  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.

Quantum Algebra;Quantum Algebra