Emmy Noether Lectures

2011 Lecturer: Susan Montgomery

Orthogonal Representations: From Groups to Hopf Algebras

In recent years Hopf algebras have had applications in mathematical physics, in particular conformal field theory, and in other parts of mathematics, such as knot theory and operator algebras. In this talk we discuss some recent progress in the representation theory of Hopf algebras which extends classical work in group theory.
Frobenius and Schur showed in 1906 that one can decide whether or not a complex representation V of a finite group G is real by computing the value of ν(V ) = 1|G|g χ(g2), where χ is the character of Vν(V), the indicator of V, takes only three values, 0, 1, or -1. The representation is real precisely when ν(V ) = +1; equivalently V has a symmetric non-degenerate ℂ-bilinear G-invariant form. Thus the elements of G act as orthogonal transformations on V.
In the last decade, Frobenius-Schur indicators have been extended to finite dimensional Hopf algebras and beyond, such as to quasi-Hopf algebras and fusion categories; they are invariants of the monoidal (tensor) category of representations. Moreover there are applications of indicators to Hopf algebras whose statements do not use indicators, such as results about classification or about the exponent of the Hopf algebra.
Brief Biography:
Susan Montgomery is a professor at the University of Southern California. She received her B.A. from the University of Michigan and her Ph.D. in Mathematics from the University of Chicago. She has been on the faculty of the University of Southern California since 1970. Montgomery has also spent sabbaticals at the Hebrew University of Jerusalem, the University of Leeds, the University of Munich, the Mittag-Leffler Institute, and MSRI.
In 1984 she was awarded a John S. Guggenheim Memorial Foundation Fellow, and in 1987 was awarded a Raubenheimer Outstanding Faculty Award by USC. She has given an AMS Invited Address at the Joint Mathematics Meeting in 1984 and at a sectional meeting in 2005. In 2009, she gave a Plenary Lecture at the summer meeting of the Canadian Math Society. She has also given numerous lectures at meetings and universities around the world.
Montgomery was the Principle Lecturer at the Conference Board of the Mathematical Sciences conference in 1992 on Hopf Algebras, and her CBMS monograph Hopf Algebras and their Actions on Rings is widely cited.
She served as an editor for the Journal of Algebra for over 20 years. She was also an editor for the AMS Proceedings, AMS Surveys and Monographs, and Advances in Math, and currently is on the editorial boards of Algebras and Representation Theory and of Algebra and Number Theory.
She has been very active in the American Mathematical Society, serving on the Board of Trustees for 10 years. She has also served on the Council, the Policy Committee on Publications, and most recently on the Nominating Committee. She was also a member of the Board on Mathematical Sciences (BMS), serving one year on the Executive Committee.
Montgomery has been active in the Association for Women in Mathematics for 35 years. She was a member of the Executive Committee from 1975-1976. She served on the Nominating Committee in 1982 (as chair) and again in 2009. She was on the committee to select the Noether Lecturer from 1990-1992. At USC, she has served on the Provost’s Committee on Women in Science and Engineering since 2000. However, she regards her main contribution to women in mathematics to be doing mathematics with her women coauthors and Ph.D. students.
Montgomery’s early research was on group action on rings, but since the 1980s, she has worked primarily in Hopf algebras, their representations, and their actions on other algebras.