DePaul University Algebra and Combinatorics Seminar
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Abstract |
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Bialgebras are algebraic structures arising in numerous parts of mathematics, as well as in physics. A bialgebra is, roughly speaking, an algebra on which there exist a dual structure (called a coalgebra structure) such that the two structures are compatible in the appropriate sense. Perhaps the most striking aspect of bialgebras is their extraordinary ubiquity in virtually all fields of mathematics. Examples of bialgebras include group algebras, universal envelopes of Lie algebras, algebras of representative functions on Lie groups, coordinate algebras of algebraic groups, cohomology of H-spaces, combinatorial Hopf algebras, and quantum groups. Free probability is a mathematical theory that studies non-commuting random variables. The free independence is an analogue of the classical notion of independence, and is connected with free products of algebras. It has many applications in the theory of operator algebras (such as C*-algebras and Von Neumenn algebras) as well as in the theory of random matrices. In the talk I will review the basic notions regarding bialgebras and free probability and illustrate them on several examples. I will then try to explain how bialgebras can be used to encode some of the combinatorics of free multiplicative convolution of distributions (the free analogue of the classical convolution of distributions). This part of the talk is based on joint work with A. Nica. |