dorsal/arxiv
View SchemaOn Skein Algebras And Sl_2(C)-Character Varieties
| Authors | Jozef H. Przytycki, Adam S. Sikora |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9705011 |
| URL | https://arxiv.org/abs/q-alg/9705011 |
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.
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"abstract": "This paper gives insight into intriguing connections between two apparently\nunrelated theories: the theory of skein modules of 3-manifolds and the theory\nof representations of groups into special linear groups of 2 by 2 matrices.\n Let R be a ring with an invertible element A. For any 3-manifold M one can\nassign an R-module called the Kauffman bracket skein module of M. If A^2=1 then\nthis module has a structure of an R-algebra. We investigate this structure and,\nin particular, we prove that if R is the field of complex numbers then this\nalgebra is isomorphic to the (unreduced) coordinate ring of the SL_2-character\nvariety of pi_1(M). Using that result we develop a theory of Sl_2-character\nvarieties by use of topological methods.\n We also assign to any surface a relative Kauffman bracket skein algebra. We\nprove several results about this non-commutative algebra.\n Our work should be considered in the context of the book of Brumfiel and\nHilden `SL(2) Representations of Finitely Presented Groups,\u0027 Cont. Math 187. In\nparticular we give a topological interpretation to algebraic objects considered\nin that book.",
"arxiv_id": "q-alg/9705011",
"authors": [
"Jozef H. Przytycki",
"Adam S. Sikora"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "On Skein Algebras And Sl_2(C)-Character Varieties",
"url": "https://arxiv.org/abs/q-alg/9705011"
},
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