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Unicity for Representations of the Kauffman Bracket Skein Algebra
by
Joanna Kania-Bartoszynska
National Science Foundation
Coauthors: Charles Frohman and Thang Le
Let F be a connected, closed, oriented surface, with finitely many (possibly none) points removed, and let ζ be a root of unity. The Kauffman bracket skein algebra of F, denoted Kζ(F), is formed by taking C-linear combinations of isotopy classes of links in a cylinder over F, and modding out by the Kauffman bracket skein relation, with the parameter ζ. Multiplication is given by placing one link over another and extending linearly. Bonahon and Wong associate to each irreducible representation of Kζ(F) a classical shadow, which is determined by its central character, and conjecture that there is a generic family of classical shadows for which there is a unique irreducible representation of the skein algebra realizing that classical shadow. In joint work with Charles Frohman and Thang Le we resolve this conjecture. The proof is a consequence of a general unicity theorem that says that the irreducible representations of a prime affine k-algebra over an algebraically closed field k, that is finitely generated as a module over its center, are generically classified by their central characters.
Date received: December 5, 2017
Copyright © 2017 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cboj-25.