# Hochschild cohomology of a Sullivan algebra

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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

Algebra
Vector spaces

### Cite this

@article{134f8495cfa94e2093b7398f8a1d7092,
title = "Hochschild cohomology of a Sullivan algebra",
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, $$Sp(5)/Sp(2) \times Sp(3).$$",
author = "Oteng Maphane",
year = "2016",
language = "English",
volume = "1",
journal = "Petroleum Abstracts",
issn = "0031-6423",
number = "1",

}

In: Petroleum Abstracts, Vol. 1, No. 1, 2016.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Hochschild cohomology of a Sullivan algebra

AU - Maphane, Oteng

PY - 2016

Y1 - 2016

N2 - 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, $$Sp(5)/Sp(2) \times Sp(3).$$

AB - 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, $$Sp(5)/Sp(2) \times Sp(3).$$

M3 - Article

VL - 1

JO - Petroleum Abstracts

JF - Petroleum Abstracts

SN - 0031-6423

IS - 1

ER -