Mathematics – Quantum Algebra
Scientific paper
1997-05-15
Mathematics
Quantum Algebra
47 pages, Latex, 34 figures, to appear in Topology
Scientific paper
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.
Przytycki Jozef H.
Sikora Adam S.
No associations
LandOfFree
On Skein Algebras And Sl_2(C)-Character Varieties does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On Skein Algebras And Sl_2(C)-Character Varieties, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Skein Algebras And Sl_2(C)-Character Varieties will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-271214