Abstracts and Available Videos
Lecture 1. Orthogonal representations: from groups to Hopf algebras to tensor categories.
Abstract: For a finite group G, a representation V of G over C is called orthogonal if V admits a bilinear, non-degenerate, symmetric G-invariant form. Frobenius and Schur determined when this happens, by defining an "indicator" function on V. This function has values 1, -1 or 0, depending on whether V is orthogonal, symplectic, or does not admit a non-degenerate bilinear form. The so-called higher indicators may also be defined on V; their values still are integers. This idea has been extended to Hopf algebras, and more recently to fusion categories, a special kind of tensor category, although the values may no longer be integers. The indicators are very useful since they are an invariant of the category (a "gauge invariant"). The question then arises as to when the representation ring together with the indicators determines the category. We survey some recent results in this area.
Lecture 2. Actions of Hopf algebra on matrices.
Abstract: Classically the actions of the Lie algebra sl2 on a polynomial ring C[x, y] were known. This was extended to the (Hopf) actions of the quantized enveloping algebra Uq(sl2) and its finite-dimensional analog H = uq(sl2). Note these algebras are generated by a group element and two skew-primitive elements. In recent joint work with Bahturin we have extended this work to actions of H on Mn(C), n x n matrices over C, inspired by the fact that the matrix ring is also generated by two elements. Crucial ingredients are Masuoka's theorem (1990) that Hopf actions on matrices are "inner", as well as the determination of group gradings on matrices by Bahturin, Sehgal, and Zaicev (2001). A first step is to determine actions of the Taft Hopf algebra, generated by a group element and a single skew-primitive element.
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