Enveloping Algebras, Differential Operators and Quantum Groups

University Of Wisconsin-Milwaukee, Milwaukee WI

Investigators

Abstract

This proposal is for research into three areas of current interest in noncommutative algebra. First the PI plans to study the enveloping algebra of a classical simple Lie superalgebra. Specifically he is interested in primitive ideals and problems concerning the center of the enveloping algebra. Secondly the PI intends to investigate differential operators on toric varieties and maps between toric varieties arising from refinements of fans. Finally the PI proposes the study of certain finite dimensional Hopf algebras arising from quantized enveloping algebras of classical Lie superalgebras. The methods to be used come from representation theory and algebraic geometry. This research is concerned with some particular aspects of modern abstract algebra, as an area of mathematics. In recent years a number of algebraic constructions have been increasingly important in Physics. These include Lie superalgebras and quantum groups, two constructions which will be investigated in this proposal. It is a widely held belief that a better understanding of the algebraic structure of these objects could lead to further significant advances in Physics.