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On the structure of split regularHom-Lie–Rinehart algebras

Volume 160 / 2020

Shengxiang Wang, Xiaohui Zhang, Shuangjian Guo Colloquium Mathematicum 160 (2020), 1-18 MSC: 17A60, 17B22, 17B60, 17B65. DOI: 10.4064/cm7903-9-2019 Published online: 21 January 2020

Abstract

The aim of this paper is to study the structures of split regular Hom-Lie–Rinehart algebras. Let $(L,A)$ be a split regular Hom-Lie–Rinehart algebra. We first show that $L$ is of the form $L=U+\sum _{[\gamma ]\in \Gamma /\thicksim }I_{[\gamma ]}$ with $U$ a vector space complement of $\sum _{\gamma \in \Gamma ,-\gamma \in \Lambda }A_{-\gamma }L_{\gamma }+\sum _{\gamma \in \Gamma }[L_{-\gamma },L_{\gamma }]$ in $H$ and with each $I_{[\gamma ]}$ being a well defined ideal of $L,$ satisfying $[I_{[\gamma ]},I_{[\delta ]}]=0$ if $I_{[\gamma ]}\neq I_{[\delta ]}$. Also, we discuss the weight spaces and decompositions of $A$ and present the relation between the decompositions of $L$ and $A$. Finally, we consider the structures of tight split regular Hom-Lie–Rinehart algebras.

Authors

  • Shengxiang WangSchool of Mathematics and Finance
    Chuzhou University
    Chuzhou, 239000, P.R. China
    e-mail
  • Xiaohui ZhangSchool of Mathematical Sciences
    Qufu Normal University
    Qufu, 273165, P.R. China
    e-mail
  • Shuangjian GuoSchool of Mathematics and Statistics
    Guizhou University of Finance and Economics
    Guiyang, 550025, P.R. China
    e-mail

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