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A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras

Yangyon Hong, Chengming Bai, Li Guo

Abstract

Gel'fand-Dorfman algebras (GD algebras) give a natural construction of Lie conformal algebras and are in turn characterized by this construction. In this paper, we define the Gel'fand-Dorfman bialgebra (GD bialgebras) and enrich the above construction to a construction of Lie conformal bialgebras by GD bialgebras. As a special case, Novikov bialgebras yield Lie conformal bialgebras. We further introduce the notion of the Gel'fand-Dorfman Yang-Baxter equation (GDYBE), whose skew-symmetric solutions produce GD bialgebras. Moreover, the notions of $\mathcal{O}$-operators on GD algebras and pre-Gel'fand-Dorfman algebras (pre-GD algebras) are introduced to provide skew-symmetric solutions of the GDYBE. The relationships between these notions for GD algebras and the corresponding ones for Lie conformal algebras are given. In particular, there is a natural construction of Lie conformal bialgebras from pre-GD algebras. Finally, GD bialgebras are characterized by certain matched pairs and Manin triples of GD algebras.

A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras

Abstract

Gel'fand-Dorfman algebras (GD algebras) give a natural construction of Lie conformal algebras and are in turn characterized by this construction. In this paper, we define the Gel'fand-Dorfman bialgebra (GD bialgebras) and enrich the above construction to a construction of Lie conformal bialgebras by GD bialgebras. As a special case, Novikov bialgebras yield Lie conformal bialgebras. We further introduce the notion of the Gel'fand-Dorfman Yang-Baxter equation (GDYBE), whose skew-symmetric solutions produce GD bialgebras. Moreover, the notions of -operators on GD algebras and pre-Gel'fand-Dorfman algebras (pre-GD algebras) are introduced to provide skew-symmetric solutions of the GDYBE. The relationships between these notions for GD algebras and the corresponding ones for Lie conformal algebras are given. In particular, there is a natural construction of Lie conformal bialgebras from pre-GD algebras. Finally, GD bialgebras are characterized by certain matched pairs and Manin triples of GD algebras.
Paper Structure (10 sections, 39 theorems, 122 equations)

This paper contains 10 sections, 39 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. GD bialgebras corresponding to a class of Lie conformal bialgebras
  3. A correspondence between GD (co)algebras and Lie conformal (co)algebras
  4. GD bialgebras and the relationship with Lie conformal bialgebras
  5. Gel'fand-Dorfman Yang-Baxter equation and the relationship with classical conformal Yang-Baxter equation
  6. GDYBE, $\mathcal{O}$-operators and pre-GD algebras
  7. GDYBE and classical conformal Yang-Baxter equation
  8. Characterizations of GD bialgebras and relations with Lie conformal bialgebras
  9. Equivalent characterizations of GD bialgebras
  10. Correspondences between GD bialgebras and Lie conformal bialgebras in terms of matched pairs and Manin triples

Key Result

Proposition 2.4

X1 Let $(A, \circ, [\cdot,\cdot])$ be a ${\bf k}$-vector space with binary operations $\circ$ and $[\cdot,\cdot]$. Equip the free ${\bf k}[\partial]$-module $R:={\bf k}[\partial]A(={\bf k}[\partial]\otimes_{\bf k} A)$ with the bilinear map where $a\star b=a\circ b+b\circ a$. Then $(R,[\cdot_\lambda\cdot])$ is a Lie conformal algebra if and only if $(A, \circ, [\cdot,\cdot])$ is a GD algebra. We c

Theorems & Definitions (99)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • Remark 2.9
  • ...and 89 more