By Shahn Majid

Here's a self-contained advent to quantum teams as algebraic items. in line with the author's lecture notes for the half III natural arithmetic direction at Cambridge college, the publication is appropriate as a major textual content for graduate classes in quantum teams or supplementary examining for contemporary classes in complicated algebra. the cloth assumes wisdom of uncomplicated and linear algebra. a few familiarity with semisimple Lie algebras could even be priceless. the amount is a primer for mathematicians however it may also be beneficial for mathematical physicists.

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Then Rg13Rg23egq-q-1)E®g®Fe(q (A®id)R = 2q 1)1®E®F - R13R23, where we use the gFg-1 = q-2F relation to move Rg23 to the right, which also has the effect of turning E ®g ® F into E ®1® F as required. Here Rg(g ® F) = (1® F) R. along the same lines as the similar identity for 10 E already used. We need one more detail to complete the proof, namely that R is invertible. Actually, the inverse is R-1 = g e2(q-q-1)E®FR91 by an elementary property of q-exponentials which we will see later. 3, R(S ®id)R = (e ®id) (R) = ((S ®id)R)R once axiom 3 of a quasitriangular structure holds, so we can take R-1 = (S ® id)R as a definition.

A dual quasitriangular Hopf algebra is dual triangular if Q(hog) = e(h)e(g). It is `factorisable' if Q is nondegenerate in the sense that Q(h 0 g) = 0 for all g implies that h = 0. , in the convolution algebra of maps H -# k. As with comodules, working with these dual quasitriangular structures often keeps things algebraic. For example, the group algebra is defined for any group G, not only finite ones. Clearly in this case the definition of a dual quasitriangular structure reduces to R being the extension by linearity of a bicharacter R : G x G -p k and the requirement that G be Abelian.

3. (A ®id)R = R13R23, (id (9 A)R = R13R12, where the product is in H ®H ®H and R12 = R ®1, R23 = 1®R as elements of this. The numerical suffixes denote the embedding of R in higher tensor powers of H. The specific meaning of such a quasitriangular structure R will emerge during the next several lectures. However, there are many examples and indeed, most self-respecting quantum groups are either quasitriangular or dual to a quasitriangular one. 2 Let (H, R) be a quasitriangular bialgebra. Then 29 30 5 Quasitriangular structures 1.

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