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

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

Research output: Contribution to journal › Article

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