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 3-manifolds 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 3-manifold M one can assign an R-module called the Kauffman bracket skein module of M. If A^2=1 then this module has a structure of an R-algebra. 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_2-character variety of pi_1(M). Using that result we develop a theory of Sl_2-character varieties by use of topological methods. We also assign to any surface a relative Kauffman bracket skein algebra. We prove several results about this non-commutative 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:q-alg/9705011
- Pub Date:
- May 1997
- DOI:
- 10.48550/arXiv.q-alg/9705011
- arXiv:
- arXiv:q-alg/9705011
- Bibcode:
- 1997q.alg.....5011P
- Keywords:
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- Mathematics - Quantum Algebra
- E-Print:
- 47 pages, Latex, 34 figures, to appear in Topology