Extending Structures for Lie Conformal Algebras

The ℂ[∂]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}[\partial ]$\end{document}-split extending structures problem for Lie conformal algebras is studied. In this paper, we introduce the definition of unified product of a given Lie conformal algebra R and a given ℂ[∂]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}[\partial ]$\end{document}-module Q. This product includes some other interesting products of Lie conformal algebras such as twisted product, crossed product, and bicrossed product. Using this product, a cohomological type object is constructed to provide a theoretical answer to the ℂ[∂]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}[\partial ]$\end{document}-split extending structures problem. Moreover, using this general theory, we investigate crossed product and bicrossed product in detail, which give the answers for the ℂ[∂]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}[\partial ]$\end{document}-split extension problem and the ℂ[∂]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}[\partial ]$\end{document}-split factorization problem respectively.