A finite dimensional algebra approach to quantum groups

Abstract 

These lectures will explore connections between (generalized) q-Schur algebras and the quantized enveloping algebra Uq(g) associated with a simple Lie algebra g. These connections are facilitated by a certain completion of Lusztig’s modified form of Uq(g). Although the q-Schur algebras arose initially as quotients of Uq(g) it is possible to reverse history and use them as a tool to reconstruct Uq(g). In type A, q-Schur algebras arise naturally in connection with a q-analogue of Schur–Weyl duality discovered by Jimbo (1986) which links the representation theories of Uq(gl_n) and the Iwahori–Hecke algebra Hq(S_r) of the symmetric group S_r.

About the Speaker 

Stephen Doty

Loyola University Chicago

Stephen Doty holds a Ph.D. in mathematics from the University of Notre Dame. He joined the Loyola faculty in 1987, after postdoctoral appointments at the Universities of Washington (Seattle) and Notre Dame. Dr Doty holds a joint appointment between the Departments of Mathematics & Statistics and Computer Science. He is an avid proponent of open-source software, and his current favorite programming language is Python.

His research interests include representation theory, algebraic groups, quantum groups, and finite-dimensional algebras. Dr Doty has held visiting fellowships at a number of international research institutes, including the Mathematical Sciences Research Institutein Berkeley, California and the Universities of Aarhus, Cambridge, Leicester, Manchester, Oxford, and Queen Mary College (London). His research travels have included extended visits to Canada, China, Denmark, Germany, India, Japan, and the United Kingdom. His research has been supported by the National Science Foundation and the National Security Agency in the USA. In 2007 Dr Doty was appointed Yip Fellow of Magdalene College, Cambridge and in 2009 he was awarded a Mercator Visiting Professorship, funded by the German Research Council, to lecture and research in Germany for seven months.