Hochschild Cohomology Of Algebras Research Articles

Hochschild Cohomology for Algebras of Semidihedral Type. X. Cohomology Algebra for the Exceptional Local Algebras

Hochschild cohomology algebra is described in term of generators and relations for a family of local algebras of semidihedral type. This family appears in famous K. Erdmann’s classification only if the characteristic of the base field is equal to 2.

Invariance of the Goresky–Hingston algebra on reduced Hochschild homology

We prove that two quasi-isomorphic simply connected differential graded associative Frobenius algebras have isomorphic Goresky–Hingston algebras on their reduced Hochschild homology. Our proof is based on relating the Goresky–Hingston algebra on reduced Hochschild homology to the singular Hochschild cohomology algebra. For any simply connected oriented closed manifold M $M$ of dimension k $k$ , the Goresky–Hingston algebra on reduced Hochschild homology induces an algebra structure of degree k − 1 $k-1$ on H ¯ ∗ ( L M ; Q ) $\overline{\mathrm{H}}^*(LM;\mathbb {Q})$ , the reduced rational cohomology of the free loop space of M $M$ . As a consequence of our algebraic result, we deduce that the isomorphism class of the induced algebra structure on H ¯ ∗ ( L M ; Q ) $\overline{\mathrm{H}}^*(LM;\mathbb {Q})$ is an invariant of the homotopy type of M $M$ .

Superfiltered A∞$A_\infty$‐deformations of the exterior algebra, and local mirror symmetry

The exterior algebra $E$ on a finite-rank free module $V$ carries a $\mathbb{Z}/2$-grading and an increasing filtration, and the $\mathbb{Z}/2$-graded filtered deformations of $E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on $V$. We extend this result to $A_\infty$-algebra deformations $\mathcal{A}$, showing that they are classified by formal functions on $V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho-Hong-Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such $\mathcal{A}$. By applying these ideas to a related construction of Cho-Hong-Lau we prove a local form of homological mirror symmetry: the Floer $A_\infty$-algebra of a monotone Lagrangian torus is quasi-isomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential.

Homotopy liftings and Hochschild cohomology of some twisted tensor products

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov’s homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.

Book Review: Hochschild cohomology for algebras

Book Review: Hochschild cohomology for algebras

Hochschild Cohomology of Algebras of Semidihedral Type. IX: Exceptional Local Algebras

The Hochschild cohomology groups for a family of local algebras of semidihedral type are calculated. The family occurs in the Erdmann classification only in the case where the characteristic of underlying field equals 2.

Hochschild Cohomology of Algebras of Quaternion Type. IV: Cohomology Algebra for Exceptional Local Algebras

The description of the Hochschild cohomology algebra for a family of local algebras of quaternion type is given in terms of generators and relations. This family appears in the famous K. Erdmann’s classification only in the case where the characteristic of the base field is equal to 2.

Hochschild Cohomology of Algebras of Dihedral Type. VIII. Hochshild Cohomology Algebra for the Family D(2ℬ)(k, s, 0) in Characteristic 2

The Hochschild cohomology algebra for the algebras of dihedral type in the subfamily of the family D(2ℬ), for which the parameter c is equal to 0, are described. The calculation of multiplication in this cohomology algebra, uses the minimal bimodule projective resolution for algebras under consideration, that was constructed in the previous paper of the authors. The obtained results allow to describe the Hochschild cohomology algebra also for algebras with c = 0 in the family D(2$$ \mathcal{A} $$).

Hochschild Cohomology for Algebras of Dihedral Type. VII. The Family D(3R)

The Hochschild cohomology groups for algebras of dihedral type which are contained in the family D(3R) (from the famous K.Erdmann’s classification) are calculated. In the calculation, a construction of the minimal bimodule resolution for algebras from the family under discussion, that is defined in the present paper, is used.

The Hochschild cohomology of square-free monomial complete intersections

We determine the Hochschild cohomology algebras of the square-free monomial complete intersections. In particular we provide an explicit formula for the cup product which gives the cohomology module an algebra structure and then we describe this structure in terms of generators and relations. In addition, we compute the Hilbert series of the Hochschild cohomology of these algebras.

Hochschild cohomology of polynomial representations of GL2

We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of GL2 over an algebraically closed field of characteristic p>2 , that is, of any block whose number of simple modules is a power of p>2. These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications.

Hochschild Cohomology for Algebras of Semidihedral Type. VII. Algebras with a Small Parameter

Hochschild Cohomology for Algebras of Semidihedral Type. VII. Algebras with a Small Parameter

Homological invariants relating the super Jordan plane to the Virasoro algebra

Nichols algebras are an important tool for the classification of Hopf algebras. Within those with finite GK dimension, we study homological invariants of the super Jordan plane, that is, the Nichols algebra A=B(V(−1,2)). These invariants are Hochschild homology, the Hochschild cohomology algebra, the Lie structure of the first cohomology space – which is a Lie subalgebra of the Virasoro algebra – and its representations Hn(A,A) and also the Yoneda algebra. We prove that the algebra A is K2. Moreover, we prove that the Yoneda algebra of the bosonization A#kZ of A is also finitely generated, but not K2.

Hochschild Cohomology of Polynomial Representations of $\mathrm{GL}_2$

We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of \mathrm{GL}_2 over an algebraically closed field of characteristic p>2 , that is, of any block whose number of simple modules is a power of p . These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications.

Hochschild Cohomology for Algebras of Dihedral Type. V. The Family D(3K) in Characteristic Different from 2

Hochschild Cohomology for Algebras of Dihedral Type. V. The Family D(3K) in Characteristic Different from 2

Homological epimorphisms, recollements and Hochschild cohomology – with a conjecture by Snashall–Solberg in view

We show that recollements of module categories give rise to homomorphisms between the associated Hochschild cohomology algebras which preserve the strict Gerstenhaber structure, i.e., the cup product, the graded Lie bracket and the squaring map. We review various long exact sequences in Hochschild cohomology and apply our results in order to realise that the occurring maps preserve the strict Gerstenhaber structure as well. As a byproduct, we generalise a known long exact cohomology sequence of Koenig–Nagase to arbitrary surjective homological epimorphisms. We use our observations to motivate and formulate a variation of the finite generation conjecture by Snashall–Solberg.

BV differential on Hochschild cohomology of Frobenius algebras

For a finite-dimensional Frobenius k-algebra R with a Nakayama automorphism ν, we define an algebra HH⁎(R)ν↑. If the order of ν is not divisible by the characteristic of k, this algebra is isomorphic to the Hochschild cohomology algebra of R. We prove that this algebra is a BV algebra. Moreover, we construct a BV differential on it in a canonical way. We use this fact to calculate the Gerstenhaber algebra structure and a BV structure on the Hochschild cohomology algebras of a family of self-injective algebras of tree type Dn.

BASIC MORITA EQUIVALENCES AND A CONJECTURE OF SASAKI FOR COHOMOLOGY OF BLOCK ALGEBRAS OF FINITE GROUPS

We show, using basic Morita equivalences between block algebras of nite groups, that the Conjecture of H. Sasaki from [9] is true for a new class of blocks called nilpotent covered blocks. When this Conjecture is true we dene some transfer maps between cohomology of block algebras and we analyze them through transfer maps between Hochschild cohomology algebras.

Hochschild Cohomology for Algebras of Semidihedral Type. IV. The Cohomology Algebra for the Family SD(2ℬ)2(k, t, c) in the Case c = 0

The paper continues the previous author’s paper in which the bimodule resolution was constructed for algebras of semidihedral type from the family SD(2ℬ)2. In the present paper, using this resolution the multiplicative structure of the Hochschild cohomology algebra for algebras in the mentioned family is described over a base field of characteristic 2 under an additional assumption that the parameter c contained in the defining relations of the algebras is equal to zero.

A category of kernels for equivariant factorizations and its implications for Hodge theory

We provide a factorization model for the continuous internal Hom, in the homotopy category of k-linear dg-categories, between dg-categories of equivariant factorizations. This motivates a notion, similar to that of Kuznetsov, which we call the extended Hochschild cohomology algebra of the category of equivariant factorizations. In some cases of geometric interest, extended Hochschild cohomology contains Hochschild cohomology as a subalgebra and Hochschild homology as a homogeneous component. We use our factorization model for the internal Hom to calculate the extended Hochschild cohomology for equivariant factorizations on affine space. Combining the computation of extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov recovers and extends Griffiths’ classical description of the primitive cohomology of a smooth, complex projective hypersurface in terms of homogeneous pieces of the Jacobian algebra. In the process, the primitive cohomology is identified with the fixed subspace of the cohomological endomorphism associated to an interesting endofunctor of the bounded derived category of coherent sheaves on the hypersurface. We also demonstrate how to understand the whole Jacobian algebra as morphisms between kernels of endofunctors of the derived category. Finally, we present a bootstrap method for producing algebraic cycles in categories of equivariant factorizations. As proof of concept, we show how this reproves the Hodge conjecture for all self-products of a particular K3 surface closely related to the Fermat cubic fourfold.