Gradings on simple algebras of finitary matrices

Journal of Algebra 324 (2010) 1279–1289

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Journal of Algebra www.elsevier.com/locate/jalgebra

Gradings on simple algebras of finitary matrices Yuri Bahturin a,b,∗,1 , Mikhail Zaicev b,2 a b

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C5S7, Canada Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992, Russia

a r t i c l e

i n f o

Article history: Received 11 November 2008 Available online 26 May 2010 Communicated by Vera Serganova

a b s t r a c t We describe gradings by finite abelian groups on the associative algebras of infinite matrices with finitely many nonzero entries, over an algebraically closed field of characteristic zero. © 2010 Elsevier Inc. All rights reserved.

MSC: 17B20 17B40 16W10 16W50 Keywords: Matrix algebras Graded algebras Automorphisms

1. Introduction This paper is devoted to the extension of the results of [1–3] about the group gradings on finite-dimensional matrix algebras to the case of infinite-dimensional simple algebras of finitary linear transformations. After reminding the main results in the case of finite dimensions, we describe the G-graded embeddings of one finite-dimensional graded matrix algebra into another (Theorem 3), with G a finite abelian group. Our next result says that if a simple locally finite algebra with minimal one-sided ideals is graded by G as above then it can be presented as the direct limit of finitedimensional G-graded matrix algebras (Theorem 4). This allows us to describe in Theorem 5 the gradings on the simple algebra of finitary matrices that is, the algebra of infinite matrices such that

*

Corresponding author at: Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C5S7, Canada. E-mail addresses: [email protected] (Y. Bahturin), [email protected] (M. Zaicev). 1 Work is partially supported by NSERC grant #227060-04 and URP grant, Memorial University of Newfoundland. 2 Work is partially supported by RFBR, grant 06-01-00485a, and SSH-1983.2008.1. 0021-8693/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2010.05.008

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each matrix has only finitely many nonzero entries. Finally, in Theorem 6, we give a necessary and sufficient condition for the equivalence of elementary gradings on the above algebra of infinite matrices. 2. Some notation and simple facts Let F be an arbitrary field, R a not necessarily associative algebra over an F and G a group. We say that R is a G-graded algebra, if there is a vector space sum decomposition

R=



R (g),

(1)

for all g , h ∈ G .

(2)

g ∈G

such that

R ( g ) R (h) ⊂ R ( gh) Two G-gradings

R=



R (g)

and

R=

g ∈G

 ( g )  R

(3)

g ∈G

are called isomorphic if there is an automorphism ϕ of R such that ϕ ( R ( g ) ) = ( R  )( g ) , for all g ∈ G.  ( g ) ). An element a ∈ R is A subspace V ⊂ R is called graded (or homogeneous) if V = g ∈G ( V ∩ R ( g) called homogeneous of degree g if a ∈ R . We also write deg a = g. The support of the G-grading is a subset







Supp R = g ∈ G  R ( g ) = 0 . 3. Reminder: Group gradings on matrix algebras Below we briefly recall the results of [1–3], where the full description of a finite group gradings on the full matrix algebra has been given. ( g ) on the matrix algebra R = M ( F ) is called elementary if there exists an nA grading R = n g ∈G R

tuple ( g 1 , . . . , gn ) ∈ G n , such that the matrix units E i j , 1  i , j  n are homogeneous and E i j ∈ R ( g ) ⇔

g = g i−1 g j . If R is a matrix algebra with an elementary G-grading defined by a tuple ( g 1 , . . . , gn ) ∈ G n and B an algebra with a G-grading then the tensor product R = A ⊗ B will be given a grading if, given a homogeneous element x of degree h, we set E i j ⊗ x ∈ R ( g ) provided that g = g i−1hg j , for any 1  i , j  n. This grading of the tensor product is called induced. A grading is called fine if dim R ( g ) = 1 for any g ∈ Supp R. In this case T = Supp R is always a subgroup of G [3]. In this case if V is a natural R-module then V is the space of a faithful irreducible representation of T (see [2]). If we denote by X t the image of t ∈ T in R corresponding to this representation then X t is a basis of R t and there is a 2-cocycle α : G × G → F ∗ such that X t X s = α (t , s) Xts , for any t , s ∈ T . This makes R isomorphic to a twisted group algebra F α G. The main result of [1, Theorem 6] can be formulated as follows. Theorem 1. Let G be a group of order d, F an algebraically closed field and R = M n ( F ). Then, as a G-graded algebra, R is isomorphic to the tensor product with induced grading R ∼ = A ⊗ B where A = M k ( F ) has an elementary G-grading, with support S, B = M l ( F ) has a fine grading, with support T , and S ∩ T = {e }.

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A particular case of the fine gradings is a so-called ε -grading where ε is an nth primitive root of 1. Let G = a n × b n be the direct product of two cyclic groups of order n and

⎛

ε n −1

⎜ 0 ···

Xa = ⎝

0

0

ε

n −2

··· 0

⎞ ... 0 ... 0 ⎟ ⎠, ··· ··· ... 1

⎞ ... 0 ⎜··· ··· ··· ···⎟ Xb = ⎝ ⎠. 0 0 ... 1 1 0 ... 0

(4)

X an = X bn = I

(5)

⎛

0

1

Then

X a X b X a−1 = ε X b , j

j

and all X ai X b , 1  i , j  n, are linearly independent. Clearly, the elements X ai X b , i , j = 1, . . . , n, form a basis of R and all the products of these basis elements are uniquely defined by (5). j Now for any g ∈ G, g = ai b j , we set X g = X ai X b and denote by R ( g ) a one-dimensional subspace



j

R ( g ) = X ai X b .

(6)



( g ) is a G-grading on M ( F ) which is called an ε Then from (5) it follows that R = n g ∈G R grading. Now let R = M n ( F ) be the full matrix algebra over F graded by an abelian group G. The following result has been proved in [3, Section 4, Theorems 5, 6] and [1, Section 2.2, Theorem 6, Section 2.3, Theorem 8].

Theorem 2. Let F be an algebraically closed field of characteristic zero. Then as a G-graded algebra R is isomorphic to the tensor product

R 0 ⊗ R 1 ⊗ · · · ⊗ Rk where R 0 = M n0 ( F ) has an elementary G-grading, Supp R 0 = S is a finite subset of G, R i = M ni ( F ) has the εi grading, εi being a primitive ni th root of 1, Supp R i = H i ∼ = Zni × Zni , i = 1, . . . , k. Also H = H 1 · · · H k ∼ = H 1 × · · · × H k and S ∩ H = {e } in G. 4. Embeddings of graded matrix algebras To describe the gradings on the algebra of finitary matrices we will need to consider the embedding of one G-graded finite-dimensional simple algebra into another. It follows from Theorem 1, if R∼ = Mn ( F ) is G-graded then, as a graded algebra, R is isomorphic to a tensor product C ⊗ D where C = M p ( F ), D = M q ( F ), n = pq, M p ( F ) is a matrix algebra with elementary G-grading, M q ( F ) a matrix algebra with a fine T -grading where T = Supp D is a subgroup in G such that T ∩ Supp C = {1} where 1 is the identity element of G. Thus we may think that R = C D ∼ = C ⊗ D. Let us notice that the subalgebra D is not defined uniquely and once D has been fixed, C is uniquely defined as the centralizer of D in R. Theorem 3. Let F be an algebraically closed field and G a finite abelian group. Let R 1 ∼ = M k ( F ) and R 2 ∼ = M n ( F ) be two G-graded matrix algebras with identity elements e 1 and e 2 , respectively, R 1 = C 1 D 1 , R 2 = C 2 D 2 their decompositions in which C 1 , C 2 have elementary grading while D 1 , D 2 have fine grading. Let also D1 ∼ = D 2 as G-graded algebras and ϕ : R 1 → R 2 be an injective homomorphism of graded algebras. Then there exists a decomposition R 2 =  C2 D2 ∼ C2 ⊗  D 2 such that  C 2 is a matrix algebra with elementary grading,  D 2 is a = matrix algebra with fine grading,  D2 ∼ C 2 . If ϕ (e 1 ) R 2 ϕ (e 1 ) = ϕ ( R 1 ) then = D 2 as graded algebras and ϕ (C 1 ) ⊂  C 2 ϕ (e 1 ) = ϕ (C 1 ). Besides, there is an isomorphism ψ : D 1 →  D 2 such that ϕ (a)ϕ (d) = ϕ (a)ψ(d), also ϕ (e 1 ) for any a ∈ R 1 , d ∈ D 1 .

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Proof. Since D 1 ∼ = D 2 , in particular, Supp D 1 ∼ = Supp D 2 . We denote T = Supp D 1 . Then for any t ∈ T there exist invertible matrices X t ∈ R 1 and X t ∈ R 2 such that

D 1 = Span{ X t | t ∈ T }, In addition, for any t , s ∈ T there is







D 2 = Span X t  t ∈ T .

α (t , s) ∈ F such that  X t X s = α (t , s) X ts ,

X t X s = α (t , s) X ts ,

(7)

following because D 1 and D 2 are isomorphic. One may also assume that X 1 and X 1 are the identity elements of R 1 and R 2 , respectively. Since T ∩ Supp C 2 = {1} it follows that ϕ ( X t ) = A t X t for some matrix A t ∈ C 2 , deg A t = 1 in the G-grading. We then set

A t = A t ϕ (e 1 ) + e 2 − ϕ (e 1 ) and

X t = A t X t ,

where e 2 is the identity of R 2 . We will first show that

    D 2 = Span X t  t ∈ T is a graded subalgebra in R 2 isomorphic to D 1 (or D 2 ). Now since

ϕ (e1 ) At Xt = ϕ (e1 )ϕ ( Xt ) = ϕ ( Xt ) = At Xt and X t is nondegenerate, it follows that

ϕ (e 1 ) A t = A t . Since A t and Xt commute in R 2 , it follows that

X t A t ϕ (e 1 ) = ϕ ( X t )ϕ (e 1 ) = ϕ ( X t ) = X t A t , and so

ϕ (e1 ) At = At ϕ (e1 ) = At .

(8)

In particular, (e 2 − ϕ (e 1 )) A t = 0. Now let us recall that ϕ (e 1 ) = ϕ ( X 1 ) = A 1 X 1 = A 1 ∈ C R 2 ( D 2 ) and so

ϕ (e1 ) Xt = Xt ϕ (e1 ) for any t ∈ T .

(9)

If we use (7), (8), and (9) we will obtain the following









X t X s = A t + e 2 − ϕ (e 1 ) X t A s + e 2 − ϕ (e 1 ) X s

  = A t X t A s X s + e 2 − ϕ (e 1 ) X t X s   = ϕ ( X t )ϕ ( X s ) + e 2 − ϕ (e 1 ) X t X s   = ϕ ( X t X s ) + e 2 − ϕ (e 1 ) X t X s      = ϕ α (t , s) X ts + e 2 − ϕ (e 1 ) α (t , s) X ts     = α (t , s) A ts X ts + α (t , s) e 2 − ϕ (e 1 ) X ts  = α (t , s) X ts .

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Since the elements X t , t ∈ T are linearly independent, it follows that the mapping X t → X t is a (graded) isomorphism of algebras D 1 and  D2. Now we denote by  C 2 the centralizer C R 2 ( D 2 ) of  D 2 in R 2 . Then  C 2 is a graded subalgebra of R 2 . D 2 since X 1 = e 1 and The identity element e 2 of R 2 is in 

X 1 = ϕ ( X 1 ) + e 2 − ϕ (e 1 ) = ϕ (e 1 ) + e 2 − ϕ (e 1 ) = e 2 . In this case R 2 =  C2 D2 ∼ C2 ⊗  D 2 (see, for instance, [3]). = Now we would like to show that  C 2 is an algebra with elementary G-grading. Since  C 2 is a central simple F -algebra, by the main result of [3]  C2 =  C0 D0 ∼ C0 ⊗  D 0 where the grading on =  C 0 is elementary while on  D 0 fine. We set T 0 = Supp  D 0 . Then T 0 is a subgroup in G such that T 0 ∩ T = {1}. It follows that  D0 D2 ∼ D0 ⊗  D 2 is a graded subalgebra in R 2 with fine grading such = that Supp  D0 D2 = T0 T ∼ C 0 ⊗ ( D0 ⊗  D 2 ) is another decomposition of R 2 as the = T 0 × T . Besides, R 2 ∼ = tensor product of algebras with elementary and fine grading. From the above mentioned property of the supports, the identity component R e of R = C ⊗ D, C elementary, D fine, is C e ⊗ I . The centralizer of R e is a graded subalgebra C e ⊗ D which has the same support as D. This uniquely defines the D 2 = Supp  D0 D 2 = T 0 T . Then T 0 = {1} support of the fine component. As a result, we have T = Supp    implying that C 2 = C 0 is an algebra with elementary grading. C 2 = C R 2 ( D 2 ). Let a ∈ C 1 = C R 1 ( D 1 ). Then a X t = X t a for any Now we need to show that ϕ (C 1 ) ⊂  t ∈ T . Then we have the following





ϕ (a) Xt = ϕ (a) ϕ ( Xt ) + e2 − ϕ (e1 )

  = ϕ (a)ϕ ( X t ) + ϕ (a)ϕ (e 1 ) e 2 − ϕ (e 1 ) = ϕ (a X t )   = ϕ ( X t a) = ϕ ( X t ) + e 2 − ϕ (e 1 ) ϕ (a) = X t ϕ (a),

proving that ϕ (a) ∈ C R 2 ( D 2 ), that is, ϕ (C 1 ) ⊂  C2. Finally, let us assume ϕ (e 1 ) R 2 ϕ (e 1 ) = ϕ ( R 1 ). Since ϕ (C 1 ) ⊂  C 2 the containment ϕ (e 1 ) C 2 ϕ (e 1 ) is obvious. To prove the converse, we notice that

ϕ (C 1 ) ⊂

ϕ(R1) ∩  C 2 = ϕ (e 1 ) C 2 ϕ (e 1 ). Now C 1 = C R 1 ( D 1 ) and so b ∈ R 1 satisfies ϕ (b) ∈ C R 2 ( D 2 ) if and only if b ∈ C 1 , that is, ϕ ( R 1 ) ∩  C2 = ϕ (C 1 ) which now implies ϕ (C 1 ) = ϕ (e 1 ) C 2 ϕ (e 1 ). It remains to look at the homomorphism ψ : D 1 →  D 2 given by ψ( X t ) = X t . We have the following





ϕ (e1 ) Xt = ϕ (e1 ) At Xt + ϕ (e1 ) e2 − ϕ (e1 ) Xt   = ϕ (e 1 ) A t X t = ϕ (e 1 )ϕ X t

so that

ϕ (e 1 )ψ(d) = ϕ (e 1 )ϕ (d) for any d ∈ D 1 and thus

ϕ (a)ψ(d) = ϕ (a)ϕ (e1 )ϕ (d) = ϕ (a)ϕ (d). Now the proof is complete.

2

To describe the elementary gradings on infinite-dimensional simple algebras we first consider the case of one finite-dimensional matrix algebra embedded in another, both having elementary gradings. Notice that in the claims to follow the grading group G may be infinite and nonabelian.

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To start with we notice that if R = M n ( F ) is an algebra with an elementary grading given by a tuple (h1 , . . . , hn ), n = km + r for some k, m  1, r  0 and h1 , . . . , hn satisfy the conditions 1 1 1 h− h = h− h = · · · = h− h i +1 i +2 i +k+1 i +k+2 i +(m−1)k+1 i +(m−1)k+2

for 1  i  k − 2

(10)

then the subalgebra C consisting of all block-diagonal matrices of the form diag{ X , . . . , X , 0} where X is an arbitrary (k × k)-matrix repeated m times on the diagonal is G-graded and isomorphic to a matrix algebra M k ( F ) with an elementary grading given by the tuple (h1 , . . . , hk ). This easily follows because by (10) all matrix units E α +ik,β+ik , i = 0, 1, . . . , m − 1, have the same degree for fixed 1  α , β  k. We would like now to prove that any embedding of simple algebras with elementary gradings amounts to this construction.  Let us recall that if V = g ∈G V g is a G-graded space then R = End V canonically becomes Ggraded if, given a G-graded basis { v 1 , . . . , v n } of V with deg v i = g i−1 , 1  i  n, one gives the matrix

unit E i j the degree equal g i−1 g j . Thus the grading of M n ( F ) induced from V is elementary.



( g ) the algebra Lemma 1. Let V be an n-dimensional G-graded space over a field F and End V = R = g ∈G R of all linear transformations of V with induced elementary grading. Let C be a graded subalgebra in R which is isomorphic to the matrix algebra M k ( F ) with an elementary grading given by the tuple ( g 1 , . . . , gk ). Then V splits as the sum of C -invariant subspaces

V = V1 ⊕ · · · ⊕ Vm ⊕ V0

(11)

where dim V 1 = · · · = dim V m = k, V 1 , . . . , V m are faithful irreducible C -modules while C V 0 = {0}. Besides, there is a homogeneous basis of V in which all matrices of the transformations in C have the block-diagonal form diag{ X , . . . , X , 0} where X is a (k × k)-matrix and the tuple (h1 , . . . , hn ) giving the induced elementary grading on R = M n ( F ) satisfies (10). Proof. Since the grading on C ∼ = M k ( F ) is elementary, any subspace spanned by a set of matrix units is graded. In particular, this is true for any minimal left ideal spanned by all matrix units in a fixed column. Let L be one of such minimal ideals, corresponding to the last, kth column of C . If we fix any v ∈ V then the left C -module L v is either irreducible or equal zero. Moreover, if v is homogeneous then the C -submodule L v is also G-graded. These remarks are sufficient to prove the existence of the decomposition (11). Now let E i j , 1  i , j  k be the set of all matrix units of C . Since V 1 in (11) is a faithful C module, there exists a homogeneous element v ∈ V such that E 1k v = 0. In this case the vectors v i = E ik v = E i ,i +1 · · · E k−1,k v, i = 1, . . . , k − 1, v k = v, form a homogeneous basis of V 1 and the elementary grading on C induced from this grading is given by the tuple ( g 1 , . . . , gk ). Indeed, if deg v = h then 1 −1 deg v i = g i−1 gk h, deg v j = g − j gk h and so deg E i j = g i g j and still E i j v j = v i . If we choose the bases in other V 2 , . . . , V m and an arbitrary homogeneous basis in V 0 then we obtain a realization of C by the block-diagonal matrices of the form diag{ X , . . . , X , 0}. It remains to consider the tuple (h1 , . . . , hn ) which defines the elementary grading on R = M n ( F ) induced from the graded basis of V just constructed. If we denote by  E st , 1  s, t  n, the matrix E st = g s−1 gt . Also, for any 1  i , j  k we units of R corresponding to this basis, then, as usual, deg  will have

Eij =  Eij +  E i +k, j +k + · · · +  E i +(m−1)k, j +(m−1)k in R and deg E i j = g i−1 g j in C , hence in R, since the embedding of C in R is graded. Now all  E st are homogeneous and so the conditions (10) must be satisfied. Now the proof is complete. 2

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Example 1. The condition of C having an elementary grading is essential. For example, suppose C∼ = Mn ( F ) with a fine grading. Let V be C itself as a graded vector space and let us assume that C acts on itself by multiplication on the left. Then R = End V is an algebra with elementary grading induced from V and C a graded subalgebra. So, C is a graded matrix subalgebra of a matrix algebra with an elementary grading but the grading of C is not elementary. So the conclusion of the previous lemma cannot hold for C . Lemma 1 enables one to describe the gradings on all possible direct  limits of matrix algebras with elementary gradings. Here we will need a special case where C = i 1 C i where C 1 ⊂ C 2 ⊂ · · · is an ascending chain of matrix algebras and C i = e i C j e i , for any 1  i  j, where e i is the identity element of C i . To start with we generalize the notion of the elementary grading to the case of finitary matrices. Definition 1. Let R be the algebra of finitary and g = ( g 1 , g 2 , . . .) a sequence of ele matrices ( g ) is called elementary defined by g if R ( g ) = ments in a group G. Then a grading R = g ∈G R Span{ E i j | g i−1 g j = g }.





( g ) be a G-graded algebra over a field F which is the union C = Lemma 2. Let C = g ∈G C i 1 C i of an ascending chain of graded matrix subalgebras of orders n1 , n2 , . . . , with identity elements e 1 , e 2 , . . . . Suppose all the gradings on the subalgebras C 1 , C 2 , . . . are elementary and C i = e i C j e i for all i , j with 1  i  j. Then C is isomorphic to the algebra R of finitary matrices with elementary grading given by a sequence g = ( g 1 , g 2 , . . .) of elements of G in which every C i is embedded as a graded subalgebra of all matrices with zero entries in all rows and columns whose numbers are greater than ni , i = 1, 2, . . . . The G-grading on C i is elementary given by an n-tuple ( g 1 , . . . , gni ).

Proof. By Lemma 1, we may adjust our graded embeddings in the sequence C 1 ⊂ C 2 ⊂ · · · in such a way that each C i can be viewed as a graded subalgebra of C i +1 consisting of all ni × ni matrices in the left upper corner. These adjustments do not change the isomorphism class of the limit since this depend only on the module structure of C i +1 over C i , for each i (see [4]). But then the set of all matrices L i in C i with zeros outside the first column is a graded subspace of the similar subspace L i +1 in C i +1 . If {e = g 1−1 , g 2−1 , . . . , gn−i 1 } is the set of degrees of the matrix units spanning L i then the elementary grading ∞ of C i is defined by the tuple ( g 1 , . . . , gni ). Then the set of degrees of the matrix units in L = i =1 is the desired sequence of elements of G defining the elementary grading on the algebra of finitary matrices C . 2 5. Gradings on simple algebras with minimal one sided ideals In this section we consider the gradings by finite abelian groups on simple locally finite algebras with minimal one sided ideals. Suppose that R is such an algebra. Using the Structure Theorem in [7, Chapter 4, Section 9], we find a pair of mutually dual spaces V and Π ⊂ V ∗ such that R ∼ = V ⊗Π with the product given by

( v 1 ⊗ π1 )( v 2 ⊗ π2 ) = π1 ( v 2 )( v 1 ⊗ π2 ) where v 1 , v 2 ∈ V , π1 , π2 ∈ Π and the kernel of the bilinear mapping ( v , π ) → π ( v ) is trivial. If dim V = dim Π = n < ∞ we have R ∼ = Mn ( F ), the matrix algebra of order n over F . The linear mapping S : V → V and T : Π → Π are called adjoint if ( T (π ))( v ) = π ( S ( v )). Actually, T is completely defined by S and we write T = S ∗ . The Isomorphism Theorem [7, Chapter 4, Section 11] describes the automorphisms of V ⊗ Π with the help of the automorphisms of V in the following way. If ϕ ∈ Aut( V ⊗ Π) then there exists a linear automorphism S : V → V , for which there exists the adjoint automorphism S ∗ : Π → Π , such that

ϕ ( v ⊗ π ) = S −1 ( v ) ⊗ S ∗ (π ) for any v ∈ V , π ∈ Π. The automorphism S is defined by

ϕ uniquely up to a nonzero scalar multiple.

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The finite-dimensional subspaces V  ⊂ V and Π  ⊂ Π are called compatible if they are of the same dimension n and the annihilator of V  in Π  is zero. As mentioned above, in this case V  ⊗ Π  ∼ = M n ( F ). A simple remark is that V  ⊗ Π  ⊂ V  ⊗ Π  if and only if V  ⊂ V  and Π  ⊂ Π  . It is shown in [7, Chapter 4, Section 16] that R = V ⊗ Π has a local system of matrix subalgebras of such form. It will be convenient to label the subalgebras in this local system by the elements of a directed set I , that is, an ordered set such that for any α , β ∈ I there is γ ∈ I with α ≺ γ and β ≺ γ . We will have V α ⊗ Πα ⊂ V β ⊗ Πβ if and only if V α ⊂ V β and Πα ⊂ Πβ . The latter holds if and only if α ≺ β . Our aim is to prove the following. Theorem 4. Let a simple locally finite algebra R with minimal one sided ideals over an algebraically closed field of characteristic zero be given a grading by a finite abelian group G. Then R has a local system of graded finite-dimensional matrix algebras. Proof. Using Litoff’s Theorem [7, Chapter 4, Section 15], we find that R is locally matrix, that is, there a local system { V α ⊗ Πα | α ∈ I } of matrix subalgebras in a G-graded algebra R = V ⊗ Π . We need to prove that there is another local system whose terms are G-graded matrix subalgebras of the form { V α ⊗ Π α | α ∈ I }. Now the conditions imposed on the field allow one to replace the graded subspaces by the invariant subspaces with respect to the automorphisms corresponding to the multiplicative characters χ ∈ G, given by χ ∗ r = χ ( g )r, for any r ∈ R ( g ) . As mentioned before, to each such χ one can associate ∗ : Π → Π so that χ ∗ ( v ⊗ π ) = S −1 ( v ) ⊗ S ∗ (π ), for an automorphism S χ : V → V and its adjoint S χ χ χ ∗ are any v ∈ V and π ∈ Π . Since S χ is defined up to scalar, the mappings χ → S χ and χ → S χ

projective representations of  G by linear transformations of V and Π . It is obvious that given a  Ginvariant subspace U in V , the annihilator U ⊥ in Π is also  G-invariant. With these facts in mind, we first pick, for each α ∈ I a subspace of finite codimension Πα⊥ . Set

Uα =







S χ Πα⊥ ⊂ Πα⊥ .

χ ∈ G



 This subspace in V of finite codimension. Since α ∈ I Πα = Π we must have  is ⊥a G-invariant  Π = 0, hence α∈I α α ∈ I U α = 0. Note that U γ ⊂U β as soon as β ≺ γ . Let us now consider a finite-dimensional  G-invariant subspace  G ( V α ) = χ ∈ G χ ( V α ). Then there exists U β such that

 G ( V α ) ∩ U β = 0. Since I is a directed set, there is γ ∈ I such that α , β ≺ γ hence  G ( V α ) ∩ U γ = 0. Since the projective representation of a finite group is fully reducible there is a  G-invariant subspace L G ( V α ) ⊕ U γ ) = 0. We then set V α = L ⊕  G ( V α ). Also, we set Π α = U γ⊥ . Since in V such that V = L ⊕ ( ⊥ U γ ⊂ U α ⊂ Πα , we have that Πα ⊂ Π α . Being an orthogonal complement to a  G-invariant space, Π α is  G-invariant. By construction, dim Π α = dim V α and also Π ⊥ = U has trivial intersection with V α . γ α This proves that V α and Π α are compatible invariant spaces so that V α ⊗ Π α is a  G-invariant matrix G-invariant local system, hence a local system of subalgebra. Since V ⊗ Πα ⊂ V α ⊗ Π α , we obtain a  graded matrix subalgebras. 2 6. Gradings on simple algebras of finitary matrices Now we are ready to prove our main result.



( g ) be a G-graded algebra of infinite matrices each Theorem 5. Let G be a finite abelian group, R = g ∈G R having only finitely many nonzero entries over an algebraically closed field F of characteristic zero. Then R is isomorphic to a graded tensor product C ⊗ D where C is such with an elementary grading and D = M n ( F ) is a matrix algebra of order n with a fine grading. Additionally, we have Supp C ∩ Supp D = {1}.

Proof. According to [7, Chapter 4, Section 15] R is the same as the simple algebra with minimal one sided ideals since in our case dim R is countable. Clearly, in this case we can remove unnecessary

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terms from the local system provided by Theorem 4 and conclude that R is the union of the ascending chain R 1 ⊂ R 2 ⊂ · · · of graded simple finite-dimensional subalgebras. Each R i decomposes as the tensor product R i = C i D i ∼ = C i ⊗ D i of a simple subalgebra C i with an elementary grading and a simple subalgebra D i with a fine grading. The support T i = Supp D i is a subgroup in G. Since the number of subgroups in G is finite, by excluding unnecessary subalgebras R i we may assume that T 1 = T 2 = · · · is the same subgroup T of G. In particular, dim D i = | T | for all i and that the D i as ungraded algebras all isomorphic to the same M n ( F ). Since the number of different fine gradings is also finite, we may, as before, assume that all D i are isomorphic as graded algebras. Let ϕi +1,i be the graded embedding of R i in R i +1 . We also set ϕ ji = ϕ j , j −1 · · · ϕi +1,i . If we apply Theorem 3 to each embedding R i ⊂ R i +1 then we may assume that ϕi +1,i (C i ) ⊂ C i +1 and ϕi+1,i (C i ) = ϕi+1,i (e i )C i+1 ϕi+1,i (e i ) where e i is the identity element of C i and that there is an isomorphism ψi +1,i ( D i ) → D i +1 such that

ϕi+1,i (a)ϕi+1,i (d) = ϕi+1,i (a)ψi+1,i (d) for all a ∈ C i , d ∈ D i .

(12)

We set ψ1 = id D 1 and ψi = ψi ,i −1 · · · ψ2,1 : D 1 → D i , for all i  2. Then ψ j (d) = ψ j ,i (ψi (d)) for any i , j with 1  i  j, and any d ∈ D 1 . Besides, using (12), we may write





ϕ j,i (a)ϕ j,i (d) = ϕi, j (a)ψ j ψi−1 (d) Let us set C =



i 1

C i and construct an isomorphism

for all a ∈ C i , d ∈ D i .

(13)

ρ : R → C ⊗ D 1 . If a ∈ C i , d ∈ D i then we set

ρ (ad) = a ⊗ ψi−1 (d) for any a ∈ C i , d ∈ D i .

(14)

Clearly, (14) defines an injective homomorphism of R i = C i D i into C ⊗ D 1 . Actually, the same formula defines an isomorphism of R to C ⊗ D 1 . To prove this we only need to check that ρ is well defined on R. Indeed, if a ∈ C i , d ∈ D i and i < j then ϕ j ,i (ad) = ϕ j ,i (a)ϕ j ,i (d) in R and a = ϕ j ,i (a) in C since we identify a ∈ C i with its image ϕ j ,i (a) in C j . But then, according to (13) we should have











ρ ϕ j,i (a)ϕ j,i (d) = ρ ϕi, j (a)ψ j ψi−1 (d)

= ϕ j ,i (a) ⊗ ψi−1 (d) = a ⊗ ψi−1 (d), proving that, indeed, ρ is defined correctly. By Lemma 2C is isomorphic to the algebra of finitary matrices with an elementary G-grading. Since Supp C = i 1 Supp C i and T ∩ Supp C i = {1}, for all i  1, we have T ∩ Supp C = {1}, and the proof is complete. 2 7. The uniqueness theorem for the elementary gradings of simple algebras of finitary matrices The defining sequence g of an elementary grading is not defined uniquely. In what follows we prove a theorem that gives necessary and sufficient conditions for two sequences to define isomorphic gradings. It will be convenient to denote such sequence as a function τ : I → G such that τ (i ) = g i . Here I is the sequence of natural numbers or any initial segment of this. In the latter case we simply deal with R = M n ( F ) for a natural number n. With each such function we associate a function S τ : G → N ∪ {∞} given by S τ ( g ) = Card(τ −1 ( g )). Further notice that for each elementary grading defined by a function τ there is a graded vector space V with a basis { v i | i ∈ I } such that deg v i = g i−1 . We denote the subspace spanned by all v i with τ (i ) = g by V g −1 . In this case the algebra of finitary matrices can be identified with the set all linear transformations of V spanned by the linear transformations with matrices E i j with respect to the above basis. The homogeneous component R ( g ) is then the set of all linear transformations ϕ such that ϕ ( V h ) ⊂ V gh .

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Theorem 6. Let G be a group,  R and R  the algebras of finitary matrices endowed by two elementary grad( g ) and R  =  ( g ) defined by the tuples τ and τ  , respectively. Then R and R  are R ings R = g ∈G g ∈G ( R ) isomorphic as graded algebras if and only if there is an element g 0 ∈ G such that S τ ( g ) = S τ  ( g 0 g ), for all g ∈ G. Proof. First we assume that the gradings defined by τ and τ  are isomorphic. Note that two sequences g = ( g 1 , g 2 , . . .) and h = (ag 1 , ag 2 , . . .) define the same gradings on R and S τ ( g ) = S ρ (ag ) for all g ∈ G where ρ (i ) = ag. Hence we can suppose that ρ (1) = e that is g 1 = 1 in g. Let f : R → R  be the graded isomorphism of R and R  , that is, f ( R ( g ) ) = ( R  )( g ) , for all g ∈ G. Let us consider the identity components R (e) and ( R  )(e) . Each of these algebras is the sum of simple ideals M ( g ) and ( M  )( g ) each defined as the linear span of the set of matrix units E i j or E i j , respectively, such that τ (i ) = τ ( j ) = τ  (i ) = τ  ( j ) = g. Since f ( R (e) ) = ( R  )(e) we must have f ( M ( g ) ) = ( M  )(σ ( g )) for a bijective map σ : Supp R → Supp R  = Supp R on G. Let us also recall [7, Corollary 2, Section 4.11] that there is a linear bijective map α : V → V  such that f (ϕ ) = αϕα −1 for any ϕ ∈ R. Let us notice first that such α must satisfy the equation α ( V g −1 ) = V σ ( g )−1 . Indeed, we have

M (g) =





ϕ ∈ R (e)  ϕ ( V ) ⊂ V g −1

We have α M ( g ) α −1 = ( M  )(σ ( g )) and so mind the equations

α(V ) = V ,



and



M

( g )

   (e)      ϕ V ⊂ V  −1 . = ϕ ∈ R g

α M ( g ) = ( M  )(σ ( g )) α . Applying both sides to V and having in

M ( g ) ( V ) = V g −1

and



M

(σ ( g )) 



V  = V σ ( g )−1

α ( V g −1 ) = V σ ( g )−1 . Now let us use α R ( g ) α −1 = ( R  )( g ) or α R ( g ) = ( R  )( g ) α , for all g ∈ Supp R ⊂ G. Applying both sides of this equation to any V h , h ∈ Supp V ⊂ G, we obtain α R ( g ) ( V h ) = ( R  )( g ) α ( V h ) and so α ( V gh ) = we obtain

( R  )( g ) ( V σ (h−1 )−1 ). In other words, V σ (h−1 g −1 )−1 = V g σ (h−1 )−1 and



σ h −1 g −1

−1

 − 1 = g σ h −1

(15)

for any h ∈ Supp R, g ∈ Supp R. Recall that g 1 = e in g that is e −1 = e ∈ Supp V . Substituting h = e in (15) and setting g 0 = σ (e ), we obtain σ ( g −1 ) = g 0 g −1 , for any g ∈ Supp R. Note that for any elementary grading oh g −1 ∈ Supp R if and only if g ∈ Supp R. Hence also σ ( g ) = g 0 g for all g ∈ Supp R. So we have dim V g = dim V g 0 g . Since S τ ( g ) = dim V g , we easily obtain the desired condition: there is g 0 ∈ G such that S τ ( g ) = S τ  ( g 0 g ) for all g ∈ G.  (g) , To prove the converse, we consider two G-graded finitary matrix algebras R = g ∈G R



 ( g ) and assume that there is g ∈ G such that S ( g ) = S  ( g g ), for any g ∈ R = 0 τ 0 τ g ∈G ( R ) Supp R ⊂ G. We define an isomorphism f : R → R  in the following way. For each g ∈ G, let the ordered subset I g label the elements v i of the basis of V ∩ V g −1 . Let I g be the same thing for V  . Then there is an ordered map β g : I g → I g0 g . We extend it to a bijection β of I into itself. Then β satisfies the following condition. If g = ( g 1 , g 2 , . . .) and h = (h1 , h2 , . . .) then

hβ(i ) = g 0 g i .

(16)

Denote by f the linear map R → R  such that f ( E i j ) = E β(i )β( j ) . Then f is an isomorphism and f ( R ( g ) ) = ( R  )( g ) due to (16). 2

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Remark 1. The theorem above is no longer true if we replace the algebra of finitary matrices by the other direct limits of matrix algebras. For example, suppose an algebra R is the direct limit of the algebras R i = M 2i , i = 1, 2, . . . , where the structure mappings ϕi : R i → R i +1 are given by X → diag{ X , X }. Then the elementary grading by G = a 2 of R i given by a tuple τ can be extended to the grading of R i +1 defined by τ  to make ϕi graded if we either choose τ  = (τ , τ ) or τ  = (τ , aτ ). If we start with the grading of R 1 defined by τ = (e , a) and consider the identity component of the grading in each of the two cases then we will see the limits of semisimple algebras, each of which is the sum of two isomorphic matrix subalgebras. But the Bratteli diagrams [5] of these limits are different and so the limits are not isomorphic. At the same time the “Steinitz numbers” S τ and S τ are the same and both equal to e ∞ a∞ . Remark 2. A uniqueness theorem for the G-gradings of matrix algebras over algebraically closed field F of characteristic zero has been established by A.A. Chasov [6]. References [1] Y. Bahturin, M. Zaicev, Graded algebras and graded identities, in: Polynomial Identities and Combinatorial Methods, Pantelleria, 2001, in: Lect. Notes Pure Appl. Math., vol. 235, Dekker, New York, 2003, pp. 101–139. [2] Y. Bahturin, M. Zaicev, Group gradings on matrix algebras, Canad. Math. Bull. 45 (2002) 499–508. [3] Y. Bahturin, S. Sehgal, M. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2001) 677–698. [4] A. Baranov, A. Zhilinskii, Diagonal direct limits of simple Lie algebras, Comm. Algebra 27 (1999) 2749–2766. [5] O. Bratteli, Inductive limits of finite dimensional C ∗ algebras, Trans. Amer. Math. Soc. 171 (1972) 195–234. [6] A.A. Chasov, Isomorphisms of graded matrix algebras, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 70 (5) (2004) 12–18; transl. in: Moscow Univ. Math. Bull. 59 (2004) 12–18, (2005). [7] N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Publ., vol. 37, American Mathematical Society, 1964.