BIMODULES OVER SIMPLE FINITE-DIMENSIONAL JORDAN SUPERALGEBRAS Consuelo Mart¶³nezL¶opez Workshop on Nonassociative Algebras Toronto, 12-14 May 2005 SUPERALGEBRA : A = A0¹ + A1¹, A¹i ¢ A¹j ⊆ Ai+¹j a Z=2Z-graded algebra EX. V vector space of countable dimension, G(V ) = G(V )0¹ + G(V )1¹ Grassmann algebra over V , G(A) = A0¹ ­ G(V )0¹ + A1¹ ­ G(V )1¹ · A ­ G(V ) Grassmann enveloping algebra of A V a variety of algebras (associative, Lie, Jordan,...) DEF. A = A0¹+A1¹ is a V-superalgebra if G(A) 2 V. J = J0¹ + J1¹ is a Jordan superalgebra if it satis¯es SJ1. Supercommutativity a ¢ b = (¡1)jajjbjb ¢ a, SJ2. Super Jordan identity (a ¢ b) ¢ (c ¢ d) + (¡1)jbjjcj(a ¢ c) ¢ (b ¢ d)+ (¡1)jbjjdj+jcjjdj(a ¢ d) ¢ (b ¢ c) = ((a ¢ b) ¢ c) ¢ d + (¡1)jcjjdj+jbjjc((a ¢ d) ¢ c) ¢ b+ (¡1)jajjbj+jajjcj+jajjdj+jcjjdj((b ¢ d) ¢ c) ¢ a. JORDAN SUPERALGEBRAS A = A0¹ + A1¹ associative superalgebra (+) 1 jajjbj A = (A; a¢b = 2 (ab+(¡1) ba) Jordan super- algebra (+) J = J0¹ + J1¹ · A special. Otherwise excep- tional (+) (A) A , A = Mm+n(F ) full linear superalgebra µ ¶ a b (Q) A(+) , A = f j a; b 2 M (F )g b a n If ? : A ! A is an involution : (a?)? = a,(ab)? = (¡1)jajjbjb?a?. H(A; ?) = fa 2 Aja? = ag · A(+) µ ¶ I 0 (BC) M (F ), Q = m , m+2n 0 S 0 12n 0 1 ::: B ¡1 0 ::: C B C S = B ::::: C 2n @ A ::: 0 1 µ :::¶ ¡µ1 0 ¶ a b aT ¡cT ? : ! Q¡1 Q, a 2 M (F ), c d bT dT m d 2 M2n(F ), H(A; ?) = ospm;2n(F). µ ¶ µ ¶ a b dT ¡bT (P) A = M (F ), ? : ! , n+n c d cT aT µ ¶ a b H(A; ?) = f j a; b; c 2 M (F ); bT = ¡b; c aT n cT = cg. (D) A Superalgebra of a superform V = V0¹ + V1¹, <; >: V £ V ! F a supersymmetric bilinear form J = F 1 + V = (F 1 + V0¹) + V1¹,(®1 + v)(¯1 + w) = (®¯+ < v; w >)1 + (®w + ¯v). (Dt) Jt = (F e1 + F e2) + (F x + F y), t 6= 0 2 1 1 ei = ei; e1e2 = 0; eix = 2 x; eiy = 2 y; [x; y] = e1 + te2. (J) All simple Jordan algebras (F) The 10-dimensional exceptional Kac superalgebra P4 P2 K10 = [(F e1 + i=1 F vi)+F e2]+( i=1 F xi +F yi) 2 ei = ei; e1e2 = 0; e1vi = vi; e2vi = 0; v1v2 = 2e1 = v3v4, 1 1 eixj = 2 xj; eiyj = 2 yj; i; j = 1; 2 y1v1 = x2; y2v1 = ¡x1; x1v2 = ¡y2; x2v2 = y1, x2v3 = x1; y1v3 = y2; x1v4 = x2; y2v4 = y1, [xi; yi] = e1 ¡ 3e2; [x1; x2] = v1; [y1; y2] = v2, [x1; y2] = v3; [x2; y1] = v4. (K) The 3-dimensional Kaplansky superalgebra 2 1 K3 = F e + (F x + F y); e = e; ex = 2 x; 1 ey = 2 y; [x; y] = e. Theorem. (Kac 77, Kantor 89) A simple ¯nite dimen- sional Jordan superalgebra over an algebraically closed ¯eld of zero characteristic is isomorphic to one of the su- peralgebras A, BC, D, P, Q, Dt, 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 ¯eld F , chF = 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 de¯ned by Brackets ¡ = ¡0¹ + ¡1¹ an associative commutative superalgebra f; g :¡£¡ ! ¡ a Poisson bracket if f¡¹i; ¡¹jg ⊆ ¡i+¹j and (1) (¡; f; g) is a Lie superalgebra, (2) fab; cg = afb; cg+(¡1)jbjjcjfa; cgb (Leibniz iden- tity) Kantor Double Superalgebra J = ¡ + ¡x, a(bx) = (ab)x,(bx)a = (¡1)jaj(ba)x, jbj (ax)(bx) = (¡1) fa; bg, J0¹ = ¡0¹ + ¡1¹x, J1¹ = ¡1¹ + ¡0¹x. Theorem. (Kantor 1992) Let f; g be a Poisson bracket =) J = ¡ + ¡x is a Jordan superalgebra. Kantor superalgebra ¡ = Grassman algebra on »1;:::;»n Pn jfj @f @g ¡ = ¡0¹ + ¡1¹, ff; gg = (¡1) ½ i=1 @»i @»i n = 1 J ' D(¡1) J = ¡ + ¡x n ¸ 2 J0¹ is not semisimple CHENG-KAC JORDAN SUPERALGEBRAS Z unital associative commutative algebra, d : Z ! Z a derivation, P3 CK(Z; d) = J0¹ + J1¹, J0¹ = Z + i=1 wiZ, J1¹ = xZ + P3 i=1 xiZ free Z-modules of rank 4. 2 2 2 Even part wiwj = 0; i 6= j; w1 = w2 = 1; w3 = ¡1, Notation: xi£i = 0; x1£2 = ¡x2£1 = x3 x1£3 = ¡x3£1 = x2; ¡x2£3 = x3£2 = x1. Module action f; g 2 Z g wjg d xf x(fg) xj(fg ) xif xi(fg)xi£j(fg) Bracket on M xg xjg d d xf f g ¡ fg ¡wj(fg) xif wi(fg) 0 CK(Z; d) is simple () Z does not contain proper d-invariant ideals. p B(m) = F [a1; : : : ; am j ai = 0] B(m; n) = B(m) ­ G(n) G(n) =< 1;»1;:::;»n > Theorem. (M., Zelmanov, J. of Algebra 236, 2001) Let J = J0¹ +J1¹ be a ¯nite dimensional simple unital Jordan superalgebra over an algebraically closed ¯eld F , chF = p > 2, J0¹ 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 ¯eld type (fa; bg = a0b ¡ ab0 0 a derivation) is special. @f @g @f @g - The Kantor Double of ff; gg = @x @y ¡ @y @x on F [x; y] is exceptional. Shestakov (1993) - A Kantor Double of Poisson bracket <; >:¡£¡ ! ¡ is special i® << ¡; ¡ >; ¡ >= (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 = ¡ + ¡x does not contain 6= (0) nilpo- tent ideals - If ¡ = ¡0¹ then J is special i® <; > is of vector ¯eld type. - If ¡1¹¡1¹ 6= (0) (at least 2 Grassmann variables) then J is exceptional. - If ¡ = ¡0¹ + ¡0¹»1; < ¡0¹;»1 >= (0); < »1;»1 >= ¡1 then J is special i® <; >:¡0¹ £ ¡0¹ ! ¡0¹ is of vector ¯eld type. Theorem. (M., Shestakov, Zelmanov) The Cheng-Kac superalgebra CK(Z; d) is special The embedding extends McCrimmon embedding for vector ¯eld type brackets. W =< R(a); a 2 Z; d > - di®erential operators on Z R = R0¹ + R1¹ = M4£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 ' : J ! A there exists a homomorphism of associative algebras » : U ! A such that ' = » ¢ 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) ¯xed, i.e., u(J) ⊆ H(U; ¤). We call a special Jordan superalgebra reflexive if u(J) = H(U; ¤). (+) Theorem. U(Mm;n(F )) ' Mm;n(F ) © Mm;n(F ) for (m; n) 6= (1; 1); U(Q(+)(n)) = Q(n) © Q(n), n ¸ 2; U(osp(m; n)) ' Mm;n(F ), (m; n) 6= (1; 2); U(P (n)) ' Mn;n(F ), n ¸ 3. Theorem. The embedding σ of the Cheng-Kac super- » algebra is universal, that is, U(CK(Z; D)) = M2;2(W ). The restriction of the embedding u (see above) to P (2) is a universal specialization; U(P (2)) ' M2;2(F [t]), where F [t] is a polynomial algebra in one variable. The Jordan superalgebra of a superform Let V = V0¹+V1¹ be a Z=2Z-graded vector space, dim V0¹ = m; dimV1¹ = 2m; let <; >: V £ V ! F be a super- symmetric bilinear form on V . The universal associative enveloping algebra of the Jordan algebra F 1 + V0¹ is the Cli®ord algebra Cl(m) =< 1; e1; : : : ; emjeiej + ejei = 2 0; i 6= j; ei = 1 >. Consider the Weyl algebra Wn =< 1; xi; yi; 1 · i · nj[xi; yj] = ±ij; [xi; xj] = [yi; yj] = 0 >. Assuming xi; yi; 1 · i · n to be odd, we make Wn a superalgebra. The universal associative enveloping algebra of F 1+V is isomorphic to the (super)tensor product Cl(m) ­F Wn. Specializations of M1;1(F ) µ ¶ AM12 Theorem. U(M1;1(F )) ' . The map- M21 A ping µ ¶ µ ¶ ® ® ® ® + ® a¡1z u : 11 12 ! 11 12 21 2 ®21 ®22 ®12z1 + ®21a ®22 is a universal specialization. 2 Here a is root of the equation a +a¡z1z2 = 0, A = F [z1; z2]+F [z1; z2]a is a subring of K a quadratic exten- sion of F (z1; z2) generated by a and M12 = F [z1; z2] + ¡1 F [z1; z2]a z2, M21 = F [z1; z2]z1 + F [z1; z2]a are sub- spaces of K. Let V be a Jordan bimodule over the (super)algebra J V is a one-sided bimodule if fJ; V; Jg = (0). Then, the mapping a ! 2RV (a) 2 EndF V is a specialization. The category of one-sided bimodules over Jis equiv- alent to the category of right (left) U(J)-modules. Let e be the identity of J and let V = fe; V; eg + f1 ¡ e; V; eg + f1 ¡ e; V; 1 ¡ eg be the Peirce decomposition. Then fe; V; eg is a unital bimod- ule over J, that is, e is an identity of fe; V; eg + J.