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The second cohomology group of a left skew brace with coefficients in a trivial left brace with non-trivial actions is defined, its connection with extensions of a left skew brace by a trivial brace is established and a Wells' like exact sequence relating the second cohomology group with inducible automorphisms of an extension of left skew braces is constructed. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give a cohomology theory for any operad equipped with a (higher) derivation. Our construction is based on an operadic level correspondence between higher derivations and sequences of derivations and the legitimacy of this indirect approach is justified by an illustration of how the formal deformation is controlled by the cohomology. The results are generalized to the bihom-context at the end of the paper. [ABSTRACT FROM AUTHOR]

We consider local Gorenstein duality for cochain spectra C ⁎ (B G ; R) on the classifying spaces of compact Lie groups G over complex orientable ring spectra R. We show that it holds systematically for a large array of examples of ring spectra R , including Lubin-Tate theories, topological K -theory, and various forms of topological modular forms. We also prove a descent result for local Gorenstein duality which allows us to access further examples. [ABSTRACT FROM AUTHOR]

Let a and b be two coprime positive integers and k an arbitrary field. We determine the ring structure of the Hochschild cohomology of the numerical semigroup algebras k [ s a , s b ] of embedding dimension two (thus also complete intersections) in terms of generators and relations. In addition, we compute the Hilbert series of these cohomology rings. [ABSTRACT FROM AUTHOR]

For a compact Lie group G we show that the representing spectrum for Borel cohomology generates its category of modules if G is connected or if coefficients are of characteristic p and π 0 (G) is a p -group. For a closed subgroup H of G we consider the map C ⁎ (B G) ⟶ C ⁎ (B H) and establish the sense in which it is relatively Gorenstein. Throughout, we pay careful attention to the importance of connectedness of the groups. [ABSTRACT FROM AUTHOR]

Let p be a prime number and let c , d be natural numbers. Then, the number of possible isomorphism types for the mod p cohomology algebra of a d -generated finite p -group of nilpotency class c is bounded by a function depending only on p , c and d. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we attempt to study the structure of multiplicative Lie algebras, the theory of extensions, the second cohomology groups of multiplicative Lie algebras, and in turn the Schur multipliers. The Schur–Hopf formula is established for multiplicative Lie algebras. We also introduce the group of nontrivial relations satisfied by the Lie product in a multiplicative Lie algebra, and study it as a functor arising from the presentations of multiplicative Lie algebras. Some applications in K-theory are also discussed. [ABSTRACT FROM AUTHOR]

Abstract We classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero such that its coradical is isomorphic to the algebra of functions k D m over a dihedral group D m , with m = 4 a ≥ 12. We obtain this classification by means of the lifting method, where we use cohomology theory to determine all possible deformations. Our result provides an infinite family of new examples of finite-dimensional copointed Hopf algebras over dihedral groups. [ABSTRACT FROM AUTHOR]

We study T. Willwacher's twisting endofunctor tw in the category of dg prop(erad)s P under the operad of (strongly homotopy) Lie algebras, i : L ie → P. It is proven that if P is a properad under properad of Lie bialgebras L ieb , then the associated twisted properad tw P becomes in general a properad under quasi-Lie bialgebras (rather than under L ieb). This result implies that the cyclic cohomology of any cyclic homotopy associative algebra has in general an induced structure of a quasi-Lie bialgebra. We show that the cohomology of the twisted properad tw L ieb is highly non-trivial — it contains the cohomology of the so called hairy graph complex introduced and studied recently in the context of the theory of long knots and the theory of moduli spaces M g , n of algebraic curves of arbitrary genus g with n punctures. Using a polydifferential functor from the category of props to the category of operads, we introduce and study two new twisting endofunctors, one in the category dg prop(erad)s P under the minimal resolution of L ieb , and one for the involutive version of L ieb. We compute the cohomology of the associated deformation complexes, and discuss their applications in string topology. [ABSTRACT FROM AUTHOR]

Abstract We provide some new computations of Farrell–Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell–Tate or Bredon homology, one needs cell complexes where cell stabilizers fix their cells pointwise. We provide two algorithms computing an efficient subdivision of a complex to achieve this rigidity property. Applying these algorithms to available cell complexes for PSL 4 (Z) provides computations of Farrell–Tate cohomology for small primes as well as the Bredon homology for the classifying spaces of proper actions with coefficients in the complex representation ring. [ABSTRACT FROM AUTHOR]

Abstract The category of rational G -equivariant cohomology theories for a compact Lie group G is the homotopy category of rational G -spectra and therefore tensor-triangulated. We show that its Balmer spectrum is the set of conjugacy classes of closed subgroups of G , with the topology corresponding to the topological poset of [7]. This is used to classify the collections of subgroups arising as the geometric isotropy of finite G -spectra. The ingredients for this classification are (i) the algebraic model of free spectra of the author and B. Shipley [14] , (ii) the Localization Theorem of Borel–Hsiang–Quillen [21] and (iii) tom Dieck's calculation of the rational Burnside ring [4]. [ABSTRACT FROM AUTHOR]

Abstract Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic. Furthermore, given a flat morphism of Cohen–Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is "additive". Finally, over a ring that is a homomorphic image of a regular local ring, a formula for local entropy in terms of an asymptotic partial Euler characteristic is given. [ABSTRACT FROM AUTHOR]

Abstract We introduce the notion of noncompact (partial) silting and (partial) tilting sets and objects in any triangulated category D with arbitrary (set-indexed) coproducts. We show that equivalence classes of partial silting sets are in bijection with t-structures generated by their co-heart whose heart has a generator, and in case D is compactly generated, this restricts to: i) a bijection between equivalence classes of self-small partial silting objects and left nondegenerate t-structures in D whose heart is a module category and whose associated cohomological functor preserves products; ii) a bijection between equivalence classes of classical silting objects and nondegenerate smashing and co-smashing t-structures whose heart is a module category. We describe the objects in the aisle of the t-structure associated to a partial silting set T as Milnor (or homotopy) colimits of sequences of morphisms with successive cones in Sum (T) [ n ]. We use this fact to develop a theory of tilting objects in very general AB3 abelian categories, a setting and its dual in which we show the validity of several well-known results of tilting and cotilting theory of modules. Finally, we show that if T is a bounded tilting set in a compactly generated algebraic triangulated category D and H is the heart of the associated t-structure, then the inclusion H ↪ D extends to a triangulated equivalence D (H) ⟶ ∼ D which restricts to bounded levels. [ABSTRACT FROM AUTHOR]

Abstract In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer–Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the fundamental group of a compact aspherical 3-manifold is a profinite Poincaré duality group relative to the profinite completions of the fundamental groups of its boundary components. [ABSTRACT FROM AUTHOR]

Abstract We describe the dimensions of low Hochschild cohomology spaces of exceptional periodic representation-infinite algebras of polynomial growth. As an application we obtain that an indecomposable non-standard periodic representation-infinite algebra of polynomial growth is not derived equivalent to a standard self-injective algebra. [ABSTRACT FROM AUTHOR]

Abstract We define a de Rham cohomology theory for analytic varieties over a valued field K ♭ of equal characteristic p with coefficients in a chosen untilt of the perfection of K ♭ by means of the motivic version of Scholze's tilting equivalence. We show that this definition generalizes the usual rigid cohomology in case the variety has good reduction. We also prove a conjecture of Ayoub yielding an equivalence between rigid analytic motives with good reduction and unipotent algebraic motives over the residue field, also in mixed characteristic. [ABSTRACT FROM AUTHOR]

Abstract In this paper we generalize the concepts of good and elementary gradings for an associative algebra A with a fixed multiplicative basis B. When the group G considered in the grading is abelian, we equip the set of good G -gradings of A with a structure of abelian group, which is denoted by G (B , G). Moreover, when A admits elementary G -gradings, we show the set E (B , G) of all elementary G -gradings of A is a subgroup of G (B , G). In this case, we introduce a cohomology for the pair (A , B) and we show that G (B , G) / E (B , G) is isomorphic to the first cohomology group of (A , B) with coefficients in G. [ABSTRACT FROM AUTHOR]

Extending the Eilenberg–Mac Lane approach, we introduce and explore higher-level cohomology theories for commutative monoids and compare them with pre-existing theories (Leech, Grillet, etc.). We offer a cohomological classification of symmetric monoidal groupoid structures and work out some explicit computations for cyclic monoids. [ABSTRACT FROM AUTHOR]

In earlier work the authors determined the Brauer kernel of extensions of degree p in characteristic p > 2 where the Galois group is a semidirect product of order ps for s | ( p − 1 ) . This result is extended here and tools are developed to compute the cohomological kernels H p m n + 1 ( E m / F ) for all n ≥ 0 where [ E m : F ] = p m and the Galois closure is a semidirect product of cyclic groups order p m and s where s | ( p − 1 ) . A six-term exact sequence describing the K-theory and cohomology of the extension is obtained. As an application it is shown that any F -division p -algebra of index p m split in E m is cyclic; a characteristic p analogue of a result of Vishne. The proofs use the de Rham Witt complex and Izhboldin groups, extending techniques developed earlier for the study of degree 4 extensions in characteristic two. The paper also provides background on the de Rham Witt Complex and Izhboldin groups difficult to track down in the literature. [ABSTRACT FROM AUTHOR]

Given a prime number p , a field F with char ( F ) = p and a positive integer n , we study the class-preserving modifications of Kato–Milne classes of decomposable differential forms. These modifications demonstrate a natural connection between differential forms and p -regular forms. A p -regular form is defined to be a homogeneous polynomial form of degree p for which there is no nonzero point where all the order p − 1 partial derivatives vanish simultaneously. We define a C ˜ p , m field to be a field over which every p -regular form of dimension greater than p m is isotropic. The main results are that for a C ˜ p , m field F , the symbol length of H p 2 ( F ) is bounded from above by p m − 1 − 1 and for any n ⩾ ⌈ ( m − 1 ) log 2 ⁡ ( p ) ⌉ + 1 , H p n + 1 ( F ) = 0 . [ABSTRACT FROM AUTHOR]

Let G be a central product of two groups H and K . We study second cohomology group of G , having coefficients in a divisible abelian group D with trivial G -action, in terms of the second cohomology groups of certain quotients of H and K . In particular, for D = C ⁎ , some of our results provide a refinement of results from Wiegold (1971) [10] and Eckmann et al. (1973) [2] . [ABSTRACT FROM AUTHOR]

We extend the notion of a partial cohomology group H n ( G , A ) to the case of non-unital A and find interpretations of H 1 ( G , A ) and H 2 ( G , A ) in the theory of extensions of semilattices of abelian groups by groups. [ABSTRACT FROM AUTHOR]

We obtain an explicit formula for the best lower bound for the higher topological complexity, TC k ( R P n ) , of real projective space implied by mod 2 cohomology. [ABSTRACT FROM AUTHOR]

We provide a cohomological interpretation of the zeroth stable A 1 -homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a certain sheaf closely related to the first Milnor–Witt K-theory sheaf. This cohomology group can be computed using an explicit Gersten-type complex. We show that if the base field is algebraically closed then the zeroth stable A 1 -homotopy group of a smooth curve coincides with the zeroth Suslin homology group that was identified by Suslin and Voevodsky with a relative Picard group. As a consequence we reobtain a version of Suslin's rigidity theorem. [ABSTRACT FROM AUTHOR]

We introduce the relative units-Picard complex of an arbitrary morphism of schemes and apply it to the problem of describing the (cohomological) Brauer group of a (fiber) product of schemes in terms of the Brauer groups of the factors. Under appropriate hypotheses, we obtain a five-term exact sequence involving the preceding groups which enables us to solve the indicated problem for, e.g., certain classes of varieties over a field of characteristic zero. [ABSTRACT FROM AUTHOR]

Let G be a simple algebraic group over the field of complex numbers. Fix a maximal torus T and a Borel subgroup B of G containing T . Let w be an element of the Weyl group W of G , and let Z ( w ˜ ) be the Bott–Samelson–Demazure–Hansen (BSDH) variety corresponding to a reduced expression w ˜ of w with respect to the data ( G , B , T ) . In this article we give complete characterization of the expressions w ˜ such that the corresponding BSDH variety Z ( w ˜ ) is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results. [ABSTRACT FROM AUTHOR]

We describe a duality phenomenon for cohomology theories with the character of Gorenstein rings. [ABSTRACT FROM AUTHOR]

A typical example of a Cartan calculus consists of the Lie derivative and the contraction with vector fields of a manifold on the derivation ring of the de Rham complex. In this manuscript, a second stage of the Cartan calculus is investigated. In a general setting, the stage is formulated with operators obtained by the André–Quillen cohomology of a commutative differential graded algebra A on the Hochschild homology of A in terms of the homotopy Cartan calculus in the sense of Fiorenza and Kowalzig. Moreover, the Cartan calculus is interpreted geometrically with maps from the rational homotopy group of the monoid of self-homotopy equivalences on a space M to the derivation ring on the loop cohomology of M. We also give a geometric description to Sullivan's isomorphism, which relates the geometric Cartan calculus to the algebraic one, via the Γ 1 map due to Félix and Thomas. [ABSTRACT FROM AUTHOR]

In the present paper we completely classify locally free sheaves of rank 2 with vanishing intermediate cohomology modules on the image of the Segre embedding P 2 × P 2 ⊆ P 8 and its general hyperplane sections. Such a classification extends similar already known results regarding del Pezzo varieties with Picard numbers 1 and 3 and dimension at least 3. [ABSTRACT FROM AUTHOR]

We use the abstract framework constructed in our earlier paper [8] to study local duality for Noetherian E ∞ -ring spectra. In particular, we compute the local cohomology of relative dualizing modules for finite morphisms of ring spectra, thereby generalizing the local duality theorem of Benson and Greenlees. We then explain how our results apply to the modular representation theory of compact Lie groups and finite group schemes, which recovers the theory previously developed by Benson, Iyengar, Krause, and Pevtsova. [ABSTRACT FROM AUTHOR]

Let k be a field of odd prime characteristic p . We calculate the Lie algebra structure of the first Hochschild cohomology of a class of quantum complete intersections over k . As a consequence, we prove that if B is a defect 2-block of a finite group algebra kG whose Brauer correspondent C has a unique isomorphism class of simple modules, then a basic algebra of B is a local algebra which can be generated by at most 2 I elements, where I is the inertial index of B , and where we assume that k is a splitting field for B and C . [ABSTRACT FROM AUTHOR]

Using the theory of D -modules, we prove some finiteness properties for local cohomology modules with respect to a family of supports on the spectrum of a k -algebra R which satisfies certain conditions introduced by L. Núñez-Betancourt in [15] . These results generalize in a sense most of the finiteness properties established by G. Lyubeznik in [9] . We also introduce the class of quasi-holonomic D -modules which properly contains the class of holonomic D -modules and is closed by taking submodules, quotients, extensions, direct limits, localizations at multiplicative subsets of R and local cohomology. [ABSTRACT FROM AUTHOR]

Given a Jordan algebra A and a vector space V , we describe and classify all Jordan algebras containing A as a subalgebra of codimension dim k (V) in terms of a non-abelian cohomological type object J A (V , A). Any such algebra is isomorphic to a newly introduced object called unified product A ♮ V. The crossed/twisted product of two Jordan algebras are introduced as special cases of the unified product and the role of the subsequent problem corresponding to each such product is discussed. The non-abelian cohomology H nab 2 (V , A) associated to two Jordan algebras A and V which classifies all extensions of V by A is also constructed. Several applications and examples are given: we prove that H nab 2 (k , k n) is identified with the set of all matrices D ∈ M n (k) satisfying 2 D 3 − 3 D 2 + D = 0 , where we consider the abelian Jordan algebra structure on k and k n. [ABSTRACT FROM AUTHOR]

We continue our investigation of a variation of the group ring isomorphism problem for twisted group algebras. Contrary to previous work, we include cohomology classes which do not contain any cocycle of finite order. This allows us to study the problem in particular over any field of characteristic 0. We prove that there are finite groups G and H which can not be distinguished by their rational twisted group algebras, while G and H can be identified by their semi-simple twisted group algebras over other fields. This is in contrast with the fact that the structural information on G which can be obtained from all the semi-simple group algebras of G is already encoded in its rational group algebra. We further show that for an odd prime p there are groups of order p 4 which can not be distinguished by their twisted group algebras over F for any field F of characteristic different from p. On the other hand we prove that the groups constructed by E. Dade, which have isomorphic group algebras over any field, can be distinguished by their rational twisted group algebras. We also answer a question about sufficient conditions for the twisted group ring isomorphism problem to hold over the complex numbers. [ABSTRACT FROM AUTHOR]

We introduce a tower of localizing subcategories in Voevodsky's big (closed under infinite coproducts) triangulated category of motives. We show that the tower induces a finite filtration on the motivic cohomology groups of smooth schemes over a perfect field. With rational coefficients, this finite filtration satisfies several of the properties of the still conjectural Bloch–Beilinson–Murre filtration. [ABSTRACT FROM AUTHOR]

We discuss the structure of integral étale motivic cohomology groups of smooth and projective schemes over algebraically closed fields, finite fields, local fields, and arithmetic schemes. [ABSTRACT FROM AUTHOR]

Let R be a semiperfect commutative Noetherian ring and C a semidualizing R -module. We study the theory of linkage for modules of finite G C -dimension. For a horizontally linked R -module M of finite G C -dimension, the connection of the Serre condition ( S n ) with the vanishing of certain relative cohomology modules of its linked module is discussed. [ABSTRACT FROM AUTHOR]

Given an acyclic twisting cochain π : C → A , from a curved dg coalgebra C to a dg algebra A , we show that the associated twisted hom complex Hom k π ( C , A ) has cohomology equal to the Hochschild cohomology of A , as a graded ring. As a corollary we find that the Hochschild cohomology of a Koszul algebra A , along with its cup product, is a subquotient of the tensor product algebra A ! ⊗ A of A with its Koszul dual. [ABSTRACT FROM AUTHOR]

Let ( A , m ) be a Noetherian local ring, let M be a finitely generated Cohen–Macaulay A -module of dimension r ≥ 2 and let I be an ideal of definition for M . Set L I ( M ) = ⨁ n ≥ 0 M / I n + 1 M . In part one of this paper we showed that L I ( M ) is a module over R ( I ) , the Rees algebra of I and we gave many applications of L I ( M ) to study the associated graded module, G I ( M ) . In this paper we give many further applications of our technique; most notable is a complete characterization of good behavior of the Ratliff–Rush filtration modulo a superficial element. [ABSTRACT FROM AUTHOR]

Let A be a unital associative algebra over a field k , E a vector space and π : E → A a surjective linear map with V = Ker ( π ) . All algebra structures on E such that π : E → A becomes an algebra map are described and classified by an explicitly constructed global cohomological type object G H 2 ( A , V ) . Any such algebra is isomorphic to a Hochschild product A ⋆ V , an algebra introduced as a generalization of a classical construction. We prove that G H 2 ( A , V ) is the coproduct of all non-abelian cohomologies H 2 ( A , ( V , ⋅ ) ) . The key object G H 2 ( A , k ) responsible for the classification of all co-flag algebras is computed. All Hochschild products A ⋆ k are also classified and the automorphism groups Aut Alg ( A ⋆ k ) are fully determined as subgroups of a semidirect product A ⁎ ⋉ ( k ⁎ × Aut Alg ( A ) ) of groups. Several examples are given as well as applications to the theory of supersolvable coalgebras or Poisson algebras. In particular, for a given Poisson algebra P , all Poisson algebras having a Poisson algebra surjection on P with a 1-dimensional kernel are described and classified. [ABSTRACT FROM AUTHOR]

Adem and Reichstein introduced the ideal of truncated symmetric polynomials to present the permutation invariant subring in the cohomology of a finite product of projective spaces. Building upon their work, I describe a generating set of the ideal of truncated symmetric polynomials in arbitrary positive characteristic, and offer a conjecture for minimal generators. [ABSTRACT FROM AUTHOR]

In this paper, we compute the integral singular cohomology ring of homology split polyhedral product spaces and the singular cohomology algebra over a field of polyhedral product spaces. As an application, we give two polyhedral product spaces Z ( K ; X 1 , A 1 ) and Z ( K ; X 2 , A 2 ) such that the cohomology homomorphisms i k ⁎ : H ⁎ ( X k ) → H ⁎ ( A k ) induced by the inclusions are the same, but the cohomology rings of the two polyhedral product spaces are not isomorphic. [ABSTRACT FROM AUTHOR]

For a set theoretical solution of the Yang–Baxter equation ( X , σ ) , we define a d.g. bialgebra B = B ( X , σ ) , containing the semigroup algebra A = k { X } / 〈 x y = z t : σ ( x , y ) = ( z , t ) 〉 , such that k ⊗ A B ⊗ A k and Hom A − A ( B , k ) are respectively the homology and cohomology complexes computing biquandle homology and cohomology defined in [2,5] and other generalizations of cohomology of rack-quandle case (for example defined in [4] ). This algebraic structure allows us to show the existence of an associative product in the cohomology of biquandles, and a comparison map with Hochschild (co)homology of the algebra A . [ABSTRACT FROM AUTHOR]

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 D n . [ABSTRACT FROM AUTHOR]

We prove Eilenberg–Watts Theorem for 2-categories of the representation categories C - Mod of finite tensor categories C . For a consequence we obtain that any autoequivalence of C - Mod is given by tensoring with a representative of some class in the Brauer–Picard group BrPic ( C ) . We introduce bialgebroid categories over C and a cohomology over a symmetric bialgebroid category. This cohomology turns out to be a generalization of the one we developed in a previous paper and moreover, an analogous Villamayor–Zelinsky sequence exists in this setting. In this context, for a symmetric bialgebroid category A , we interpret the middle cohomology group appearing in the third level of the latter sequence. We obtain a group of quasi-monoidal structures on the representation category A - Mod . [ABSTRACT FROM AUTHOR]

We define the notion of the canonical module of a complex. We then consider Serre's conditions for a complex and study their relationship to the local cohomology of the canonical module and its ring of endomorphisms. [ABSTRACT FROM AUTHOR]

Given a graded complete intersection ideal J = ( f 1 , … , f c ) ⊆ k [ x 0 , … , x n ] = S , where k is a field of characteristic p > 0 such that [ k : k p ] < ∞ , we show that if S / J has an isolated non-F-pure point then the Frobenius action on top local cohomology H m n + 1 − c ( S / J ) is injective in sufficiently negative degrees, and we compute the least degree of any kernel element. If S / J has an isolated singularity, we are also able to give an effective bound on p ensuring the Frobenius action on H m n + 1 − c ( S / J ) is injective in all negative degrees, extending a result of Bhatt and Singh in the hypersurface case. [ABSTRACT FROM AUTHOR]

Free and cofree equivariant spectra are important classes of equivariant spectra which represent equivariant cohomology theories on free equivariant spaces. Greenlees-Shipley [24,26] and Pol and the author [45] have given an algebraic model for rational (co)free equivariant spectra. In this paper, we extend this framework by proving that the Quillen functors of induction-restriction-coinduction between categories of (co)free rational equivariant spectra correspond to Quillen functors between the algebraic models in the case of connected compact Lie groups. This is achieved using new abstract techniques regarding correspondences of Quillen functors along Quillen equivalences, which we expect to be of use in other applications. [ABSTRACT FROM AUTHOR]

Let A be a Koszul Calabi–Yau algebra. We show that there exists an isomorphism of Batalin–Vilkovisky algebras between the Hochschild cohomology ring of A and that of its Koszul dual algebra A ! . This confirms (a generalization of) a conjecture of R. Rouquier. [ABSTRACT FROM AUTHOR]

Let B be the split extension of a finite dimensional algebra C by a C - C -bimodule E . We define a morphism of associative graded algebras φ ⁎ : HH ⁎ ( B ) → HH ⁎ ( C ) from the Hochschild cohomology of B to that of C , extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler. In the case of a trivial extension B = C ⋉ E , we give necessary and sufficient conditions for each φ n to be surjective. We prove the surjectivity of φ 1 for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of φ 1 for any trivial extension, and give a more precise description of this kernel in the case of relation extensions. [ABSTRACT FROM AUTHOR]