BIMODULES OVER SIMPLE FINITE-DIMENSIONAL JORDAN SUPERALGEBRAS - PDF document

BIMODULES OVER SIMPLE FINITE-DIMENSIONAL JORDAN SUPERALGEBRAS Consuelo Mart´ ınez L´ opez Workshop on Nonassociative Algebras Toronto, 12-14 May 2005

SUPERALGEBRA : A = A ¯ 0 + A ¯ 1 , A ¯ i · A ¯ j ⊆ A ¯ i + j a Z/ 2 Z -graded algebra EX. V vector space of countable dimension, G ( V ) = G ( V ) ¯ 0 + G ( V ) ¯ 1 Grassmann algebra over V , G ( A ) = A ¯ 0 ⊗ G ( V ) ¯ 0 + A ¯ 1 ⊗ G ( V ) ¯ 1 ≤ A ⊗ G ( V ) Grassmann enveloping algebra of A V a variety of algebras (associative, Lie, Jordan,...) DEF. A = A ¯ 0 + A ¯ 1 is a V -superalgebra if G ( A ) ∈ V . J = J ¯ 0 + J ¯ 1 is a Jordan superalgebra if it satisfies SJ1. Supercommutativity a · b = ( − 1) | a || b | b · a , SJ2. Super Jordan identity ( a · b ) · ( c · d ) + ( − 1) | b || c | ( a · c ) · ( b · d )+ ( − 1) | b || d | + | c || d | ( a · d ) · ( b · c ) = (( a · b ) · c ) · d + ( − 1) | c || d | + | b || c (( a · d ) · c ) · b + ( − 1) | a || b | + | a || c | + | a || d | + | c || d | (( b · d ) · c ) · a .

JORDAN SUPERALGEBRAS A = A ¯ 0 + A ¯ 1 associative superalgebra A (+) = ( A, a · b = 1 2 ( ab +( − 1) | a || b | ba ) Jordan super- algebra 1 ≤ A (+) special . Otherwise excep- J = J ¯ 0 + J ¯ tional (A) A (+) , A = M m + n ( F ) full linear superalgebra � � a b (Q) A (+) , A = { | a, b ∈ M n ( F ) } b a If ⋆ : A → A is an involution : ( a ⋆ ) ⋆ = a , ( ab ) ⋆ = ( − 1) | a || b | b ⋆ a ⋆ . H ( A, ⋆ ) = { a ∈ A | a ⋆ = a } ≤ A (+) � � I m 0 (BC) M m +2 n ( F ), Q = , 0 S 2 n 0 1 . . .   − 1 0 . . .   S 2 n = . . . . .     . . . 0 1   . . . − 1 0 � � � a T − c T � a b → Q − 1 ⋆ : Q , a ∈ M m ( F ), b T d T c d d ∈ M 2 n ( F ), H ( A, ⋆ ) = osp m , 2n ( F ).

� � � d T − b T � a b (P) A = M n + n ( F ), ⋆ : → , c T a T c d � � a b | a, b, c ∈ M n ( F ) , b T = − b, H ( A, ⋆ ) = { a T c c T = c } . (D) A Superalgebra of a superform V = V ¯ 0 + V ¯ 1 , <, > : V × V → F a supersymmetric bilinear form J = F 1 + V = ( F 1 + V ¯ 0 ) + V ¯ 1 , ( α 1 + v )( β 1 + w ) = ( αβ + < v, w > )1 + ( αw + βv ). ( D t ) J t = ( Fe 1 + Fe 2 ) + ( Fx + Fy ), t � = 0 i = e i , e 1 e 2 = 0 , e i x = 1 2 x, e i y = 1 e 2 2 y, [ x, y ] = e 1 + te 2 . (J) All simple Jordan algebras

(F) The 10-dimensional exceptional Kac superalgebra K 10 = [( Fe 1 + � 4 i =1 Fv i )+ Fe 2 ]+( � 2 i =1 Fx i + Fy i ) e 2 i = e i , e 1 e 2 = 0 , e 1 v i = v i , e 2 v i = 0 , v 1 v 2 = 2 e 1 = v 3 v 4 , e i x j = 1 e i y j = 1 2 x j , 2 y j , i, j = 1 , 2 y 1 v 1 = x 2 , y 2 v 1 = − x 1 , x 1 v 2 = − y 2 , x 2 v 2 = y 1 , x 2 v 3 = x 1 , y 1 v 3 = y 2 , x 1 v 4 = x 2 , y 2 v 4 = y 1 , [ x i , y i ] = e 1 − 3 e 2 , [ x 1 , x 2 ] = v 1 , [ y 1 , y 2 ] = v 2 , [ x 1 , y 2 ] = v 3 , [ x 2 , y 1 ] = v 4 . (K) The 3-dimensional Kaplansky superalgebra e 2 = e, ex = 1 K 3 = Fe + ( Fx + Fy ) , 2 x, ey = 1 2 y, [ x, y ] = e . Theorem. (Kac 77, Kantor 89) A simple finite dimen- sional Jordan superalgebra over an algebraically closed field of zero characteristic is isomorphic to one of the su- peralgebras A, BC, D, P, Q, D t , F, K, J listed above or to a superalgebra obtained by the Kantor-double process

Theorem. (Racine, Zelmanov, J. of Algebra 270, 2003) Every simple Jordan superalgebra over an algebraically closed field F , ch F = p > 2 , with its even part semisim- ple is isomorphic to one of the superalgebras mentioned above + Some additional examples in char 3 Jordan Superalgebras defined by Brackets Γ = Γ ¯ 0 + Γ ¯ 1 an associative commutative superalgebra { , } : Γ × Γ → Γ a Poisson bracket if { Γ ¯ i , Γ ¯ j } ⊆ Γ ¯ i + j and (1) (Γ , { , } ) is a Lie superalgebra, (2) { ab, c } = a { b, c } +( − 1) | b || c | { a, c } b ( Leibniz iden- tity ) Kantor Double Superalgebra ( bx ) a = ( − 1) | a | ( ba ) x , J = Γ + Γ x , a ( bx ) = ( ab ) x , ( ax )( bx ) = ( − 1) | b | { a, b } , J ¯ 0 = Γ ¯ 0 + Γ ¯ 1 x , J ¯ 1 = Γ ¯ 1 + Γ ¯ 0 x . Theorem. (Kantor 1992) Let { , } be a Poisson bracket = ⇒ J = Γ + Γ x is a Jordan superalgebra. Kantor superalgebra Γ = Grassman algebra on ξ 1 , . . . , ξ n 1 , { f, g } = � n i =1 ( − 1) | f | ∂f ∂g Γ = Γ ¯ 0 + Γ ¯ ∂ξ i ∂ξ i � n = 1 J ≃ D ( − 1) J = Γ + Γ x n ≥ 2 J ¯ is not semisimple 0

CHENG-KAC JORDAN SUPERALGEBRAS Z unital associative commutative algebra, d : Z → Z a derivation, 0 = Z + � 3 CK ( Z, d ) = J ¯ 0 + J ¯ 1 , J ¯ i =1 w i Z , J ¯ 1 = xZ + � 3 i =1 x i Z free Z-modules of rank 4. Even part w i w j = 0 , i � = j, w 2 1 = w 2 2 = 1 , w 2 3 = − 1, Notation: x i × i = 0 , x 1 × 2 = − x 2 × 1 = x 3 x 1 × 3 = − x 3 × 1 = x 2 , − x 2 × 3 = x 3 × 2 = x 1 . Module action f, g ∈ Z g w j g xf x ( fg ) x j ( fg d ) x i fx i ( fg ) x i × j ( fg ) Bracket on M xg x j g xf f d g − fg d − w j ( fg ) x i f w i ( fg ) 0 CK ( Z, d ) is simple ⇐ ⇒ Z does not contain proper d-invariant ideals.

B ( m ) = F [ a 1 , . . . , a m | a p i = 0] B ( m , n ) = B ( m ) ⊗ G ( n ) G ( n ) = < 1 , ξ 1 , . . . , ξ n > Theorem. (M., Zelmanov, J. of Algebra 236, 2001) Let J = J ¯ 0 + J ¯ 1 be a finite dimensional simple unital Jordan superalgebra over an algebraically closed field F , ch F = p > 2, J ¯ 0 not semisimple. Then J ≃ B ( m , n ) + B ( m , n ) x a Kantor double or J ≃ CK ( B ( m ) , d ) . SPECIALITY King, McCrimmon (J. Algebra 149, 1995) - The Kantor Double of a bracket of vector field type ( { a, b } = a ′ b − ab ′ ′ a derivation) is special. - The Kantor Double of { f, g } = ∂f ∂g ∂y − ∂f ∂g ∂x on ∂x ∂y F [ x, y ] is exceptional. Shestakov (1993) - A Kantor Double of Poisson bracket <, > : Γ × Γ → Γ is special iff << Γ , Γ >, Γ > = (0). - A Kantor Double of a Poisson bracket is i-special (homomorphic image of a special superalgebra)

Theorem. (M., Shestakov, Zelmanov) A Kantor Dou- ble of a Jordan bracket is i-special. Assumption: J = Γ + Γ x does not contain � = (0) nilpo- tent ideals - If Γ = Γ ¯ 0 then J is special iff <, > is of vector field type. - If Γ ¯ 1 Γ ¯ 1 � = (0) (at least 2 Grassmann variables) then J is exceptional. - If Γ = Γ ¯ 0 + Γ ¯ 0 ξ 1 , < Γ ¯ 0 , ξ 1 > = (0) , < ξ 1 , ξ 1 > = − 1 then J is special iff <, > : Γ ¯ 0 × Γ ¯ 0 → Γ ¯ 0 is of vector field type. Theorem. (M., Shestakov, Zelmanov) The Cheng-Kac superalgebra CK ( Z, d ) is special The embedding extends McCrimmon embedding for vector field type brackets. W = < R ( a ) , a ∈ Z, d > - differential operators on Z R = R ¯ 0 + R ¯ 1 = M 4 × 4 ( W )

Let J be a special Jordan superalgebra. A specialization u : J − → U into an associative algebra U is said to be universal if U = < u ( J ) > and for an arbitrary specialization ϕ : J → A there exists a homomorphism of associative algebras ξ : U → A such that ϕ = ξ · u . The algebra U is called the universal associative enveloping algebra of J . An arbitrary special Jordan superalgebra contains a unique universal specialization u : J → U . U is equipped with a superinvolution * having all elements from u ( J ) fixed, i.e., u ( J ) ⊆ H ( U, ∗ ). We call a special Jordan superalgebra reflexive if u ( J ) = H ( U, ∗ ). Theorem. U ( M (+) m,n ( F )) ≃ M m,n ( F ) ⊕ M m,n ( F ) for ( m, n ) � = (1 , 1) ; U ( Q (+) ( n )) = Q ( n ) ⊕ Q ( n ) , n ≥ 2 ; U ( osp ( m, n )) ≃ M m,n ( F ) , ( m, n ) � = (1 , 2) ; U ( P ( n )) ≃ M n,n ( F ) , n ≥ 3 . Theorem. The embedding σ of the Cheng-Kac super- algebra is universal, that is, U ( CK ( Z, D )) ∼ = M 2 , 2 ( W ) . The restriction of the embedding u (see above) to P (2) is a universal specialization; U ( P (2)) ≃ M 2 , 2 ( F [ t ]) , where F [ t ] is a polynomial algebra in one variable.

The Jordan superalgebra of a superform Let V = V ¯ 0 + V ¯ 1 be a Z/ 2 Z -graded vector space, dim V ¯ 0 = m, dimV ¯ 1 = 2 m ; let <, > : V × V → F be a super- symmetric bilinear form on V . The universal associative enveloping algebra of the Jordan algebra F 1 + V ¯ 0 is the Clifford algebra Cl ( m ) = < 1 , e 1 , . . . , e m | e i e j + e j e i = 0 , i � = j, e 2 i = 1 > . Consider the Weyl algebra W n = < 1 , x i , y i , 1 ≤ i ≤ n | [ x i , y j ] = δ ij , [ x i , x j ] = [ y i , y j ] = 0 > . Assuming x i , y i , 1 ≤ i ≤ n to be odd, we make W n a superalgebra. The universal associative enveloping algebra of F 1+ V is isomorphic to the (super)tensor product Cl ( m ) ⊗ F W n . Specializations of M 1 , 1 ( F ) � � A M 12 Theorem. U ( M 1 , 1 ( F )) ≃ . The map- M 21 A ping � � � α 12 + α 21 a − 1 z 2 � α 11 α 12 α 11 u : → α 21 α 22 α 12 z 1 + α 21 a α 22 is a universal specialization. Here a is root of the equation a 2 + a − z 1 z 2 = 0, A = F [ z 1 , z 2 ]+ F [ z 1 , z 2 ] a is a subring of K a quadratic exten- sion of F ( z 1 , z 2 ) generated by a and M 12 = F [ z 1 , z 2 ] + F [ z 1 , z 2 ] a − 1 z 2 , M 21 = F [ z 1 , z 2 ] z 1 + F [ z 1 , z 2 ] a are sub- spaces of K .