Quantum orthogonal and symplectic groups and their embedding into quantum $GL$

Abstract

The article contains constructions, without proofs, of quantumanalogues of the orthogonal and symplectic groups. These quantumgroups are attained via Jimbo's R matrices [M. Jimbo, Comm.Math. Phys. 102 (1986), no. 4, 537 547; MR 87h:58086] which yieldq deformations of the coordinate Hopf algebras. The classicalgroups can be recovered by letting q=1. Naturally, thesequantum groups should be embeddable in some quantum versions ofthe general linear group, and the author suggests separatecandidates in the orthogonal and symplectic cases. Thesecandidates are formulated using analogues for the quantumexterior and symmetric algebras in the two cases. The embeddingof the orthogonal group \roman O\sb q(n) into the generallinear group of orthogonal type \roman{GL}\sb q(n) is explainedin detail for n=3.\par Some of the results here overlap withwork of N. Yu. Reshetikhin, L. Takhtadzhyan and L. D. Faddeev[Algebra i Analiz 1 (1989), no. 1, 178 206; MR\Cite{Reshetikhin89:Quantization:178--206}[90j:17039]]