# Hochschild cohomology of a Sullivan algebra

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## Abstract

A derivation $\theta$ is a $k$-linear map $\theta :A^{n}\rightarrow A^{n-k}$ such that $\theta (ab)=\theta (a)b+(-1)^{k|a|}a\theta (b),$ where $A=\underset{n\geq 0}{\oplus}A^{n}$ is a commutative graded algebra over a commutative ring $\Bbbk.$ Let $\der _{k}A$ denote the vector space of all derivations of degree $k$ and $$\der A=\underset{k}{\oplus}\der _{k}A.$$ If $A=(\wedge V,d)$ is a minimal Sullivan algebra, then there is a homomorphism $\phi : (\wedge _{A}L,d_{0})\rightarrow C^{\ast}(A;A)$ which induces an isomorphism of graded Gerstenhaber algebras in homology, where $L=s^{-1}(\der A).$ The latter shows that the Hochschild cochain complex of $A$ with coefficients in $A$ can be computed in terms of derivations of $A.$ In this talk we shall use this method to compute Hochschild cohomology of the minimal Sullivan algebra of a formal homogeneous space, the Grassmannian over the quaternion division algebra, $$Sp(5)/Sp(2) \times Sp(3).$$
Original language English Petroleum Abstracts 1 1 Published - 2016

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