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Deformation families of Novikov bialgebras via differential antisymmetric infinitesimal bialgebras

Yanyong Hong, Chengming Bai, Li Guo

Abstract

Generalizing S. Gelfand's classical construction of a Novikov algebra from a commutative differential algebra, a deformation family $(A,\circ_q)$, for scalars $q$, of Novikov algebras is constructed from what we call an admissible commutative differential algebra, by adding a second linear operator to the commutative differential algebra with certain admissibility condition. The case of $(A,\circ_0)$ recovers the construction of S. Gelfand. This admissibility condition also ensures a bialgebra theory of commutative differential algebras, enriching the antisymmetric infinitesimal bialgebra. This way, a deformation family of Novikov bialgebras is obtained, under the further condition that the two operators are bialgebra derivations. As a special case, we obtain a bialgebra variation of S. Gelfand's construction with an interesting twist: every commutative and cocommutative differential antisymmetric infinitesimal bialgebra gives rise to a Novikov bialgebra whose underlying Novikov algebra is $(A,\circ_{-\frac{1}{2}})$ instead of $(A,\circ_0)$. The close relations of the classical bialgebra theories with Manin triples, classical Yang-Baxter type equations, $\mathcal{O}$-operators, and pre-structures are expanded to the two new bialgebra theories, in a way that is compatible with the just established connection between the two bialgebras. As an application, Novikov bialgebras are obtained from admissible differential Zinbiel algebras.

Deformation families of Novikov bialgebras via differential antisymmetric infinitesimal bialgebras

Abstract

Generalizing S. Gelfand's classical construction of a Novikov algebra from a commutative differential algebra, a deformation family , for scalars , of Novikov algebras is constructed from what we call an admissible commutative differential algebra, by adding a second linear operator to the commutative differential algebra with certain admissibility condition. The case of recovers the construction of S. Gelfand. This admissibility condition also ensures a bialgebra theory of commutative differential algebras, enriching the antisymmetric infinitesimal bialgebra. This way, a deformation family of Novikov bialgebras is obtained, under the further condition that the two operators are bialgebra derivations. As a special case, we obtain a bialgebra variation of S. Gelfand's construction with an interesting twist: every commutative and cocommutative differential antisymmetric infinitesimal bialgebra gives rise to a Novikov bialgebra whose underlying Novikov algebra is instead of . The close relations of the classical bialgebra theories with Manin triples, classical Yang-Baxter type equations, -operators, and pre-structures are expanded to the two new bialgebra theories, in a way that is compatible with the just established connection between the two bialgebras. As an application, Novikov bialgebras are obtained from admissible differential Zinbiel algebras.
Paper Structure (10 sections, 44 theorems, 131 equations)

This paper contains 10 sections, 44 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Deformation families of Novikov bialgebras via admissible commutative differential bialgebras
  3. Novikov algebras via admissible commutative differential algebras
  4. Novikov bialgebras and commutative and cocomutative differential ASI bialgebras
  5. Construction of Novikov bialgebras from commutative and cocommutative differential ASI bialgebras
  6. Characterizations in terms of Manin triples
  7. Novikov Yang-Baxter equation via admissible associative Yang-Baxter equation
  8. NYBE via AYBE: the general case
  9. NYBE via AYBE: $\mathcal{O}$-operators
  10. NYBE via AYBE: pre-Novikov algebras via admissible differential Zinbiel algebras

Key Result

Proposition 2.1

(GD1, S. Gelfand) Let $(A, \cdot, D)$ be a commutative differential algebra. Define a binary operation $\circ$ on $A$ by Then $(A,\circ)$ is a Novikov algebra, called the Novikov algebra induced from $(A, \cdot, D)$.

Theorems & Definitions (108)

  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • Proposition 2.8
  • proof
  • ...and 98 more