On Skein Algebras And Sl_2(C)Character Varieties
Abstract
This paper gives insight into intriguing connections between two apparently unrelated theories: the theory of skein modules of 3manifolds and the theory of representations of groups into special linear groups of 2 by 2 matrices. Let R be a ring with an invertible element A. For any 3manifold M one can assign an Rmodule called the Kauffman bracket skein module of M. If A^2=1 then this module has a structure of an Ralgebra. We investigate this structure and, in particular, we prove that if R is the field of complex numbers then this algebra is isomorphic to the (unreduced) coordinate ring of the SL_2character variety of pi_1(M). Using that result we develop a theory of Sl_2character varieties by use of topological methods. We also assign to any surface a relative Kauffman bracket skein algebra. We prove several results about this noncommutative algebra. Our work should be considered in the context of the book of Brumfiel and Hilden `SL(2) Representations of Finitely Presented Groups,' Cont. Math 187. In particular we give a topological interpretation to algebraic objects considered in that book.
 Publication:

eprint arXiv:qalg/9705011
 Pub Date:
 May 1997
 arXiv:
 arXiv:qalg/9705011
 Bibcode:
 1997q.alg.....5011P
 Keywords:

 Mathematics  Quantum Algebra
 EPrint:
 47 pages, Latex, 34 figures, to appear in Topology