Sbornik: 199:7 965–983 ⃝c 2008 RAS(DoM) and LMS Matematicheski˘ıSbornik 199:7 21–40 DOI 10.1070/SM2008v199n07ABEH003949

Finite-dimensional simple graded algebras

Yu. A. Bahturin, M. V. Zaicev and S. K. Sehgal

Abstract. Let R be a finite-dimensional algebra over an algebraically closed F graded by an arbitrary group G. In the paper it is proved that if the characteristic of F is zero or does not divide the order of any finite subgroup of G, then R is graded simple if and only if it is isomorphic to a matrix algebra over a finite-dimensional graded skew field. Bibliography: 24 titles.

§ 1. Introduction Graded rings and graded algebras were actively investigated during the last decades. One of the important directions of these investigations is the development of structure theory for graded rings and algebras and, in particular, the descrip- tion of radicals and of simple and semisimple objects. One of the first steps along the way is the description of all possible gradings on simple algebras. For exam- ple, all gradings by cyclic groups on complex simple finite-dimensional Lie algebras were described in [1] and [2]. In [3] all possible gradings on the Cayley-Dickson alge- bra were classified. In [4]–[7] the authors described all Abelian gradings on finite- dimensional classical simple Lie algebras and on special simple Jordan algebras. Several papers devoted to group gradings on simple associative algebras have been published during the last few years. The author of [8] showed that finite Z-gradings on simple associative algebras (not necessarily finite-dimensional) can be obtained from Peirce decompositions. In [9] all gradings of a matrix algebra by torsion-free groups were described. The last result was extended in [10] to the case of an arbitrary simple Artinian . In [11] all Abelian gradings on a matrix algebra over an algebraically closed field of characteristic zero were completely described. Later on, in [12], this result was generalized to the case of non-Abelian gradings on matrix algebras. As is well known, a graded simple algebra is not simple in the ordinary sense in the general case. For instance, the group algebra F [G] of any non-trivial group over a field is not simple, but it is graded simple in the canonical G-grading. Thus, the description of all possible gradings on simple algebras is only the first step in the classification of graded simple algebras.

The work of the first author was carried out with the partial support ofNSERC (grant no. 227060-04). The work of the second author was carried out with the partial support of RFBR (grant no. 06-01-00485) and the Programme for the Support of Leading Scientific Schools (grant no. НШ-5666.2006.1). The work of the third author was partially supported by NSERC (grant no. A-5300). AMS 2000 Mathematics Subject Classification. Primary 16W50; Secondary 12E15, 17A35. 966 Yu. A. Bahturin, M. V. Zaicev and S. K. Sehgal

One of the first results in this area was the classification of finite-dimensional associative simple superalgebras (that is, associative algebras with Z2-gradings) over an algebraically closed field (see [13]). Now this result is almost obvious, and any superalgebra of this kind is either a matrix algebra with Z2-grading or the of a matrix algebra and the group algebra of the group Z2. The recent progress in the description of all Abelian gradings on full matrix algebras made it possible to complete the description of all graded simple algebras. In the paper [11] cited above the authors classified the finite-dimensional graded simple algebras over an algebraically closed field of characteristic zero, provided that the grading group is Abelian. In fact, for a wide class of Abelian groups the result looks like that for the superalgebras, namely, a G-graded algebra A is graded simple if and only if A is the tensor product of a matrix algebra with some G-grading and the group algebra of some finite subgroup of G. Some partial results concerning graded simple algebras with non-Abelian torsion free grading were also obtained in [11]. In the present paper we study finite-dimensional graded simple algebras over an algebraically closed field F under some natural restrictions on the characteristic of F . In § 2 we recall all necessary definitions and concepts. In § 3 we prove that the graded simple algebras are unitary. The main result of § 4 states that each graded skew field is a twisted group algebra of some finite group G. In § 5 we prove the semiprimitivity of graded simple algebras under weak assumptions concerning char F . Theorem3, which is the main result of this paper, is presented in § 6 and asserts that any graded simple algebra is a matrix algebra over a graded skew field under the same conditions on the characteristic of the ground fieldasin § 5. A preliminary version of this paper in English has been put on an open Internet archive since 2005 (see [14]).

§ 2. Preliminary facts Here we recall the main definitions and constructions. Let R be an over a field F and let G be a multiplicative group with identity element e. In this case R is said to be a G-graded algebra (or simply a graded algebra if G is fixed) if there is a vector-space decomposition M R = Rg, g∈G where the subspaces Rg satisfy the conditions RgRh ⊆ Rgh for any g, h ∈ G. The subspaces Rg are referred to as homogeneous components of the grading. In particular, Re is called the identity component. Clearly, Re is a subalgebra of R. For each g ∈ G an element a ∈ Rg is said to be homogeneous of degree g, and we write deg a = g. By the definition of the grading, any element x ∈ R can be written uniquely in the form X x = xg, where xg ∈ Rg. g∈G A subspace V ⊆ R is said to be a graded subspace (or a homogeneous subspace) if M V = (V ∩ Rg). g∈G Finite-dimensional simple graded algebras 967

P Equivalently, a subspace V is graded if and only if for any x = g∈G xg ∈ V we have xg ∈ V for any g ∈ G. A subalgebra A is said to be graded if it is a graded subspace. A graded (left, right or two-sided) ideal is defined in a similar L way. Following this general approach we say that R = g∈G Rg is a graded simple algebra if R2 ̸= 0 and R has no non-trivial proper graded ideals. Similarly, R is a graded skew field if R is unitary, that is, it has a multiplicative identity element 1, and any non-zero homogeneous element of R is invertible. L L Let A = g∈G Ag and B = g∈G Bg be two G-graded algebras. A homomor- phism (isomorphism) of algebras, f : A → B, is called a homomorphism (iso- morphism) of graded algebras if f preserves the graded structure, that is, f(Ag)⊆Bg for any g ∈ G. For an arbitrary algebra R the simplest way to define a G-grading on R is to set Re = R and Rg = 0 for any g, e ̸= g ∈ G. This is the so-called trivial G-grading. L Thus, some homogeneous components Rg of a G-graded algebra R = g∈G Rg can be zero. We set Supp R = {g ∈ G | Rg ̸= 0} and refer to this set as the support of the grading. The support of a G-grading is not a subgroup of G in general. For example, if R2 = 0, then Supp R can be an arbitrary subset of the grading group. As we shall see below, Supp R is not necessarily a subgroup even if multiplication in R is non-trivial. As an illustration of the above definitions, we present several examples of graded algebras that are of importance for our work.

2.1. Elementary gradings on full matrix algebras. Let R = Mn(F ) be the n × n matrix algebra over F and let G be a group. Fix an arbitrary n-tuple n g = (g1, . . . , gn) ∈ G of elements of G. Then the n-tuple g defines a G-grading on R as follows. Let Eij, 1 6 i, j 6 n, be the set of all matrix units of the algebra R. Then we set −1 Rg = Span{Eij | gi gj = g}.

Immediate manipulations show that RgRh ⊆ Rgh for any g, h ∈ G, and therefore the decomposition M Mn(F ) = R = Rg g∈G is a G-grading, which is referred to as the elementary grading defined by the n-tuple g. ′ We note that if g = (ag1, . . . , agn) is another n-tuple with an arbitrary a ∈ G, −1 −1 then it defines the same G-grading because (agi) (agj) = gi gj. In particular, we can always assume that g1 is the identity element of G. For any permutation σ of the {1, . . . , n} we can consider the n-tuple gσ = (gσ(1), . . . , gσ(n)). In this case g and gσ define the structure of two G-graded algebras R(g) and R(gσ) on R = Mn(F ). However, these are isomorphic as G-graded algebras. This can be seen, for instance, as follows. Let V be an n-dimensional over F . We say that V is a G-graded L space if V = g∈G Vg. A linear transformation f : V → V is said to be homo- geneous of degree deg f = h if f(Vg) ⊆ Vhg for any g ∈ G. In this way the algebra End V of all linear transformations of V becomes a G-graded algebra, and any 968 Yu. A. Bahturin, M. V. Zaicev and S. K. Sehgal matrix algebra Mn(F ) with an elementary G-grading is isomorphic to the algebra End V for some G-graded vector space (see, for example, [12], Proposition 2.2). In fact, any isomorphism of this kind can be obtained by fixing a homogeneous basis of V . It is now clear that any permutation σ of the elements of a homogeneous basis of V leads to a graded isomorphism R(g) → R(gσ). In the present paper, keeping in mind this isomorphism, we identify the G-gradings on Mn(F ) defined by the n-tuples g = (g1, . . . , gn) and gσ = (gσ(1), . . . , gσ(n)). 2.2. The tensor product of graded algebras. Our second example enables us to construct new graded algebras using given ones. Let G and H be two groups, L L and let A = g∈G Ag and B = h∈H Bh be a G-graded and an H-graded algebra, respectively. In this case the tensor product C = A ⊗ B can be equipped with a G × H-grading in the following natural way. For any t = gh ∈ G × H we can set

Ct = Ag ⊗ Bh. Obviously, this is a well-defined G × H-grading on C. This approach can be L generalized as follows. Consider two G-graded algebras, A = Ag and B = L g∈G g∈G Bg, and let us write S = Supp A and T = Supp B. If the elements of the subsets S and T commute pairwise, then we can introduce a G-grading on C = A ⊗ B by setting X Cr = Ag ⊗ Bh gh=r for any r ∈ ST . In particular, this definition is consistent if G is an . We cannot extend this definition to the general non-Abelian case; however, there is a special case in which such an extension is possible, and this definition is of importance for our classification.

2.3. Induced gradings on a tensor product. Let A = Mn(F ) be a matrix algebra with an elementary G-grading defined by an n-tuple g = (g1, . . . , gn) and let L B = g∈G Bg be an arbitrary G-graded algebra. We recall the concept of induced grading on C = A ⊗ B introduced in [12]. We note that the same construction was studied in [15], § 1.5 for the case when B is a graded skew field. As in the commutative case, each homogeneous component of the algebra C is defined as the span of the tensor products of homogeneous elements. However, their degrees are defined in a different way, namely,

−1 Cg = Span{Eij ⊗ b | deg b = h, gi hgj = g}. L Immediate computations show that the decomposition C = g∈G Cg is a G-grading, and B is a graded subalgebra of C. Moreover, A is a graded subalgebra of C if B is an algebra with identity element. Obviously, A ⊗ B is isomorphic to the matrix algebra Mn(B) if we identify the product Eij ⊗ b with the matrix Eij(b) the only non-zero entry b ∈ B of which is located at the intersection of the ith row and the jth column. Moreover, this is a graded algebra isomorphism if we define a G-grading on Mn(B) by setting −1 deg Eij(b) = gi hgj = g for any homogeneous b ∈ Bh. Finite-dimensional simple graded algebras 969

Every grading on a matrix algebra makes of it a graded simple algebra. In the next subsection we present examples of graded algebras that are graded skew fields. 2.4. Twisted group algebras. Let R = F [G] be the group algebra of a group G over a field F , that is, R is the linear span of its basis elements rg, g ∈ G, with the product rgrh = rgh. In this case R is equipped with the canonical G-grading L R = g∈G Rg, where Rg = Span{rg} is a one-dimensional vector space and all homogeneous non-zero elements are invertible. Hence F [G] is a graded skew field. We assume now that the same vector space Span{rg | g ∈ G} is equipped with another product, rgrh = σ(g, h)rgh, (1) where σ(g, h) ∈ F ∗ is a non-zero scalar for any g, h ∈ G. This product is completely determined by the map σ : G × G → F ∗. For the product to be associative this map must satisfy the relation

σ(x, y)σ(xy, z) = σ(y, z)σ(x, yz) for any x, y, z ∈ G (2)

(for details, see, for instance, [16], Ch. 1). A map σ : G × G → F ∗ satisfying (2) is called a 2-cocycle on G with values in F ∗, and the associative algebra

σ F [G] = Span{rg | g ∈ G} with product (1) is referred to as the twisted group algebra defined by σ. For example, if σ ≡ 1, then F σ[G] is the ordinary group algebra. Twisted group algebras inherit some properties of ordinary group algebras. For σ example, if we set deg rg = g, then F [G] obviously becomes a G-graded algebra. This grading is said to be canonical. Another important property is Maschke’s theorem (see, for instance, [17], Theorem 4.4). Proposition. Let G be a finite group such that the order |G| is invertible in F . Then the algebra F σ[G] is semisimple for any 2-cocycle σ.

§ 3. Unitarity of graded simple algebras Throughout this paper G will be the grading group and e the identity element of this group. The main aim of this section is to prove that every graded simple algebra over an arbitrary field F has an identity element. As is known, every semisimple Artinian ring has an identity element (see [18], § 1.4). Following the lines of the proof of the theorem in [18], we prove that every graded simple finite-dimensional algebra is unitary. We also refer to homogeneous idempotents of a graded algebra as graded idempotents. L Lemma 1. Let R = g∈G Rg be a finite-dimensional graded simple algebra and let I ⊂ R be a minimal graded right ideal of R. Then I = aR for some graded idempotent a. Proof. We assume first that I2 = 0. Since (RI)2 = R(IR)I ⊂ RI2 = 0, the two-sided ideal RI of R is nilpotent. Obviously, RI is a graded ideal; thus, we have RI = 0 by the assumption of the lemma. However, this means that I is a non-trivial two-sided graded ideal of R: a contradiction. Hence I2 ̸= 0, and there 970 Yu. A. Bahturin, M. V. Zaicev and S. K. Sehgal exists a homogeneous element x ̸= 0 such that xI ̸= 0. Since the ideal I is minimal, it follows that xI = I. In particular, xa = x for some non-zero homogeneous element a ∈ I. Further, the right annihilator X of an element x in R is a graded right ideal and either I ∩ X = 0 or I ∩ X = I. Since xa = x ̸= 0, we have I ∩ X = 0. Therefore, it follows from xa2 = xa = x that x(a2 − a) = 0. Hence, a2 = a is a graded idempotent in I. Since aI is a non-zero graded right ideal contained in I, we conclude that aI = I. Lemma 2. Let I ⊂ R be an arbitrary non-zero graded right ideal of a graded simple finite-dimensional algebra R. Then I = bR for some graded idempotent b. Proof. Since dim R < ∞, the right ideal I contains some (non-zero) minimal ideal. In this case it contains some graded idempotent a by Lemma1. Further, if t ∈ I is a graded idempotent, then we denote by A(t) the right annihilator of the element t in I: A(t) = {x ∈ I | tx = 0}. We claim that there exists an idempotent b ∈ A such that A(b) = 0. To prove this fact, it is sufficient to show that if A(a) ̸= 0, then we can find another graded idempotent t ∈ I such that dim A(t) < dim A(a). Since a is homogeneous, the right annihilator T of the element a in R is a graded right ideal. Hence, A(a) = T ∩ I is also a graded right ideal. As above, applying Lemma1, we can find a homogeneous idempotent f ∈ A(a). In this case f 2 = f and af = 0. We write t = a + f − fa. Then t2 = (a + f − fa)(a + f − fa) = a2 + fa + f 2 − f 2a − fa2 = a + f − fa = t, that is, t is an idempotent. We observe that every graded idempotent in a graded algebra R belongs to the identity component Re. Hence a, f ∈ Re, and t is also a graded idempotent. Let us show now that A(t) is a proper subspace of A(a). We choose x ∈ A(t) first. In this case, 0 = tx = atx = a(a + f − fa)x = a2x = ax, that is, x annihilates a. On the other hand, af = 0; however, tf = (a + f − fa)f = f 2 = f ̸= 0. Hence, A(t) ̸= A(a). Continuing this process we obtain a graded idempotent b ∈ I such that A(b) = 0, that is, bx ̸= 0 for any 0 ̸= x ∈ I. Since b(x − bx) = 0 for any x ∈ I, it follows that x = bx; in particular, bI = I. We recall that b ∈ I and I is a graded right ideal. Hence bR ⊂ I, and we have the required equality I = bR, where b = b2 is a graded idempotent. We can now prove the existence of an identity element in any finite-dimensional graded simple algebra. Theorem 1. Let R be a finite-dimensional graded simple algebra over an arbitrary field. Then R is an algebra with identity element. Finite-dimensional simple graded algebras 971

Proof. By Lemma2 there is a graded idempotent a ∈ R such that R = aR. Obviously, ax = x for any x ∈ R. Further, let I = {x − xa | x ∈ R}. Since a ∈ Re, it follows that I is a graded subspace. It is clear that I is a left ideal and Ia = 0. Hence, IR = IaR = 0, and I is a graded two-sided ideal. By assumption, in this case either I = 0 or I = R. If I = R, then R2 = IR = 0: a contradiction. Hence, I = 0, that is, xa = x for any x ∈ R. Thus, xa = ax = x for any x ∈ R, that is, a = 1, and the proof of the theorem is complete. In what follows, we denote the identity element of the algebra R by 1.

§ 4. Graded skew fields In this section we describe the finite-dimensional graded skew fields over an arbitrary algebraically closed field. Obviously, some of these results are known, and we present them here with proofs for convenience of references. L Lemma 3. Let R = g∈G Rg be a finite-dimensional graded skew field over an algebraically closed field F . Then H = supp R is a subgroup of G and dim Rh = 1 for any h ∈ H. Proof. The assertion concerning the support is obvious because an invertible ele- ment cannot be a zero divisor. Hence, Rgh ⊃ RgRh ̸= 0 whenever Rg ̸= 0 and Rh ̸= 0, that is, H is a finite multiplicatively closed set in G. Further, let R be a graded skew field over F with dim R < ∞. Then Re is a finite-dimensional skew field over the algebraically closed field F . Hence, Re = F . In particular, dim Re = 1. Finally, suppose that g ̸= e and let x ∈ Rg be a non-zero element. Then x has −1 −1 −1 an inverse x . It is clear that x is homogeneous and x ∈ Rg−1 . We choose −1 arbitrary y ∈ Rg. Then x y ∈ Re and y = λx for λ ∈ F . Thus, dim Rg = 1 and the proof of the lemma is complete. We can now characterize the graded skew fields as twisted group rings. L Theorem 2. Let R = g∈G Rg be a finite-dimensional G-graded algebra over an algebraically closed field F . Then R is a graded skew field if and only if R is isomor- phic to a twisted group algebra F σ[H] with canonical H-grading, where H is a finite subgroup of G and σ : H × H → F ∗ is a 2-cocycle on H. Proof. Obviously, every twisted group algebra is a graded skew field. Conversely, let R be a graded skew field. Then Supp R is a finite subgroup of G by Lemma3, and we also have dim Rh = 1 for any h ∈ H. We fix an arbitrary non-zero element rh ∈ Rh. Then for arbitrary fixed g, h ∈ H we have

rgrh = σ(g, h)rgh with some non-zero scalar σ(g, h) ∈ F because RgRh ⊂ Rgh and dim Rgh = 1. Since R is an associative algebra, the scalars σ(g, h) must satisfy the relations

σ(x, y)σ(xy, z) = σ(y, z)σ(x, yz)

(see the relation (2)), that is, σ : H ×H → F ∗ is a 2-cocycle. Therefore, R =∼ F σ[H], which completes the proof of the theorem. 972 Yu. A. Bahturin, M. V. Zaicev and S. K. Sehgal

At the end of the present section we characterize the graded skew fields in the class of all graded simple algebras. L Lemma 4. Let C = g∈G Cg be a finite-dimensional graded simple algebra over an arbitrary field F . If dim Ce = 1, then C is a graded skew field.

Proof. By Theorem1, C is an algebra with identity element. It is clear that 1 ∈ Ce and Ce = F . Let x ∈ Cg, g ̸= e, be a non-zero homogeneous element. Since C is graded simple, it follows that CxC = C. In particular, there exist homogeneous elements a, b ∈ C such that 0 ̸= axb ∈ Ce. We may assume without loss of generality that axb = 1 and recall that the equality uv = 1 yields vu = 1 for an arbitrary finite-dimensional algebra with identity element over the field F . Hence, bax = xba = 1, that is, ba = x−1, which completes the proof. If F is algebraically closed, then, by Theorem2, a finite-dimensional graded simple algebra C is a graded skew field if and only if dim Ce = 1.

§ 5. Semiprimitivity of graded skew fields In this section we prove that the graded simple algebras are semiprimitive when regarded as ungraded algebras. We study first some subalgebras of algebras of this kind. We begin with an observation concerning the identity component of a graded simple algebra. L Lemma 5. Let R = g∈G Rg be a finite-dimensional G-graded simple algebra over an arbitrary field. If R is graded simple, then Re is semiprimitive. L Proof. As proved in [10], for any G- R = g∈G Rg with |Supp R| < ∞ the identity component Re contains no nilpotent ideals if R does not itself have non-zero nilpotent ideals (see [10], Lemma 3). In fact, it was proved that if I is a nilpotent ideal of the algebra Re, then RIR is a two-sided nilpotent ideal of the algebra R. Obviously, RIR is a graded ideal, and therefore the condition that R is graded simple yields I = 0, which means that Re is semiprimitive. By the previous lemma every graded simple algebra R contains at least one graded idempotent. We use these idempotents to study the structure of the alge- bra R and its subalgebras. L Lemma 6. Let R = g∈G Rg be a finite-dimensional graded simple algebra and 2 t = t ∈ Re a graded idempotent. Then tRt is a graded simple algebra. Proof. We write A = tRt. By Theorem1, R is an algebra with identity element. The assertion of the lemma is obvious if t = 1. Suppose that t ̸= 1. Since t ∈ Re, it is clear that A is a graded subalgebra of R. Since t ∈ A, it follows that A is non-zero and a2 ̸= 0. It remains to prove that A is graded simple. Let I ⊂ A be a graded ideal of A. The ideal I generates a two-sided ideal T = RIR of the algebra R. We claim first that T ∩ A = I. Indeed, if y ∈ T ∩ A, then X y = rixisi ∈ A, i Finite-dimensional simple graded algebras 973 where xi ∈ I and ri, si ∈ R for all possible values of the index i. Since a = tat for any a ∈ I and the elements trit and tsit belong to A, whereas I is an ideal of A, it follows that X y = tyt = (trit)xi(tsit) ∈ I. Therefore, T ∩ A ⊂ I. The reverse inclusion is obvious. Thus, every proper graded ideal of A generates a proper graded ideal of the algebra R. It remains to observe that the algebra R is simple, which completes the proof.

We recall that, by Lemma5, the identity component Re of any graded simple algebra R is semiprimitive, and so is a sum of simple ideals. To prove that R is itself semiprimitive we consider the case of simple Re first. L Lemma 7. Let R = g∈G Rg be a finite-dimensional graded simple algebra over an algebraically closed field F such that A = Re is a simple algebra. Then R = ∼ ∼ L AC = A ⊗ C, where A = Mk(F ) with the trivial G-grading, C = g∈G Cg is the centralizer for A in R, and C is a graded skew field. Proof. By Theorem1, R is unitary. Since F is algebraically closed, A is isomorphic to some Mk(F ) and 1 ∈ A. We denote by C the centralizer of A in R. Since A = Re, it follows that C is a graded subalgebra. Moreover, A ∩ C = F . By [19], Lemma 3.11 or [20], Ch. 4, § 3 we obtain R = AC =∼ A ⊗ C. Since A ∩ C = F , we also have dim Ce = 1. Then C is a graded skew field by Lemma4. We have now all the ingredients required for the proof that the graded simple algebras are semiprimitive. We point out that if G is finite and |G| is invertible in F , then the Jacobson radical of each G-graded F -algebra with identity element is a graded ideal (see [21], Theorem 4.4). In particular, these graded simple algebras are semiprimitive. We must extend this result to the case of an arbitrary group G. L Lemma 8. Let R = g∈G Rg be a finite-dimensional graded simple algebra over an algebraically closed field F such that either char F = 0 or char F is coprime with the order of any finite subgroup of G. Then R is semiprimitive.

(1) (m) Proof. By Lemma5, Re is semiprimitive. In this case Re = A ⊕ · · · ⊕ A , (i) where the A are matrix algebras. Let e1, . . . , em be the identity elements of the algebras A(1),...,A(m), respectively. By Theorem1, R is an algebra with identity element, so that e1 + ··· + em = 1. It is clear that e1, . . . , em is a full system of orthogonal idempotents, and therefore M R = eiRej. 16i,j6m

First, we fix one of the components B = eiRei. By Lemma6, B is graded ∼ simple, and therefore, by Lemma7, B = Mk(F ) ⊗ C with some graded skew field C. By Theorem2, C is isomorphic to the twisted group algebra F σ[H] for some finite subgroup H ⊂ G and some 2-cocycle σ : H × H → F ∗. As is known, F σ[H] is semiprimitive if |H|−1 ∈ F (see, for instance, [17], Theorem 4.2). Since ∼ σ B = Mn(F [H]), it follows that the Jacobson radical of the algebra B is equal 974 Yu. A. Bahturin, M. V. Zaicev and S. K. Sehgal

σ to Mn(J), where J is the radical of the algebra F [H] (see [18]). Hence B is semiprimitive, too. We have thus proved that all components eiRei are semiprimitive subalgebras. Let us now denote by J =J(R) the Jacobson radical of R. Then J ∩eiRei =eiJei =0 L for any i = 1, . . . , m. Assume now that J is non-zero. Since J = i,j eiJej, we have ekJel ̸= 0 for some k ̸= l. Fix some x, 0 ̸= x ∈ ekJel. Since X x = ekxel = ekxgel g∈G and since ekxgel is homogeneous with respect to the G-grading with deg(ekxgel)=g, we see that xg = ekxgel, that is, all the homogeneous components of the element x belong to ekJel. In particular, eixgej = 0 if either i ̸= k or j ̸= l. Further, consider the set T = eiRekxelRei. It is clear that T ⊂ eiRei ∩ J, and therefore T = 0. However, in this case eiRekxgelRei = 0 for any component xg. We finally obtain M M RxgR = eiRejxgesRet = eiRekxgelRej, i,j,s,t i̸=j that is, xg generates (in R) some proper graded ideal: a contradiction. Hence J = J(R) = 0, which completes the proof of the lemma.

§ 6. Description of graded simple algebras First of all, we require a technical remark from linear algebra.

Lemma 9. Let B =B1⊕· · ·⊕Bn be a direct sum of matrix algebras and let z1, z2 ∈B 1 n be two orthogonal idempotents. Consider the decomposition zi = zi + ··· + zi , j i = 1, 2, with zi ∈ Bj . Then n X i i dim z1Bz2 = (rank z1)(rank z2), i=1 j where rank zi is for the ordinary matrix rank in Bj . j j j Proof. It is clear that every zi is an idempotent of the algebra Bj, and z1z2 = j j z2z1 = 0 because z1 and z2 are orthogonal. In this case, up to conjugation by an automorphism of the algebra Bj, these elements have the following form: j j z1 = E11 + ··· + Emm, z2 = Em+1, m+1 + ··· + Em+k, m+k, j where the Eii are the diagonal matrix units of the algebra Bj, rank z1 = m, and j rank z2 = k. Then we obviously have j j j j dim z1Bjz2 = mk = (rank z1)(rank z2). Our lemma follows now from the equality 1 1 n n z1Bz2 = z1 B1z2 ⊕ · · · ⊕ z1 Bnz2 . We consider further a graded simple algebra with the identity component con- sisting of two simple summands. Finite-dimensional simple graded algebras 975

L Lemma 10. Let R = g∈G Rg be a finite-dimensional graded simple algebra over an algebraically closed field F such that either char F = 0 or char F is coprime with the order of each finite subgroup of G. Assume that the identity component of R is the sum of two simple summands: Re = A1 ⊕ A2. Let e1 ∈ A1 and e2 ∈ A2 be the identity elements of the algebras A1 and A2, respectively. Setting R1 = e1Re1 and R2 = e2Re2 consider the decompositions R1 = A1C1 and R2 = A2C2 defined in Lemma 6, where C1 and C2 are graded skew fields. Let d1 ∈ A1 and d2 ∈ A2 be a pair of minimal idempotents such that M = d1Rd2 ̸= 0. Then

dim C1 = dim C2 = dim M,

M = C1x = xC2 for any homogeneous x, 0 ̸= x ∈ M.

Proof. By Lemma8 the algebra R is semiprimitive. Let R = B1 ⊕ · · · ⊕ Bn be a decomposition of the underlying ungraded algebra of R into a sum of simple ideals. In accordance with this decomposition, we can represent the identity element of the algebra A1 in the following form:

1 n i e1 = e1 + ··· + e1 , e1 ∈ Bi, i = 1, . . . , n.

In this case 1 1 n n e1Re1 = e1B1e1 ⊕ · · · ⊕ e1 Bne1 = A1C1. (3) 1 n i Since C1 commutes with A1, it follows that C1 = C1 ⊕· · ·⊕C1 , where C1 = C1 ∩Bi. We now fix an i, 1 6 i 6 n, and denote by ϕ the canonical projection of B onto Bi. i i i ∼ Let A1 = ϕ(A1). Since A1 is simple, it follows that either A1 = 0 or A1 = A1. i If A1 = 0, then

′ A1 ⊂ B1 ⊕ · · · ⊕ Bi−1 ⊕ Bi+1 ⊕ · · · ⊕ Bn = R and e1Re1 is a graded subspace of R lying in a proper ungraded ideal of R. Hence i ∼ e1Re1 generates a non-trivial graded ideal of R: a contradiction. Thus, A1 = A1 is i i a simple algebra and C1 commutes with A1. ′ i i i We write Bi = e1Bie1. Since e1 = ϕ(e1) is an idempotent, it follows that ′ i ′ ′ i i Bi is simple. Further, C1 ⊂ Bi. Then it follows from (3) that Bi = A1C1 and i i ′ i i ∼ C1 is a centralizer of A1 in Bi. Hence C1 is simple, C1 = Mpi (F ) for some pi > 1, and ′ ∼ i i Bi = A1 ⊗ C1. (4) 1 n 1 n Consider a decomposition of d1: d1 = d1 + ··· + d1 , where d1 ∈ B1, . . . , d1 ∈ Bn; i i then d1 = ϕ(d1) is a minimal idempotent of A1 and it follows from (4) that the i ′ ′ rank of the matrix d1 in Bi is equal to pi. We note that Bi ⊂ Bi are two matrix ′ i i i ′ algebras and Bi = e1Bie1, where e1 is an idempotent. Hence for any x ∈ Bi the ′ i rank of x in Bi coincides with the matrix rank of x in Bi. Thus, rank d1 = pi. Similarly,

1 n j ∼ C2 = C2 ⊕ · · · ⊕ C2 ,C2 = Mqj (F ), 1 6 j 6 n, 1 n j d2 = d2 + ··· + d2 , d2 ∈ Bj, 1 6 j 6 n, 976 Yu. A. Bahturin, M. V. Zaicev and S. K. Sehgal

i and rank d2 = qi. In this case, by Lemma9,

n X i i dim M = (rank d1)(rank d2) (5) i=1 and 2 2 2 2 dim C1 = p1 + ··· + pn, dim C2 = q1 + ··· + qn.

Since C1 is a graded simple skew field, it follows that the dimension of anarbi- trary non-trivial graded left C1-module is at least dim C1. Since d1 centralizes C1, it follows that M = d1Rd2 is a left C1-module. Hence dim M > dim C1. In a similar way dim M > dim C2. Comparing these inequalities with (5) we obtain

2 2 p1q1 + ··· + pnqn > p1 + ··· + pn, 2 2 p1q1 + ··· + pnqn > q1 + ··· + qn. Hence X X 2 2 X 2 2 piqi > (pi + qi ), (pi − qi) 6 0.

Thus, p1 = q1, ... , pn = qn, that is, dim C1 = dim C2 and it follows from (5) that dim M = dim C1 = dim C2. Finally, if we take an arbitrary homogeneous element x in M, x ̸= 0, then C1x is a non-zero graded subspace of M. Since C1 is a graded skew field, it follows that dim M = dim C1. Hence C1x = M. In a similar way xC2 = M and the proof is complete. We can now prove the main result of this paper. L Theorem 3. Let R = g∈G Rg be a finite-dimensional algebra, over an alge- braically closed field F that is graded by a group G. Suppose that either char F = 0 or char F is coprime with the order of each finite subgroup of G. Then R is a graded σ ∼ simple algebra if and only if R is isomorphic to the tensor product Mk(F )⊗F [H] = σ Mk(F [H]), that is, if and only if R is a matrix algebra over the graded skew field F σ[H], where H is a finite subgroup of G and σ : H × H → F ∗ is a 2-cocycle on H. The grading with respect to H on F σ[H] is canonical, and the G-grading on σ k −1 Mk(F [H]) is defined by a family (a1, . . . , ak) ∈ G so that deg(Eij ⊗xh) = ai haj σ for each matrix unit Eij and each homogeneous element xh ∈ F [H]h. σ Proof. The proof of the fact that Mk(F [H]) is a graded simple algebra is not complicated and is left to the reader. The main part is the proof of the fact that every graded simple algebra is a full matrix algebra over a graded skew field. Thus, suppose that R is graded simple. In this case R is semiprimitive by Lemma8, A = Re is semiprimitive by Lemma5, and we can write

A = A(1) ⊕ · · · ⊕ A(m)

(i) ∼ with A = Mni (F ), i = 1, . . . , m. We denote by e1, . . . , em the identity elements (1) (m) of the algebras A ,...,A , respectively. Then e1, . . . , em are orthogonal idem- potents and 1 = e1 + ··· + em. We find first a graded subalgebra of R isomorphic to Mm(F ) and equipped with an elementary G-grading. Finite-dimensional simple graded algebras 977

(i) (i) (i) We write R = eiRei. Then R is a graded subalgebra of R. Moreover, R is a graded simple algebra by Lemma6. By Lemma7, we have R(i) = A(i)C(i) =∼ A(i) ⊗ C(i), where C(i) is the centralizer of A(i) in R(i) and C(i) is a graded skew field. We show first that there exist homogeneous elements x12 ∈ e1Re2, . . . , xm−1, m ∈ em−1Rem, x21 ∈ e2Re1, . . . , xm, m−1 ∈ emRem−1 such that wk = ekxk, k−1ek−1 ··· e2x21e1x12e2 ··· xk−1, kek ̸= 0 (6) for any k = 2, . . . , m and deg wk = e with respect to the G-grading. Note that (k) Wk = ekR ··· Re2Re1Re2R ··· Rek = R ̸= 0 (7) for any k = 2, . . . , m. This is deduced by induction on k with an obvious base for (k−1) k = 1, where W1 = {e1}. Further, if Wk−1 = R ̸= 0, then RWk−1R is a graded (k) ideal in R, and therefore RWk−1R = R. Hence ekRWk−1Rek = ekRek = R ̸= 0 as required. We shall prove the assertion (6) by induction on k. If k = 2, then, by (7), there exist homogeneous elements x21 ∈ e2Re1, x12 ∈ e1Re2 such that w2 = (2) (2) (2) e2x21e1x12e2 ̸= 0. We set H = Supp C . By Lemma3, H is a subgroup (2) (2) (2) (2) (2) (2) of G. Moreover, Supp R = H because R = A C and A ⊂ Re. (2) Further, if deg w2 = h ̸= e, then there is an element b, 0 ̸= b ∈ C , with −1 ′ deg b = h , and we can replace x21 by the element x21 = bx21. Then the element ′ ′ w2 = bw2 = e2bx21e1x12e2 = e2x21e1x12e2 ∈ Re is non-zero because b is invertible. Similarly, if the elements wk−1 are already constructed, then there exist homoge- neous elements xk, k−1 and xk−1, k such that wk = ekxk, k−1wk−1xk−1, kek ̸= 0. The same considerations as above enable us to choose xk, k−1 in such a way that deg wk = e. In particular, we obtain

emxm, m−1em−1 ··· e2x21e1x12e2 ··· xm−1, mem ̸= 0 (8) and deg ekxk−1, k ··· xk−1, kek = e in G for k = 2, . . . , m. i (i) Further, we denote by Eα,β the matrix units of the algebra A . Then by (8), there are integers α1, . . . , αm, β1, . . . , βm such that

Em x Em−1 ··· E2 x E1 x E2 αm,αm m, m−1 αm−1,αm−1 α2,α2 21 α1,β1 12 β2,β2 ··· Em−1 x Em ̸= 0. (9) βm−1,βm−1 m−1, m βm,βm

In fact, we have here α1 = β1. Further, we define yi, i+1 and yi+1, i by setting

y = Ei Ei x Ei+1 Ei+1 , i, i+1 1,βi βi,βi i, i+1 βi+1,βi+1 βi+1,1 y = Ei+1 Ei+1 x Ei Ei i+1, i 1,αi+1 αi+1,αi+1 i+1, i αi,αi αi,1 for i = 1, . . . , m − 1. Then all the elements yi, i+1 and yi+1, i are homogeneous and

i i+1 i+1 i E11yi, i+1E11 = yi, i+1,E11 yi+1, iE11 = yi+1, i. (10) 978 Yu. A. Bahturin, M. V. Zaicev and S. K. Sehgal

Moreover, by (9) we obtain

ym, m−1 ··· y21y12 ··· ym−1, m ̸= 0. (11)

(2) Consider the element y22 = y21y12 ∈ A . It is non-zero by (11), and we have 2 2 (2) 2 −1 E11y22E11 = y22 in A . Hence y22 = λE11. If we replace y21 by λ y21, then (10) 2 and (11) still hold and y22 = E11. Further, assume that y11 = y12y21. In this case 3 2 1 y11 = y11 because y22 is an idempotent. As above, y11 = λE11, and thus either λ = 1 or λ = 0. Let us prove that y11 ̸= 0. Suppose that y11 = y12y21 = 0. The subalgebra (e1 + e2)R(e1 + e2) of R is 2 1 graded simple by Lemma6. Let d2 = E11 and d1 = E11. Then

(1) d2Rd1 = d2(e1 + e2)R(e1 + e2)d1 = y21C by Lemma 10, and therefore y12Rd1 = y12d2Rd1 = 0. If we expand e1 in a sum of minimal idempotents E1 ,...,E1 with d =E1 , then E1 ,...,E1 , e , . . . , e 11 n1,n1 1 11 11 n1,n1 2 m is a system of orthogonal idempotents with sum equal to 1. Hence,

Ry12R ∩ d1Rd1 = d1Ry12Rd1. (12)

On the other hand, the right-hand side of the equality (12) vanishes because y12Rd1 = 0. Hence Ry12R is a proper graded ideal of R. A contradiction. 1 We have thus proved that y11 = E11. Similarly, replacing y32, . . . , ym, m−1 by their scalar multiples we may assume that

i i+1 yi, i+1yi+1, i = E11, yi+1, iyi, i+1 = E11 (13) for i = 1, . . . , m − 1. Further, for 1 6 i < j 6 m we set

yij = yi, i+1 ··· yj−1, j, yji = yj, j−1 ··· yi+1, i.

In this case all the elements yij, 1 6 i, j 6 m, are non-zero by (11), and it readily follows from (10) that these elements are linearly independent. Using (13) and the definition of the elements yij, one can easily show that

yijyrt = δjryit, where δjr is the Kronecker delta. Hence, the linear span of the elements

{yij | 1 6 i, j 6 m} is a graded subalgebra of R isomorphic to Mm(F ). Consider the idempotent t = ei + ej for arbitrary i and j, i ̸= j. By Lemma6 ′ ′ (i) (j) the subalgebra R = tRt of R is graded simple and Re = A ⊕ A . Applying Lemma 10 to R′ we see that

(i) (j) C yij = yijC . (14)

Since all the homogeneous components of the algebras C(i) and C(j) are one- dimensional, formula (14) gives us a well-defined map ν : C(i) → C(j), which is an isomorphism of the underlying ungraded algebras. Moreover, for H(i) = Supp C(i) Finite-dimensional simple graded algebras 979 and H(j) = Supp C(j) we see that the subgroups H(i) and H(j) are conjugate in G, and ν becomes in fact a graded algebra isomorphism if we identify H(i) and (j) (1) (i) H by means of this conjugation. We denote by ϕi : C → C the corresponding (1) isomorphism for i = 2, . . . , m and set ϕ1 = idC(1) on C . Then

ϕ1(x)y1i = y1iϕi(x) for i = 1, . . . , m. Using this relation we shall prove that

ϕi(x)yij = yijϕj(x) (15) for any i and j. Suppose that i < j. Then xy1i = y1iϕi(x), and therefore xy1j = xy1iyij = y1iϕi(x)yij. On the other hand, xy1j = y1jϕj(x) = y1iyijϕj(x). Multiplying by (i) the element yi1 on the left and taking into account that yii ∈ A commutes (i) with ϕi(x) ∈ C , we obtain

yi1y1iϕi(x)yij = yiiϕi(x)yij = ϕi(x)yiiyij = ϕi(x)yij,

yi1y1iyijϕj(x) = yiiyijϕj(x) = yijϕj(x), that is, (15) holds for i < j. If i > j, then for any k > 2 we have

y11x = xy11 = xy1kyk1 = y1kϕk(x)yk1, and, multiplying by the element yk1 on the left we obtain

yk1x = ykkϕk(x) = ϕk(x)yk1.

The same arguments as above now give us formula (15) also for i > j. Now we extend the space Span{yij | 1 6 i, j 6 m} to a larger matrix subalgebra of R with an elementary G-grading and construct an ungraded subalgebra of C isomorphic to C(1) as ungraded algebras. For any x ∈ C(1) we write

x = x + ϕ2(x) + ··· + ϕm(x).

Since ϕi(x) ∈ eiRei and e1, . . . , em are orthogonal, it follows that

C = Span{x | x ∈ C(1)} is isomorphic to C(1). Moreover,

xzi = ϕi(x)zi, zix = ziϕi(x) (16)

(i) ∼ i for any zi ∈ eiRei. We recall that A = Mni and the elements Eα,β are the matrix units of the algebra A(i). It can readily be seen that the elements

i j Eα,1yijE1β, 1 6 α 6 ni, 1 6 β 6 nj, i, j = 1, . . . , m, (17) 980 Yu. A. Bahturin, M. V. Zaicev and S. K. Sehgal are linearly independent and homogeneous. We set k = n1 + ··· + nm and consider i j the ψ : B → Mk(F ) from B = Span{Eα,1yijE1β} onto Mk(F ) defined as follows on the basis elements:

i j ψ(Eα,1yijE1β) = Eµν (18) where µ = n1 + ··· + ni−1 + α and ν = n1 + ··· + nj−1 + β. Then ψ is an algebra isomorphism, that is, B is a graded subalgebra of R isomorphic to Mk(F ). On the other hand, it follows from (15) and (16) that

i j i j i j i j xEα,1yijE1β = ϕi(x)Eα,1yijE1β = Eα,1ϕi(x)yijE1β = Eα,1yijϕj(x)E1β i j i j = Eα,1yijE1βϕj(x) = Eα,1yijE1βx for any x ∈ C, and thus the set C centralizes B in R. In particular, C ∩ B = F , and therefore the product BC is isomorphic to B ⊗ C as ungraded algebras. We now prove that BC = R. We fix some 1 6 i, j 6 m and set

i (i) j Pijαβ = Span{Eα,1C yijE1β}.

(i) (j) We observe that Pijαβ = C Pijαβ = PijαβC . Hence X Qij = Pijαβ ⊂ eiRej 16α6ni 16β6nj

(i) (i) (j) (j) is a left A C -module and a right A C -module. We claim that Qij = eiRej. If this is not the case, then Qij generates a graded ideal of R such that

(i) (j) (i) (i) (j) (j) RQijR ∩ eiRej = (eiRei)Qij(ejRej) = R QijR = A C QijA C

= Qij ̸= eiRej.

A contradiction. Therefore, Qij = eiRej and BC = R. Using this equality, we extend the map (18) to a graded-algebra isomorphism (1) (1) ψ : R → Mk(C ), where C is a graded skew field. We describe a grading on B first. The matrix subalgebra of B that is the linear span of all elements yij = i j E11yijE11, 1 6 i, j 6 m, is graded, and all the matrix units yij are homogeneous. Hence, by [10], Lemma 1 the grading of this subalgebra is elementary, and there −1 are elements g1, . . . , gm such that deg yij = gi gj. Moreover, since the elements (ag1, . . . , agm) define the same grading for any a ∈ G, we may assume that g1 = e. i j −1 Then the G-grading on B is also elementary, and deg(Eα,1yijE1β) = gi gj for arbitrary α and β. In fact, since g1 = e, this G-grading on B is defined by a k-tuple

(a1, . . . , ak) = ( g1, . . . , g1, . . . , gm, . . . , gm ). | {z } | {z } n1 nm

−1 We can now define an elementary grading on Mk(F ) by setting deg Eµν = aµ aν . Then the map ψ : B → Mk(F ) defined by formula (18) is a graded algebra isomor- (1) (1) (1) phism. We now define a G-grading on Mk(C ). We set H = H = Supp C . Finite-dimensional simple graded algebras 981

(1) As we know, H is a subgroup of G. The underlying ungraded algebra of Mk(C ) (1) −1 is isomorphic to Mk(F ) ⊗ C . If we set deg(Eµν ⊗ yh) = aµ haν for each homo- (1) geneous yh ∈ Ch , then immediate manipulations show that this grading is well defined. As noted before the statement of the theorem, this is just theinduced (1) grading on the tensor product Mk(F ) ⊗ C . (1) In order to extend the isomorphism ψ defined in (18) to a map R=BC →Mk(C ) we set i j ψ(xhEα,1yijE1β) = Eµν ⊗ xh (19) (1) for each xh ∈ C with xh ∈ Ch , where µ = n1 + ··· + ni−1 + α and ν = n1 + ··· + nj−1 + β. It is clear that ψ is an isomorphism of the underlying ungraded algebras. We must prove that ψ preserves the grading. −1 The element on the right-hand side of (19) is homogeneous of degree aµ haν = −1 gi hgj. Although xh ∈ C is an ungraded element of R, the product

i j i j w = xhEα,1yijE1β = Eα,1ϕi(xh)yijE1β

−1 is homogeneous, and deg w = deg(ϕi(xh)) deg yij = deg(ϕi(xh))gi gj. To evaluate −1 this degree we recall that xhy1i = y1iϕi(xh) and deg y1i = g1 gi = gi because g1 = e. Hence

deg(xhy1i) = hgi = deg(y1iϕi(xh)) = gi deg ϕi(xh)

−1 and deg ϕi(xh) = gi hgi. Thus, −1 −1 −1 deg w = gi hgigi gj = gi hgj = deg(Eµν ⊗ xh) = deg ψ(w), that is, ψ is a graded algebra isomorphism, and the proof of Theorem3 is complete. Remark. The condition on the characteristic of the field F is required in the proof of Theorem3 only to ensure that every graded simple algebra R is semiprimitive. In fact, Theorem3 remains valid if we discard the condition on the characteristic of F and assume instead that R is semiprimitive. Corollary 1. Let F be an algebraically closed field and let G be a finite group such that |G|−1 ∈ F . Then the following assertions hold: (i) the number of pairwise non-isomorphic finite-dimensional G-graded skew fields is finite; (ii) for any n > 1 the number of pairwise non-isomorphic n-dimensional G-graded simple algebras is finite. Proof. (i) By Theorem2 every graded skew field is isomorphic to a twisted group ring of the form F σ[H], where H is a subgroup of G and σ is a 2-cocycle on H. As is known (see, for instance, [16], § 1), two cocycles σ and τ define the same twisted group algebra if there is a map δ : H → F ∗ such that

τ(x, y) = δ(xy)δ−1(x)δ−1(y)σ(x, y).

In particular, every 2-coboundary γ, that is, a 2-cocycle of the form

γ(x, y) = δ(xy)δ−1(x)δ−1(y), 982 Yu. A. Bahturin, M. V. Zaicev and S. K. Sehgal always defines the ordinary group ring. The cocycles form an Abelian group A, the coboundaries form a subgroup B of A, and the quotient group A/B is the second cohomology group H2(H,F ∗), which is known to be finite if so also is H (see [22], § 6.10, Exercise 3). Hence the number of non-isomorphic algebras of type F σ[H] is finite. Since the number of subgroups H ⊂ G is finite, we obtain the first assertion. (ii) By Theorem3 every G-graded simple finite-dimensional algebra is isomorphic to Mk(C), where C is a graded skew field and the grading on Mk(F ) is defined by k some k-tuple (g1, . . . , gk) ∈ G . Since the family of k-tuples is finite we can use part (i) of the corollary to complete the proof. As proved in [23], for a fixed Abelian group G a G-graded algebra R is simple if and only if R is graded simple and the centre of R is a field. In [24] the same author extended this result to an arbitrary hypercentral group. Thus, the following result is a generalization of [23] and [24]. L Corollary 2. Let R = g∈G Rg be a finite-dimensional G-graded algebra over an algebraically closed field F such that either char F = 0 or char F is coprime with the order of each finite subgroup of G. Then R is simple if and only if R is graded simple and the centre of R coincides with F .

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Yu. A. Bahturin Received 8/MAY/07 Department of Mechanics and Mathematics, Translated by A. SHTERN Moscow State University E-mail: [email protected] M. V. Zaicev Department of Mechanics and Mathematics, Moscow State University E-mail: [email protected] S. K. Sehgal University of Alberta, Edmonton, Canada E-mail: [email protected]