On quadratic Novikov algebras
Xiaofeng Dong, Yanyong Hong
TL;DR
This work studies finite-dimensional quadratic Novikov algebras, i.e., Novikov algebras with a nondegenerate symmetric invariant form $B(\cdot,\cdot)$. It develops foundational properties, including a duality between the adjoint representation and its dual, and classifies low-dimensional cases over ${\mathbb C}$, revealing a unique nontrivial 2D example and several 3D families. A central contribution is the double extension construction, showing that any quadratic Novikov algebra with a nonzero isotropic ideal arises from extending a smaller quadratic Novikov algebra by a Novikov algebra with a symmetric form, with explicit compatibility conditions. The results yield a unified framework for constructing and understanding quadratic Novikov algebras, linking to Novikov bialgebras and Lie conformal algebras, and provide concrete 4D examples illustrating the method.
Abstract
A quadratic Novikov algebra is a Novikov algebra $(A, \circ)$ with a symmetric and nondegenerate bilinear form $B(\cdot,\cdot)$ satisfying $B(a\circ b, c)=-B(b, a\circ c+c\circ a)$ for all $a$, $b$, $c\in A$. This notion appeared in the theory of Novikov bialgebras. In this paper, we first investigate some properties of quadratic Novikov algebras and give a decomposition theorem of quadratic Novikov algebras. Then we present a classification of quadratic Novikov algebras of dimensions $2$ and $3$ over $\mathbb{C}$ up to isomorphism. Finally, a construction of quadratic Novikov algebras called double extension is presented and we show that any quadratic Novikov algebra containing a nonzero isotropic ideal can be obtained by double extensions. Based on double extension, an example of quadratic Novikov algebras of dimension 4 is given.