The algebra L g,n (H) was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and quantizes the character variety of the Riemann surface Σ g,n \ D (D is an open disk). In this article we define a holonomy map in that quantized setting, which associates a tensor with components in L g,n (H) to tangles in (Σ g,n \D) × [0, 1], generalizing previous works of Buffenoir-Roche and Bullock-Frohman-Kania-Bartoszynska. We show that holonomy behaves well for the stack product and the action of the mapping class group; then we specialize this notion to links in order to define a generalized Wilson loop map. Thanks to the holonomy map, we give a geometric interpretation of the vacuum representation of L g,0 (H) on L 0,g (H). Finally, the general results are applied to the case H = U q 2 (sl 2 ) in relation to skein theory and the most important consequence is that the stated skein algebra of a compact oriented surface with just one boundary edge is isomorphic to L g,n U q 2 (sl 2 ) . Throughout the paper we use a graphical calculus for tensors with coefficients in L g,n (H) which makes the computations and definitions very intuitive.
Let Σ g,n be a compact oriented surface of genus g with n open disks removed. The graph algebra L g,n (H) was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and is a combinatorial quantization of the moduli space of flat connections on Σ g,n . We construct a projective representation of the mapping class group of Σ g,n using L g,n (H) and its subalgebra of invariant elements. Here we assume that the gauge Hopf algebra H is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We also give explicit formulas for the representation of the Dehn twists generating the mapping class group; in particular, we show that it is equivalent to a representation constructed by V. Lyubashenko using categorical methods. their regular support and their useful remarks. I thank A. Gainutdinov for inviting me to present this work at the algebra seminar of the University of Hamburg. Notations.If A is a C-algebra, V is a finite-dimensional A-module and x ∈ A, we denote by V x ∈ End C (V ) the representation of x on the module V . Similarly, if X ∈ A ⊗n and if V 1 , . . . , V n are A-modules, we denote by V 1 ...Vn X the representation of X on V 1 ⊗ . . . ⊗ V n . Here we consider only finite-dimensional representations, hence H-module implicitly means finite-dimensional H-module. IJ R 12 I T 1 J T 2 = J T 2 I T 1 IJ R 12 . Since H is finite-dimensional, it exists right and left integrals µ r , µ l ∈ O(H) defined by I B or I M , let J V be J A or J B or J M and let i < j. Then, by definition of the right action and by (18):
Let Σ g,n be a compact oriented surface of genus g with n open disks removed. The algebra L g,n (H) was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and is a combinatorial quantization of the moduli space of flat connections on Σ g,n . Here we focus on the two building blocks L 0,1 (H) and L 1,0 (H) under the assumption that the gauge Hopf algebra H is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We construct a projective representation of SL 2 (Z), the mapping class group of the torus, using L 1,0 (H) and we study it explicitly for H = U q (sl (2)). We also show that it is equivalent to the representation constructed by Lyubashenko and Majid.and this give the result sinceIt follows that ψ ∈ H * is symmetric if, and only if, J C (±) ⊲ ψ = ψI dim(J) for all J.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.