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34 GEORGIA BENKART AND EFIM ZELMANOV Recognition Theorem for Type C. Let L be a ~-graded Lie algebra. (i) If ~ = C«, n ~ 4, then there exists a unital associative algebra A with an involution * : A ~ A such that L is centrally isogenous with the algebra sP2n(A, *) of symplectic (2n) x (2n) matrices over A . (ii) If ~ = C3 , then L is centrally isogenous with the symplectic Steinberg algebra st sp6(A, *), where A is an alternative involutive algebra whose symmetric elements, {a E A I a- = a}, lie in the associative center of A.

1985 Lett. Math . Phys . 10 63; 1986 Lett . Math. Phys . , Reshetikhin N. Yu . : 1990 Leningrad Math . J. , Sorace E . : 1992 Contractions o] quan tum groups (Lecture Notes in Mathemati cs 1510) (Berlin : Springer) p . , Ruegg H. : 1992 Phys . L ett . B . , Sorace E . : 1992 Phys. Rev. Lett. 68 3178 Inonii E . : 1953 Pro c. N atl. d. Sci . U. S. J. : 1993 Proc. XIX ICGTMP CIEMAT/RSEF (Madrid) Vol. I, p. 455. : 1990 Int. J. Mod . Phys . A 5 1 QUANTUM UNIVERSAL ENVELOPING CAYLEY-KLEIN ALGEBRAS 23 13.

Fa2n and let qv be the skew bilinear form defined in 1) when z = a2n+l . We have that o([ai ,aj]) = 26 M. PI LAR BENITO ~n aqv (aj , aj )a2n+l and t he matrix B = (sv (aj , aj )) is B = (_ = 8 E Der(L) iff 8([aj ,aj]) [8(ad ,aj] A = (Tjj) , these conditions ar e th at aqv{a j , aj) = L + [aj , 8(aj )] for qv(aj , ak)Tkj + t $ k$2 n i, j L 0). 1 Clea rly, = 1, . . , 2n. If we denote Tk jqv(ak, aj) t$k$2n or in matrix form (2. 1. • aT In the sequel, for each n ~ 1 we sha ll denote by L( n) the Lie algebra with basis at , ...

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