LECTURE 15: REPRESENTATIONS OF Uq(g) AT ROOTS OF 1
نویسنده
چکیده
In Lecture 13 we have studied the representation theory of Uq(g) when q is not a root of 1. We have seen that the representation theory basically looks like the representation theory of g (over C). In this lecture we are going to study a more complicated case: when q is a root of 1 (we still need to exclude some small roots of 1 to make the algebra Uq(g) defined). When we deal with the usual definition of Uq(g) we see features of the algebra U(gF), where F is a field of positive characteristic. In particular, Uq(g) has an analog of the p-center and is finite over its center. Or we can modify our definition of Uq(g) including divided powers, this is the case we are going to mostly care about. The corresponding algebra has many features of the semisimple algebraic groups over field of positive characteristic (such as the Frobenius homomorphism). In fact, it is the connection with quantum groups that allowed to compute the characters of irreducible representations of GF when charF ≫ 0. For the most part of this lecture we consider the case of Uq(sl2) that can be treated by hand. In the second part we consider a far more complicated case of Uq(sln). We mention the connection between the representation theory of Uq(sln) and that of the affine Lie algebra ŝln discovered by Kazhdan-Lusztig. This is the main tool to study the representation theory of Uq(sln) (and of other Uq(g), there we need the affine Lie algebra ĝ).
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