Deformation theories controlled by Hochschild cohomologies
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Deformation theories controlled by Hochschild cohomologies Samuel Carolus1 · Samuel A. Hokamp2 · Jacob Laubacher2
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract We explore how the higher order Hochschild cohomology controls a deformation theory when the simplicial set models the 3-sphere. Besides generalizing to the d-sphere for any d ≥ 1 , we also investigate a deformation theory corresponding to the tertiary Hochschild cohomology, which naturally reduces to those studied for the secondary and usual Hochschild cohomologies under certain conditions. Keywords Deformations of algebras · Higher order Hochschild cohomology · Tertiary Hochschild cohomology Mathematics Subject Classification Primary 16S80 · Secondary 16E40
1 Introduction Higher order Hochschild (co)homology was implicitly defined by Anderson in [1], and was given an explicit description in [7]. The case for when the simplicial set models the d-sphere was investigated more extensively in [5]. A deformation theory of the algebra A controlled by the higher order Hochschild cohomology over the 2-sphere was studied in [3]. One of the goals of this paper is to generalize their argument.
Communicated by Iván Ezequiel Angiono. * Jacob Laubacher [email protected] Samuel Carolus s‑[email protected] Samuel A. Hokamp [email protected] 1
Department of Mathematics and Statistics, Ohio Northern University, Ada, OH 45810, USA
2
Department of Mathematics, St. Norbert College, De Pere, WI 54115, USA
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
In Sect. 3 we use the simplicial structure for the 3-sphere presented in [2], and also use their natural extension when considering the d-sphere for any d ≥ 1 . We show how the higher order Hochschild cohomology over the d-sphere controls a deformation theory, giving special attention to the case when d = 3. When the simplicial set models S1 , it is well known that one recovers the usual Hochschild cohomology, which was introduced in 1945 in [6]. Almost 20 years later in [4], Gerstenhaber used this Hochschild cohomology, denoted H∗ (A, A) , to describe deformations of the algebra A. That is, for a multiplication law mt ∶ A[[t]] ⊗ A[[t]] ⟶ A[[t]] determined mt (a ⊗ b) = ab + c1 (a ⊗ b)t + c2 (a ⊗ b)t2 + ⋯ 𝕜-linear by with maps ci ∶ A ⊗ A ⟶ A , one sees that mt is associative mod t2 if and only if c1 is a 2-cocycle. As is classical, the class of c1 is determined by the isomorphism class of mt . More generally, assuming mt is associative mod tn+1 , for mt to be associative mod tn+2 requires c1 ◦cn + c2 ◦cn−1 + ⋯ + cn ◦c1 to be 0 in H3 (A, A). In 2016, Staic showed in [8] that when one wants to study deformations of A that have a nontrivial B-algebra structure, one can use the secondary Hochschild cohomology. The B-algebra structure here is induced by a morphism of 𝕜-algebras 𝜀 ∶ B ⟶ A , where B is a commutative 𝕜-algebra. This cohomology theory has the property that when one takes B = 𝕜 , one recovers the usual Hochschild cohomology. In Sect. 4 we study
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