Duality and products in algebraic (co)homology theories

Abstract

The origin and interplay of products and dualities in algebraic\ud (co)homology theories is ascribed to a ×_A-Hopf algebra structure\ud on the relevant universal enveloping algebra. This provides a unified\ud treatment for example of results by Van den Bergh about\ud Hochschild (co)homology and by Huebschmann about Lie–Rinehart\ud (co)homology

When Ext is a Batalin–Vilkovisky algebra

“…However, it is known for the following classes of algebras that H ‚ pA, Aq does indeed admit the structure of a BV algebra: -symmetric algebras [Tr,Me1]; -Frobenius algebras with semisimple Nakayama automorphism [LaZhZi, Vo]; -Calabi-Yau algebras [Gi]; -twisted Calabi-Yau algebras [KoKr3], and probably more. These results were obtained by various different approaches, for example, those in [Gi,KoKr3,LaZhZi] by passing through a sort of Poincaré duality [VdB,La,KoKr1] and using the notion of noncommutative differential calculus [TaTs], or more precisely, the notion of a BV module structure on the respective Hochschild homology H ‚ pA, Aq, see also [Ko2] for a generalised treatment.…”

Twisted Homology of Quantum SL(2) - Part II

“…We realised subsequently [15] that the coefficients σ A also arise from Poincaré duality in Hochschild (co)homology: using the general theory of Van den Bergh [32] we showed that is the fundamental class in Hochschild homology, that is, it corresponds under (2) to 1 ∈ H 0 (A, A) (identified with the centre of A), and (2) is given by · dA [21]. A cycle in the standard Hochschild complex C • (A, σ q −2 ,1 A) = A ⊗•+1 representing dA is given explicitly in (13).…”

Koszul duality and the Hochschild cohomology of Artin–Schelter regular algebras

“…. In a similar fashion, Kowalzig and Krahmer proved in [12] that the cohomology is concentrated in the subcomplex corresponding to the eigenvalue 1, namely…”

“…[12], Theorem 1; [13] Theorem 1.5). Let ∪ 1 , ∩ 1 and {−, −} 1 be the restrictions of the cup product, cap product and Gerstenhaber bracket to the homology and cohomology spaces associated with the eigenvalue λ = 1.…”