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<p>In this paper, we show that there is a pre-Lie algebra structure on the tensor product of a pre-Novikov algebra and a right Novikov dialgebra and the tensor product of a pre-Novikov algebra and a special right Novikov algebra on the vector space of Laurent polynomials being a pre-Lie algebra characterizes the pre-Novikov algebra. The latter is called the affinization of a pre-Novikov algebra. Moreover, we extend this construction of pre-Lie algebras and the affinization of pre-Novikov algebras to the context of bialgebras. We show that there is a completed pre-Lie bialgebra structure on the tensor product of a pre-Novikov bialgebra and a quadratic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {Z}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>-graded right Novikov algebra. Moreover, a pre-Novikov bialgebra can be characterized by the fact that its affinization by a quadratic <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb {Z}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>-graded right Novikov algebra on the vector space of Laurent polynomials gives an infinite-dimensional completed pre-Lie bialgebra. Note that the reason why we choose a quadratic right Novikov algebra instead of a right Novikov dialgebra with a special bilinear form is also given. Furthermore, we construct symmetric completed solutions of the <i>S</i>-equation in the induced pre-Lie algebra by symmetric solutions of the pre-Novikov Yang-Baxter equation in a pre-Novikov algebra.</p>

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Infinite-Dimensional Pre-Lie Bialgebras via Affinization of Pre-Novikov Bialgebras

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Yue Li,
  • Yanyong Hong

摘要

In this paper, we show that there is a pre-Lie algebra structure on the tensor product of a pre-Novikov algebra and a right Novikov dialgebra and the tensor product of a pre-Novikov algebra and a special right Novikov algebra on the vector space of Laurent polynomials being a pre-Lie algebra characterizes the pre-Novikov algebra. The latter is called the affinization of a pre-Novikov algebra. Moreover, we extend this construction of pre-Lie algebras and the affinization of pre-Novikov algebras to the context of bialgebras. We show that there is a completed pre-Lie bialgebra structure on the tensor product of a pre-Novikov bialgebra and a quadratic \({\mathbb {Z}}\) Z -graded right Novikov algebra. Moreover, a pre-Novikov bialgebra can be characterized by the fact that its affinization by a quadratic \({\mathbb {Z}}\) Z -graded right Novikov algebra on the vector space of Laurent polynomials gives an infinite-dimensional completed pre-Lie bialgebra. Note that the reason why we choose a quadratic right Novikov algebra instead of a right Novikov dialgebra with a special bilinear form is also given. Furthermore, we construct symmetric completed solutions of the S-equation in the induced pre-Lie algebra by symmetric solutions of the pre-Novikov Yang-Baxter equation in a pre-Novikov algebra.