Ž . Journal of Algebra 213, 405436 1999 Article ID jabr.1998.7668, available online at http:www.idealibrary.com on Quantum Algebra Structures on n  n Matrices Louis H. Kauffman* and David E. Radford ² ( ) Department of Mathematics, Statistics, and Computer Science m c 249 , Uni  ersity of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607-7045 Communicated by Susan Montgomery Received September 16, 1997 Quantum algebras are generalizations of quasitriangular Hopf algebras and as such are used to construct invariants of 1  1 tangles and in some cases of knots and 3-manifolds. In this paper we develop a general description of quantum algebra structures whose underlying algebra is the Ž . algebra M k of all n  n matrices over a field k. An important ingredi- n ent in the structure of a quantum algebra in this case is a solution Ž . 2 R  M k to the quantum Yang Baxter equation. n We use our general results and the classification of solutions to the Ž . Ž . Ž . quantum Yang Baxter equation R  M k  M k  M k found in 4 2 2  Ž . 3 to determine the quantum algebra structures on M k when k is an 2 algebraically closed field of characteristic 0. In all but one case R is upper triangular. There are eight different classes, three of which account for the minimal structures, a concept discussed in Section 2. The invariants of 1  1 Ž . tangles arising from the quantum algebra structures on M k yield the 2 Jones polynomial in one case and variation on the writhe or Whitney degree in the seven others.  As noted in Section 8 of 6 the 1  1 tangle invariants described in this paper give rise to knot invariants under certain circumstances. It should be noted that these knot invariants are ultimately described in terms of unoriented knot diagrams. Algebra which produces invariants of oriented knot diagrams will be discussed in a future paper. *Research supported in part by NSF Grant DMS 920-5227. ² Research supported in part by NSF Grant DMS 870-1085. 405 0021-869399 $30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.