关于Hom- $F$ -流形代数与量子化
On Hom-$F$-manifold algebras and quantization
摘要：
$F$ -流形代数的概念是$F$ -流形的代数描述。本文引入了Hom- $F$ -流形代数的概念，它是$F$ -流形代数和Hom-Poisson代数的推广.本文发展了Hom- $F$ -流形代数的表示论，通过引入Hom-预$F$ -流形代数，推广了Hom-预Poisson代数的概念，并通过次相邻可换Hom-结合代数和次相邻Hom-李代数得到了Hom- $F$ -流形代数.利用Hom- $F$ -流形代数上的$\huaO$ -算子，构造了模上的Hom-准$F$ -流形代数.然后研究了交换Hom-结合代数的Hom-pre-Lie形式变形，证明了Hom- $F$ -流形代数是相应的半经典极限。最后，我们利用上同调理论研究了交换Hom-结合代数的Hom-Lie无穷小变形和Hom-pre-Lie $n$ -变形到Hom-pre-Lie $(n+1)$ -变形的推广.
英文摘要：
The notion of a $F$-manifold algebras is an algebraic description of a $F$-manifold. In this paper, we introduce the notion of Hom-$F$-manifold algebras which is generalisation of $F$-manifold algebras and Hom-Poisson algebras. We develop the representation theory of Hom-$F$-manifold algebras and generalize the notion of Hom-pre-Poisson algebras by introducing the Hom-pre-$F$-manifold algebras which give rise to a Hom-$F$-manifold algebra through the sub-adjacent commutative Hom-associative algebra and the sub-adjacent Hom-Lie algebra. Using $\huaO$-operators on a Hom-$F$-manifold algebras we construct a Hom-pre-$F$-manifold algebras on a module. Then, we study Hom-pre-Lie formal deformations of commutative Hom-associative algebra and prove that Hom-$F$-manifold algebras are the corresponding semi-classical limits. Finally, we study Hom-Lie infinitesimal deformations and extension of Hom-pre-Lie $n$-deformation to Hom-pre-Lie $(n+1)$-deformation of a commutative Hom-associative algebra via cohomology theory.
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