Bernstein Superalgebras and Supermodules

Home , Bimodule, Supermodule

Journal of Algebra 212, 119᎐131Ž. 1999 Article ID jabr.1998.7620, available online at http:rrwww.idealibrary.com on

S. Gonzalez,´´´´ M. C. Lopez-Dıaz, and C. Martınez

Departamento de Matematicas,´ Uniërsidad de Oïedo, Oïedo, Spain

and

Ivan P. Shestakov

Sobole¨ Institute of Mathematics, No¨osibirsk, Russia

Communicated by Erwin Kleinfeld

Received April 21, 1998

Bernstein superalgebras are introduced and irreducible Bernstein supermodules are classified. ᮊ 1999 Academic Press

1. INTRODUCTION

A Bernstein algebra is a commutative algebra over a field F with a nonzero algebra homomorphism ␴ : B ª F satisfying the identity

2 Ž.x 22s ␴ Ž.xx2 .

These algebras were first introduced by Lyubichwx 6 and Holgate wx 4 , in order to give an algebraic formulation of the Bernstein problem from mathematical genetics on the description of all the stationary heredity operators. Since then Bernstein algebras were studied by many authors Žseewx 2, 5, 7, 8. . The purpose of this work is to introduce Bernstein superalgebras and to describe irreducible Bernstein supermodules. Since Bernstein algebras do not form a variety of algebras, the definition of a Bernstein superalgebra is not so evident; it is not clear how ‘‘to superize’’ the defining identity which involves the homomorphism ␴ . To overcome this difficulty, we consider instead of Bernstein algebras their nucleus which does form a variety of

119 0021-8693r99 $30.00 Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved. 120 GONZALEZ´ ET AL. graded algebras with respect to the grading induced by Peirce decomposition. We call these graded algebras Ž.U, W -graded Bernstein algebras. The notions of a Ž.U, W -graded Bernstein superalgebra and of supermodule over such a superalgebra are then defined in a natural way. We show that while there are no nontrivial simple or prime Bernstein superalgebras, nontrivial irreducible Bernstein supermodules do exist, and we describe all such supermodules. Throughout the work, F denotes a field of characteristic / 2, and all the algebras are considered over F.

2. Ž.U,W -GRADED BERNSTEIN ALGEBRAS

We recall some known facts about Bernstein algebras and their nucleus Žsee, for example,wx 2, 7, 8. . Any Bernstein algebra B has an idempotent e Ž./ 0 , and with respect to e the algebra B has a Peirce decomposition, s [ [ B Fe UeeW ,1Ž. s g N s 1 s g N s s [ s where UeeÄ4Ä4x B ex 2 x , W x B ex 0 , and N U eeW ␴ s [ ker . The ideal N UeeW is called a nucleus of a Bernstein algebra B. Furthermore, the Peirce components satisfy the relations

2 : 2 : : UeeW , W eeU , UW eeeU ,2Ž. 2 u322s 0, uuwŽ.s 0, uw s 0, Ž.u s 0, Ž.uw 2 s 0, Ž. 3 g g for any u Uee, w W . If e is a fixed idempotent of B then the set IdŽ. B of all the idempotents of B is given by s q q 2 N g IdŽ. B Ä4e u u u Ue ,4 Ž. X and for any idempotent e s e q u q u2 holds

X s q N g UeeÄ4x 2ux x U ,5Ž.

X s y q 2 N g WeeÄ4y 2Ž.u u y y W .6 Ž.

Moreover, the dimensions of Ueeand W do not depend on the idempotent e. One of difficulties in the study of Bernstein algebras is caused by the fact that they do not form a variety of algebras. Now we give a definition of a Ž.U, W -graded Bernstein algebra that allows us to study instead of Bern- stein algebras their graded nucleus, which does form aŽ. graded variety. BERNSTEIN SUPERALGEBRAS 121

DEFINITION 2.1. Let A s U q W be a commutative algebra over a field F which is a direct sum of vector subspaces U and W such that

U 2 : W, W 2 : U, UW : U.

Then, A is called a Ž.U, W -graded Bernstein algebra or simply a Ž.U, W - Bernstein algebra if A satisfies the identities

2 u322s 0, uuwŽ.s 0, uw s 0, Ž.u s 0, Ž.uw 2 s 0, for any u g U, w g W. Observe that a Ž.U, W -Bernstein algebra A s U q W is graded by the commutative groupoid S s Ä s, t N s22s t, t s s, st s ts s s4, if we set s s AstU, A W. This justifies the term graded Bernstein algebra.

PROPOSITION 2.1.Ž. i For any Bernstein algebra B and any idempotent e the nucleus NŽ. B is a Ž Uee, W .-Bernstein algebra. Ž.ii Let A s U q W be a commutati¨e algebra which is a direct sum of subspaces U and W. Consider the ¨ector space direct sum AŽ. e s Fe q A and extend a multiplication from A to AŽ. e by setting

1 e2 s e, eu s u, ew s 0 for any u g U, w g W. 2

Furthermore, define the mapping ␴ : AeŽ.ª Fby␴␣ Že q u q w .s ␣. Then, AisaUŽ., W -Bernstein algebra if and only if B s A Ž. e is a Bernstein s s s algebra. In such case NŽ. B A and Uee Ž. B U, WB Ž. W. Proof. It suffices to apply propertiesŽ. 2 and Ž. 3 of the Peirce decomposition of B with respect to e and to check in a straightforward way that the algebra AeŽ.satisfies the identity Žx 22 .s ␴ Ž.xx 2 2. A commutative algebra A may carry nonisomorphic Ž.U, W -Bernstein structures, as the following examples show.

EXAMPLE 1. Let A be a commutative algebra over F with the basis Ä4a, b, c, ac, bc and the only nonzero products a и c s ac and b и c s bc. Then A3 s 0, so identitiesŽ. 3 hold in A for every grading. Consider the gradings: Ž.1 U s Ä4ac, bc and W s Ä4a, b, c . We have U 22s 0, W : U, and UW s 0, hence A is a Ž.U, W -Bernstein algebra. X X X X Ž.2 U s Ä4c, ac, bc and W s Ä4a, b . In this case U 2 s 0, W 2 s 0, XX X XX and UW : U . Thus, A is also a ŽU , W .-Bernstein algebra. 122 GONZALEZ´ ET AL.

X Since dim U s 2 and dim U s 3, the graded algebras A s U q W and X X A s U q W are not isomorphic. Moreover, the corresponding Bernstein algebras AeŽ.are not isomorphic. The next example is more delicate.

EXAMPLE 2. Let A be a commutative F-algebra with the basis s s Ä4u12, u , w 12, w and the only nonzero products uw 11u 2, uw 22u 1. Con- sider the gradings: s s 22s s : Ž.1 U Ä4u12, u , W Äw 12, w 4. Then U W 0, UW U, and identitiesŽ. 3 are satisfied, so A is a ŽU, W .-Bernstein algebra. XXXXX X Ž.2 U s U, W s Fw q Fw , where w s w y 2u , w s w y X X 12XXXX 11222 2u . Then U 2 s 0, W 2 : U , UW : U , and again identitiesŽ. 3 are 1 XX X satisfied, so A is a ŽU , W .-Bernstein algebra. But in this case W 2 / 0, so X X A s U q W and A s U q W are not isomorphic as graded algebras. Nevertheless, in this example the corresponding Bernstein algebras X X X X B s Fe q U q W and B s Fe q U q W are isomorphic: the map ␾: ¬ X y q ¬ ¬ XXq ¬ q e e Ž.u12u , u u, w 1w 12u 222, w w 2u 1is an isomorphism of algebras. The last example suggests the following definition, that will provide an equivalence relation between Ž.U, W -graded Bernstein algebras, so that two such algebras are equivalent if and only if they come from the same Bernstein algebra.

DEFINITION 2.2. Let A be a Ž.U, W -Bernstein algebra. For each u g U we define a u-isotope AŽu. s U Žu. q W Žu. of A as the same algebra A with the new grading:

U Žu. s Ä4x q 2ux N x g U , W Žu. s Ä4y y 2Ž.u q u2 y N y g W .

PROPOSITION 2.2. Let A be aŽ. U, W -Bernstein algebra and u g U. Then AisaUŽu. Ž Žu.Ž, W u..-Bernstein algebra. X Proof. Consider the Bernstein algebra AeŽ.. By Ž. 4 , we have e s e q u q u2 g IdŽŽ.. A e , and so by Ž. 1 , Ae Ž.has the following Peirce decompo- X s q XXq Xs Žu.Ž Xs u. sition: AeŽ. Fe UeeW . ByŽ. 5 and Ž. 6 , U eU and W eW . Now, by Proposition 2.1Ž. i , AŽu.Žs U u.Žq W u. is a ŽU Žu.Ž, W u..-Bernstein algebra. XX X DEFINITION 2.3. Let A s U q W and A s U q W be two Ž.U, W - X Bernstein algebras. We say that A is isotopic to A if A is isomorphic as a X X X X graded Bernstein algebra to A Žu . for some u g U . BERNSTEIN SUPERALGEBRAS 123

X PROPOSITION 2.3. Let A and A beŽ. U, W -Bernstein algebras. Then A is X X X isotopic to A if and only if AŽ. e is isomorphic to AŽ e .. X X X Proof. Let A s U q W, A s U q W . Assume first that A is isotopic X X g X ª X to A . Then there existX u U and anX isomorphism of algebras f: A A Ž.s XŽu . Ž.s XŽu . Ž. XŽ X . such that fU U , fW W . Consider AeXXand Ae. Then s XXXq q 2 g XXs XŽu . s XŽu . e e u u IdŽŽ..Ž. A e by 4 , and UeeU , W W by Ž. 5 XXs q q ª XX andŽ. 6 . Therefore, Ae Ž.Fe UeeW . Now, consider f: Ae Ž.Ae Ž. which extends f and sends e to e. Evidently, f is an isomorphism of X X algebras, so AeŽ.is isomorphic to AeŽ .. X X Conversely, let f: AeŽ.ª AeŽ . be an isomorphism of algebras. Then X X XX X feŽ.g Id Ž AŽ e ..and so by Ž 4 . there exists u g U such that feŽ.s e q X X u q u 2. Therefore, X X XŽu . s s s ŽX u . s fUŽ.fU Žef . U Že. Ue U ,

X X XŽu . s s s ŽX u . s fWŽ.fW Ž.efW Že. We W . X X X X Consider A Žu .. The restriction f N : A ª A Žu . is an isomorphism of A X graded Bernstein algebras, hence A is isotopic to A .

COROLLARY 2.1. Isotopy ofŽ. U, W -Bernstein algebras is an equi¨alence relation. Thus the classification of Bernstein algebras up to isomorphism is equivalent to the classification of Ž.U, W -Bernstein algebras up to isotopy.

3. BERNSTEIN SUPERALGEBRAS

Recall that a superalgebra, in general, is a Z2-graded algebra, that is, an s q algebra A A01A which is a direct sum of subspaces A 0Žthe even : part.Ž and A1 the odd part . , with a product satisfying AAijA iqj Žmod 2.. We will define Bernstein superalgebras according to the general definition of a superalgebra in an arbitrary homogeneous variety of algebras Žseewx 10. . ⌫ s N s Let alg²: 1, gi i 1, 2, . . . be the Grassmann Ž.or exterior F-algebra 2 s sy over a countable number of generators giiijji, with g 0, gg gg for / иии - - иии - ⌫ i j. The elements 1, ggiig i, i12i ir form an F-basis of . ⌫ 12 r ⌫ If we denote by 01Ž.respectively by the span of the products of even ⌫ s ⌫ q ⌫ lengthŽ. respectively of odd length , then 01is a superalgebra. One can easily check that this superalgebra is supercommutati¨e, that is, satisfies the identity ␦ ␦ ab syŽ.1 a b ba, g ⌫ j ⌫ g ␦ s where a, b 01and for a Ai, a i. 124 GONZALEZ´ ET AL.

s q ⌫ s ⌫ m q For a superalgebra A A01A , the subalgebra Ž.A 00A ⌫ m ⌫ m ¨ 11A of the tensor product A is called the Grassmann en elope of A. One can also consider the Grassmann envelope ⌫Ž.B of any homogeneous subspace B of A, which in general is not a subalgebra but a subspace of ⌫Ž.A . s q DEFINITION 3.1. A superalgebra A A01A , with a homogeneous s q s s q s Ž.U, W -grading AiiU W i, i 0, 1, where U U01U and W q W01W , is called a Ž.U, W -Bernstein superalgebra, or simply a Bernstein superalgebra, if its Grassmann envelope ⌫Ž.A is a ŽŽ.⌫ U , ⌫ ŽW ..-Bernstein algebra. s q s q PROPOSITION 3.1. A superalgebra A A01A , with AiiU W i for i s 0, 1, is aŽ. U, W -Bernstein superalgebra if and only if it is supercommutati¨e and satisfies the relations: : : : g Ž.1 UUijW iqjiji, WW U qjij, UW U iqj, for any i, j Ä40, 1 , 3 s 2 2 s g Ž.2 a Ža . 0, for any a U0 , ␦ ␦ и qy u2 u3 и q и s Ž.3 uu12u 3 Ž1 . uu 13u 2u 1uu 23 0, ␦ ␦ и qy u1 u2 и s Ž.4 u12uw Ž1 . u 21uw 0, и s Ž.5 u ww12 0, ␦ ␦ ␦ ␦ q␦ и qy u2 u3 и qy u4Ž u2 u3. и Ž.6 uu12uu 34 Ž1 . uu 13uu 24Ž.1 uu 14 s uu23 0, ␦ ␦ и qy w1 w 2 и s Ž.7 uw11wu 22 Ž1 . uw 12wu 12 0, g j g j for any u, ui U01U , w, wi W01W . The proof is straightforward. Note that identitiesŽ. 2 are superfluous if char F / 3. The next proposition is a superanalogue of the corresponding result fromwx 1, 3 for Bernstein algebras. s q s q PROPOSITION 3.2. Let A A01A U W be a Bernstein superalgebra. Consider L s LAŽ.s Ä4u g U N uU s 0.Then Ž.i L is a graded ideal of A and L2 s 0. Ž.ii The quotient superalgebra ArL satisfies the graded identities

a3 s 0, for any e¨en a,7Ž. ␦ ␦ и qy x 23x и q и s xx12x 3Ž.1 xx 13x 2x 1xx 23 0, Ž. 8

r for any homogeneous elements xi. In particular, the superalgebra A Lis Jordan. BERNSTEIN SUPERALGEBRAS 125

Proof. It is clear that L is homogeneous and that 0 s LU ; L. Fur- thermore, it follows from Proposition 3.1Ž. 4 that U и LW : L и UW : LU s 0, hence LW : L and L is an ideal of A. Besides, L2 : LU s 0, so Ž.i is proved. To proveŽ. ii , observe first that ArL satisfies the graded identities Ž. 7 andŽ. 8 if and only if its Grassmann envelope ⌫ ŽArL .satisfies the identity x 3 s 0. It is easily seen that ⌫Ž.ArL is isomorphic to the quotient algebra ⌫Ž.A r⌫ Ž.L . We claim that ⌫ Ž.L s L ŽŽ..⌫ A s Ä x g ⌫Ž.U N x⌫ Ž.U s 0.4 ⌫ : ⌫ s m g ⌫ In fact, clearly Ž.L L ŽŽA ... Conversely, let x Ýiig u i L ŽŽA .., g ⌫ with the elements gi homogeneous and linearly independent. For g j g ⌫ j ⌫ ␦ s ␦ any u U01U choose g 01such that u g and the elements m g ⌫ m s ggi are still linearly independent. Then g u Ž.U ,so xg Žu . m s s g Ýiigg uu i0, which yields uu i0 for every i. Therefore u i LAŽ. for all i, and x g ⌫Ž.L as required. It remains to note that the quotient algebra ⌫Ž.A rL ŽŽ..⌫ A satisfies the identity x 3 s 0 bywx 3 . It is well known that a commutative algebra satisfying this identity is Jordan, hence the algebra ⌫Ž.ArL is Jordan and so is the superalgebra ArL. Note thatŽ. 7 follows from Ž. 8 if char F / 3.

THEOREM 3.1. There are no semiprime Bernstein superalgebras o¨er a field F of characteristic / 2, 3. Proof. Assume that A is a semiprime Bernstein superalgebra. Since L s LAŽ.is a graded ideal of A with L2 s 0, we have L s 0. Therefore A satisfiesŽ. 8 and is Jordan superalgebra. Consequently ⌫ ŽA .is a Jordan algebra satisfying x 3 s 0. Bywx 9 , ⌫Ž.A 2 is nilpotent, which clearly forces that A2 is nilpotent. It is easily seen that a semiprime superalgebra satisfyingŽ. 8 does not contain nilpotent ideals, so A2 s 0 and A s 0, a contradiction.

COROLLARY 3.1. There are no simple Bernstein superalgebras o¨er a field F of characteristic / 2, 3. In the next section we will introduce Bernstein supermodules and will show that irreducible Bernstein supermodules do exist.

4. BERNSTEIN SUPERMODULES

s q Let A A01A be a superalgebra over F and M be an A-bimodule. s q Then M is a superbimodule over A if it has a Z201-grading M M M such that the corresponding split extension E s A q M is a superalgebra s q s s q with the grading EiiA M i, i 0, 1. In other words, M M01M is 126 GONZALEZ´ ET AL.

s q a superbimodule over A A01A if q : AMijMA ijM iqj Žmod 2. . A superbimodule M is supercommutati¨e if ma syŽ.1 ␦ a ␦ mam for any homogeneous m g M and a g A.

DEFINITION 4.1. Let A s U q W be a Ž.U, W -Bernstein superalgebra and M a supercommutative A-supermodule. Assume that M admits a direct vector space decomposition M s X q Y which is compatible with s q s l s l the Z2-grading on M, that is, MiiiX Y for X iX M ii, Y Y M i, i s 0, 1. Then, M is called an Ž.X, Y -Bernstein supermodule over A if the split extension E s A q M is a Ž.U q X, W q Y -Bernstein superalgebra, s q s with EiiA M i, i 0, 1. Applying the definition of Bernstein superalgebra, we get:

PROPOSITION 4.1. Let A s U q WbeaUŽ., W -Bernstein superalgebra and M a supercommutatiëA-supermodule with anŽ. X, Y -decomposition M s X q Y. Then M is anŽ. X, Y -Bernstein supermodule oër A if and only if Ž.1 XU : Y, YW : X, YU : X, XW : X, ␦ ␦ и qy u1 u2 и q и s Ž.2 xu12u Ž1 . xu 21u x uu 120, Ž.3 x и wu q xw и u s 0, ␦ ␦ и qy u1 u2 и s Ž.4 yu12u Ž1 . yu 21u 0, и s Ž.5 x ww12 0, Ž.6 yw и u s 0, ␦ ␦ ␦ ␦ q␦ и qy u1 u2 и qy u3Ž u1 u2 . и Ž.7 xu123uu Ž1 . xu 213uu Ž.1 xu 312uu s 0, ␦ ␦ и qy w1 w 2 и s Ž.8 xw12wu Ž1 . xw 21wu 0, ␦ ␦ и qy u1 u2 и s Ž.9 yu12uw Ž1 . yu 21uw 0, g g g g for any u, u123, u , u U, w, w 12, w W, x X, y Y. COROLLARY 4.1. LetAbeaUŽ., W -Bernstein superalgebra with U s 0 Ž.or, equiälently, A s W . Then any supercommutatiëA-supermodule M with anŽ. X, Y -decomposition M s X q Y such that MA : Xisan Ž. X, Y - Bernstein A-supermodule. Proof. Indeed, in this case conditionsŽ.Ž. 1 ᎐ 9 are obviously satisfied.

COROLLARY 4.2. Let M s X q YbeanXŽ., Y -Bernstein supermodule o¨er A s U q W. Denote by M s the same module M but with the opposite s s s s grading, that is, Ž M .Ž01M , M . 10M , and with the new action of A, ␦ a и m syŽ.1 a am, m и a s ma. BERNSTEIN SUPERALGEBRAS 127

Then Ms is again anŽ. X, Y -Bernstein supermodule o¨er A, which is irreducible if and only if so is M. The proof is straightforward. Notice that the supermodule M s in general may be not isomorphic to M. Before giving examples of irreducible Bernstein supermodules, we will prove a general result on irreducible supercommutative supermodules. s q ¨ PROPOSITION 4.2. Let A A01A be a supercommutati e superalgebra ¨ s q ¨ o er F and M M01M an irreducible supercommutati eA-supermodule. Then M is an irreducible A-module. Proof. Let N be a proper submodule of M. Consider the projections

␲ ª g ii: M M , i Ä40,1 , ␲ q s ␲ q s given by 00Ž.m m 1m 0, 10 Ž.m m 1m 1. Since M is irreducible l s l s as a graded module, we have N M01N M 0. It follows easily that ␲ / s ␲ q ␲ iŽ.N 0 for both i 0, 1, and the graded submodule 01 Ž.N Ž.N is ␲ q ␲ s ␲ s nonzero. Therefore 01Ž.N Ž.N M, and hence 0 Ž.N M 0and ␲ s 11Ž.N M . s q g Let n m01m be an arbitrary element from N. For any x A1 we have s q g xn xm01xm N, s q s y g nx mx01mx xm 01xm N. g l Subtracting the second equality from the first one, we get 2 xm10N M s s ␲ s s q g 0, which yields 0 A11Ž.N AM 1 1. Consequently, xn xm 0 0 l s s s N M11010, and as before, AM 0. Resuming, we have AM 0. s s Thus M is an irreducible A010-module, which yields M 0orM 0, a contradiction. Consider now two examples of irreducible Bernstein supermodules. Ž.1 Supermodules of Type M Ž K, V, ␣ .. Let K be a field extension of F, ␣ g s s 2 s V an F-subspace of K, and K. Consider A A1 Vu with A 0 s q s s and M Kx Ky, where M01Kx and M Ky. Define an action of A on M by extending K-linearly the conditions xu s ux s y, yu syuy s ␣ x. Then a trivial verification shows that M s MKŽ.Ž., V, ␣ is an X, Y - s q s s s s Bernstein supermodule over A U 0, with X X01Kx, Y Y Ky. Moreover, it can be easily seen that the corresponding split extension E s A q M satisfies superidentitiesŽ.Ž. 7 , 8 , hence E is a Jordan superalgebra and M is a Jordan A-supermodule. 128 GONZALEZ´ ET AL.

s s s s s Observe that the opposite supermodule M with M01Ky, M Kx, and yu s uy s ␣ x, xu syux s y, is isomorphic to M as a Jordan supermodule but not as Bernstein supermodule.

PROPOSITION 4.3. The supermodule M s MKŽ., V, ␣ is irreducible if and s ␣ 2 only if K alg F ² V :. s ␣ 2 / Proof. Assume that K alg F ² V :, and let N 0 be a subsuper- module of M. Obviously, there exists ␤ x g N for some 0 / ␤ g K.We have Ž.xA A s Ž␣Vx2 .Ž,so ␣V 2 .Ž.Ž.␤ x s ␤ xA A : NA A : N. Hence s и ␤ s ␣ 2 ␤ : s : Kx Ž.K x alg F ²V :Ž.x N and Ky Kx A N. Therefore M s N. Conversely, if M is irreducible then the submodule N generated by x s coincides with M, hence N0 Kx. On the other hand, it is easily seen that : ␣ 22s ␣ N0 alg FF²:Vx. Thus K alg ²:V . Remark 4.1. If F is algebraically closed then every supermodule M s MKŽ., V, ␣ of finite dimension over F and with ␣ / 0 is isomorphic to MFŽ., F,1 . Proof. Indeed, in this case dim K - ϱ and K s V s F. Moreover, XXF considering u s Ž.1r ''␣ u and y s Ž.1r ␣ y, we can always suppose ␣ s 1.

Ž.2 Supermodules of Type M Ž W, X .. Let M be a Z2-graded vector space and A a graded subspace of EndŽ.M . Consider A as a superalgebra with A2 s 0 and define an action of A on M by setting

␦ ␦ am syŽ.1 m a ma s am Ž., for any homogeneous a g A, m g M.

Then M is a supercommutative faithful A-supermodule. Set A s W, M s X. Then it follows from Corollary 4.1 that M is an Ž.X, 0 -Bernstein supermodule over theŽ. 0, W -Bernstein superalgebra A. We will call this supermodule a supermodule of type MŽ. W, X . Observe that any graded A-submodule of M s MWŽ., X is obviously also a graded Bernstein submodule, so M is irreducible as a Bernstein supermodule if and only if it is an irreducible A-supermodule, which, in turn, by Proposition 4.2 is equivalent to the irreducibility of M as A-module. The last condition is given by requiring the algebra alg EndŽ M .²:W to be dense in EndŽ.M . In particular, if M is finite dimensional over F, this is equivalent to alg²:W s End Ž.M . Recall that an A-Ž. super module M is called almost faithful if its annihilator Ann M s Ä4a g A N Ma s 0 does not contain nonzero ideals of A. BERNSTEIN SUPERALGEBRAS 129

THEOREM 4.1. Let A be aŽ. U, W -Bernstein superalgebra o¨er a field F of characteristic / 2, 3 and M be an almost faithful irreducibleŽ. X, Y -Bernstein supermodule o¨er A. Then M is isomorphic to a supermodule of one of the types: MKŽ.Ž., V, ␣ , MK, V, ␣ s, MWŽ., X . Proof. Consider the split extension E s A q M, and let LMŽ.s LEŽ.l M s Ä x g X N xUŽ.q X s 04 . By Proposition 3.2, LEŽ.is a graded ideal of E, hence LMŽ.is a graded Bernstein submodule of M. Therefore, either LMŽ.s 0orLM Ž.s M. Suppose first that LMŽ .s 0, then LEM Ž.: M l LE Ž.s LM Ž .s 0, and since M is an almost faithful supermodule, we have LEŽ.s 0. By Proposition 3.2Ž. ii , E is a Jordan superalgebra satisfying the identity Ž. 8 , and the Grassmann envelope ⌫Ž.E satisfies the identity x 3 s 0. Since charF / 2, 3, bywx 9 we have

2 ž/ž/Ž.⌫Ž.E 2222⌫ Ž.E ⌫ Ž.E ⌫ Ž.E s 0. A standard passage from E to ⌫Ž.E now implies Ž see, for example,wx 10 .

2 ž/ž/Ž.EEEE2222s 0, and so ŽŽŽ MA и AAAA2 . 2 . 2 . 2 s 0. Since AM / 0 is a Bernstein subsuper- module of M, we have AM s M and so Ž.Ž.Ž.M и A2 A2 A2 A2 s 0. Ž. 9 ByŽ. 8 , for any m g M, a g A2, b g A holds

␦ ␦ ma и b syŽ. y1 a b mb и a y m и ab. Therefore MA2 и A : MA и A2 q M и A2A and MA2 is a submodule of M which is evidently graded and Bernstein. Since M is irreducible and almost faithful, either A2 s 0orMA2 s M. If MA2 s M then M s 0 byŽ. 9 , a contradiction. Thus A2 s 0, Ann M s 0, and A satisfies

␦ ␦ ma и b qyŽ.1 a b mb и a s 0, Ž. 10 for any m g M, a, b g A. Taking here m s x g X, a s u g U, b s w g W, we obtain xu и w s "xw и u g XW и U : XU : Y. On the other hand, xu и w g XU и W : YW : X. Since X l Y s 0, we deduce that XU и W s XW и U s 0. Furthermore, by Proposition 4.1Ž. 6 , we have YW и U s 0, hence byŽ. 10 also YU и W s 0. Thus M s MA s MU q MW s MA и U q MA и W s MU и U q MW и W. 130 GONZALEZ´ ET AL.

s и s и It can be easily seen that MUWMU U and M MW W are Bernstein l s s subsupermodules of M with MUWM 0, hence either M M Uor s M MW . s s s Assume first that M MW , then MU 0 and so U 0. Moreover, s и s и q и : : s MW MW W XW W YW W XW X, hence M X and we have LMŽ.s M, a contradiction. s s Thus M MU and W 0 as above. It follows fromŽ. 10 that for any homogeneous u g U the subset Mu is a graded Bernstein submodule of M, hence either Mu s M or Mu s 0. If u is even then byŽ. 10 , Mu и u s 0, s s s hence Mu 0. By faithfulness of M, U010 and so A U . Besides, : : s q since XUi 11Y yii, YU11X yi, i 0, 1, it follows that X01Y and q s X10Y are Bernstein subsupermodules of M, and therefore either M q s q X01Y or M X 10Y . It can be now easily seen that any graded submodule of M would automatically be a graded Bernstein submodule of M, and so M is irreducible as an ordinaryŽ. not Bernstein A-supermodule. Moreover, by Proposition 4.2, M is just an irreducible A-module. In particular, the centroid CMŽ.of the A-module M is a division algebra. For any u g A s U consider the mapping R : M ª M defined by 1 u X s wxX s g RmuuuŽ.mu. It follows from Ž. 10 that R , R 0 for any u A and / therefore Ru belongs to the centroid CMŽ.. Observe that Ru 0 for any 0 / u g A. Denote by K the F-subalgebra of CMŽ.generated by the set X X N g Ä RRuu u, u A4. We claim that K is a field. Indeed, K is clearly a / ␣ g / g commutative domain. Let 0 K,0 m M0 , then it is easy to check that the submodule N generated by the element m␣ is given by N s Ž.m␣ K q Ž.m␣ K и A. Since N s M, it follows m g N, and therefore, g ␤ g ␣␤ s ␣␤ s since m M0 , there exists K such that m m. Then 1 and K is a field. / g s / 2 s ␣ g X g Fix 0 u A U1. Then 0 Ru K and for any u A holds X ␣ s X 2 s X g : RuuuuuuuR R RR R RK. Thus R AuRK, and it follows from s и q и / g the above that M m K mu K for any fixed 0 m M0. Consider s ␥ g N ␥ g V Ä4K RuAR which is clearly an F-subspace of K. Since VRu s s 22s 2␣ s s 2␣ RAAAu, we have RR VR V ,so K alg²:²RR AAalg V :. s Finally, we can identify A with RAuand get A VR . s q ␣ Now, if M X01Y then it is evidently isomorphic to MKŽ., V, , and s q if M X10Y then it is isomorphic to the opposite supermodule MKŽ., V, ␣ s. This finishes the case LMŽ.s 0. Consider now the case LMŽ.s M. By definition of LM Ž.we have M s X and MU s 0. Let I s U q U 2. Then IU : U 2 q WU : I and IW : UW q W 2 : U, hence I is an ideal of A. Furthermore, MI s MU 2 s LMUŽ.2 : ŽŽ..by Proposition 4.1 2 : LMUŽ.и U s 0, BERNSTEIN SUPERALGEBRAS 131 and since M is almost faithful, U s 0. Thus A is aŽ. 0, W -Bernstein algebra. Since A2 s 0, Ann M is an ideal of A and is therefore zero. Thus M is faithful and we can suppose without loss of generality that A is a graded subspace of End M. Therefore, M is a module of the type MWŽ., X , and the theorem is proved. The following corollary gives a description of irreducible Bernstein modules.

COROLLARY 4.3. Let A be anŽ. U, W -Bernstein algebra and M be an almost faithful irreducibleŽ. X, Y -Bernstein A-module. Then U s 0, Y s 0, and M is isomorphic to a module of type MŽ. W, X where the subalgebra alg²:W is dense in End Ž.M . Proof. The algebra A and the module M can be considered as superalgebra and supermodule, with zero odd parts. Now it follows from Theo- rem 4.1 that the only possibility for M is to be isomorphic to a module of type MWŽ., X .

REFERENCES

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