍ, φ, q) satisfies (G1) and (G2), and

q = ( \begin{matrix}1 & 1 − i + j − k & 1 − i + j + k \\ (1 − i + j − k)^{−1} & 1 & 1 − i − j + k \\ (1 − i + j + k)^{−1} & (1 − i − j + k)^{−1} & 1 \end{matrix} ) .

By Lemma 3.8, this is a division ℤ^{3}-grading triple and hence ℍ_{φ,q} is a division ℤ^{3}-graded associative algebra over ℝ.

§4 Conclusion

By 1.8, Example 2.8(c), Example 2.10, Proposition 2.13, Theorem 3.3 and Corollary 3.7, one can summarize our results as follows:

Corollary. (i) Any predivision (resp. division) (A_{l}, ℤ^{n})-graded Lie algebra over F for l ≥ 3 is an (A_{l}, ℤ^{n})-cover of psl_{l+1}(R_{φ,q}) for some (resp. division) ℤ^{n}-grading triple (R, φ, q) over F. Conversely, any psl_{l+1}(R_{φ,q}) for l ≥ 1 is a predivision (resp. division) (A_{l}, ℤ^{n})-graded Lie algebra over F.

(ii) Any predivision (resp. division) (Δ, ℤ^{n})-graded Lie algebra over F for Δ = D or E is a (Δ, ℤ^{n})-cover of 𝔤⊗_{F} K[z_{1}^{±}, …, z_{n}^{±}] where 𝔤 is a finite dimensional split simple Lie algebra over F of type D or E and K is a unital commutative associative algebra over F (resp. K is a field extension of F). Conversely, for any finite dimensional split simple Lie algebra 𝔤 over F of any type Δ, 𝔤 ⊗_{F} K[z_{1}^{±}, …, z_{n}^{±}] is a predivision (resp. division) (Δ, ℤ^{n})-graded Lie algebra over F. □

ACKNOWLEDGMENTS

The result in §3 is part of my Ph.D thesis, written at the University of Ottawa. I would like to thank my supervisor, Professor Erhard Neher, for his encouragement and suggestions.

REFERENCES

[1] B. Allison, S. Azam, S. Berman, Y. Gao and A. Pianzola, Extended affine Lie algebras and their root systems, Mem. Amer. Math. Soc. vol. 126, no. 603, Amer. Math. Soc., Providence, RI, 1997.

[2] B. Allison and Y. Gao, The root system and the core of an extended affine Lie algebra, (to appear).

[3] S. Berman and R.V. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math. 108 (1992), 323–347.

[4] S. Berman, Y. Gao and Y. Krylyuk, Quantum tori and the structure of elliptic quasisimple Lie algebras, J. Funct. Anal. 135 (1996), 339–389.

[5] G. Benkart and E. Zelmanov, Lie algebras graded by finite root systems and intersection matrix algebras, Invent. Math. 126 (1996), 1–45.

[6] R.V. Moody and A. Pianzola, Lie algebras with triangular decomposition, John Wiley, New York, 1995.

[7] E. Neher, Lie algebras graded by 3-graded root systems, Amer. J. Math. 118 (1996), 439–491.

[8] D. S. Passman, Infinite crossed products, Academic press, San Diego, 1989.

[9] Y. Yoshii, The coordinate algebra of extended affine Lie algebras of type A_{1}, (submitted).