关于色Hopf与李代数的上同调的注记
Note on The Cohomology of Color Hopf and Lie Algebras
摘要：
设$A$是具有双射对极的$(G, χ)$ -Hopf代数，$M$是$G$ -分次$A$ -双模.证明了存在同构\mathrm{HH}^*_{\rm gr}（A，M）\cong{\rm Ext}^*_{A{-}{\rm gr}}（\K，{^{ad}（M）}），其中$\K$通过$A$的可数性被看作平凡分次$A$ -模，$^{ad} M$是与分次$A$ -双模$M$相联系的伴随$A$-模，$\mathrm{HH}_{\rm gr}$表示$G$ -分次Hochschild上同调.作为应用，我们证明了色李代数$L$的上同调同构于泛包络代数$U(L)$的分次Hochschild上同调，解决了M. Scheunert
英文摘要：
Let $A$ be a $(G, χ)$-Hopf algebra with bijection antipode and let $M$ be a $G$-graded $A$-bimodule. We prove that there exists an isomorphism \mathrm{HH}^*_{\rm gr}(A, M)\cong{\rm Ext}^*_{A{-}{\rm gr}} (\K, {^{ad}(M)}), where $\K$ is viewed as the trivial graded $A$-module via the counit of $A$, $^{ad} M$ is the adjoint $A$-module associated to the graded $A$-bimodule $M$ and $\mathrm{HH}_{\rm gr}$ denotes the $G$-graded Hochschild cohomology. As an application, we deduce that the cohomology of color Lie algebra $L$ is isomorphic to the graded Hochschild cohomology of the universal enveloping algebra $U(L)$, solving a question of M. Scheunert.
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