# Mathematics > Quantum Algebra

[Submitted on 25 Feb 2024]

# Title:Deformation families of Novikov bialgebras via differential antisymmetric infinitesimal bialgebras

View PDF HTML (experimental)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.

Current browse context:

math.QA

### References & Citations

# Bibliographic and Citation Tools

Bibliographic Explorer

*(What is the Explorer?)*
Litmaps

*(What is Litmaps?)*
scite Smart Citations

*(What are Smart Citations?)*# Code, Data and Media Associated with this Article

CatalyzeX Code Finder for Papers

*(What is CatalyzeX?)*
DagsHub

*(What is DagsHub?)*
Gotit.pub

*(What is GotitPub?)*
Papers with Code

*(What is Papers with Code?)*
ScienceCast

*(What is ScienceCast?)*# Demos

# Recommenders and Search Tools

Influence Flower

*(What are Influence Flowers?)*
Connected Papers

*(What is Connected Papers?)*
CORE Recommender

*(What is CORE?)*# arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? **Learn more about arXivLabs**.