Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Biquantization of the necklace Lie bialgebra

Xiaojun Chen, Maozhou Huang, Meiliang Liu, Jun Zhang

Abstract

For the double of a quiver, the works of Ginzburg, Bocklandt-Le Bruyn and Schedler show that its closed paths, called the necklaces, have a natural Lie bialgebra structure. Schedler also constructed,in [Int. Math. Res. Notices, 2005 (12), 725-760], a Hopf algebra that quantizes this Lie bialgebra. In this paper, we pursue one more step in this direction by constructing its biquantization, in the sense of Turaev [Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 6, 635-704].

Biquantization of the necklace Lie bialgebra

Abstract

For the double of a quiver, the works of Ginzburg, Bocklandt-Le Bruyn and Schedler show that its closed paths, called the necklaces, have a natural Lie bialgebra structure. Schedler also constructed,in [Int. Math. Res. Notices, 2005 (12), 725-760], a Hopf algebra that quantizes this Lie bialgebra. In this paper, we pursue one more step in this direction by constructing its biquantization, in the sense of Turaev [Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 6, 635-704].

Paper Structure

This paper contains 26 sections, 30 theorems, 164 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main result
  3. Organization of the paper
  4. Lie bialgebras and their quantizations
  5. Lie bialgebra and its symmetric products
  6. Example: the necklace Lie bialgebra
  7. The Poisson and co-Poisson structures
  8. Biquantization of a Lie bialgebra
  9. Quantization and coquantization for $S(L)$
  10. The bialgebra $\mathbf{N}(Q)_{h,\hbar}$
  11. The algebra $\mathbf{N}(Q)_{h,\hbar}$
  12. The coproduct on $\mathbf{N}(Q)_{h,\hbar}$
  13. Construction of coproduct on $\mathbf{N}(Q)_{h,\hbar}$
  14. Well-definedness of the coproduct
  15. Bialgebra conditions for $\mathbf{N}(Q)_{h,\hbar}$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Suppose that $Q$ is a finite quiver. Let $L:=(k\overline{ Q})_{\natural}$ be the commutator quotient space of $k\overline Q$, equipped with the necklace Lie bialgebra structure. Then there is a commutative diagram of biquantization of $L$: where the vertical maps are the quantizations and the horizontal maps are the coquantization maps respectively. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (66)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: Drinfeld Drinfeld
  • Theorem 2.3: Bocklandt-Le Bruyn BocLeB, Ginzburg Ginzburg and Schedler Schedler
  • Definition 2.4
  • Definition 2.5
  • Example 2.6: Turaev Turaev
  • Definition 2.7: Quantization of a Poisson algebra
  • Definition 2.8: Coquantization of a co-Poisson coalgebra
  • Theorem 2.9: Turaev Turaev
  • ...and 56 more