Diagrams of Lie algebras.
A diagram of algebras over a small category \(\mathcal C\) is a contravariant functor \(\mathbb{A}\) from \(\mathcal C\) to some category \(\mathcal A\) of algebras (e.g. associative, Lie,\dots). Every diagram has its cohomology. A significant difference between the associative case and others is that in the associative case one has the `Cohomology Comparison Theorem' that asserts that for every diagram \(\mathbb{A}\) of associative unital algebras there is a single algebra which we denote \(\mathbb{A}!!\) and for every diagram \(\mathbb{M}\) of \(\mathbb{A}\)-bimodules there is a single \(\mathbb{A}!!\)-bimodule \(\mathbb{M}!!\) such that \(H^*(\mathbb{A},\mathbb{M})\) is canonically isomorphic to \(H^*(\mathbb{A}!!,\mathbb{M}!!)\) (recall that if we have a diagram \(\mathbb{A}\) of \(\mathcal A\)-algebras over a small category \(\mathcal C\) then by an \(\mathbb{A}\)-module \(\mathbb{M}\) we will mean a diagram of Abelian groups over \(\mathcal C\) such that for each \(i\in\text{ob\,}\mathcal C\) the group \(\mathbb{M}(i)\) is an \(\mathbb{A}(i)\)-module and for each morphism \(\phi\colon i\to j\) the map \(\mathbb{M}(\phi)\colon\mathbb{M}(j)\to\mathbb{M}(i)\) is a morphism of \(\mathbb{A}(j)\)-modules where \(\mathbb{M}(i)\) is viewed as an \(\mathbb{A}(j)\)-module by virtue of the morphism \(\mathbb{A}(\phi)\colon\mathbb{A}(j)\to\mathbb{A}(i)\)). In the Lie case, the authors show that if \(\mathbb{L}\) is a diagram of Lie algebras and \(\mathbb{M}\) a diagram of modules over it, then there is a single associative algebra \(\mathbb{A}!!\) and a single \(\mathbb{A}!!\)-bimodule \(\mathbb{M}!!\) over it such that the Lie cohomology \(H^*_{\text{Lie}}(\mathbb{L},\mathbb{M})\) is canonically isomorphic to the Hochschild cohomology \(H^*_{\text{Hoch}}(\mathbb{A}!!,\mathbb{M}!!)\).
scientific article; zbMATH DE number 2153391
scientific article