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Similar to the modular vector fields in Poisson geometry, modular derivations are defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson module with the modular derivation, the Poisson cochain complex with values in any Poisson module is proved to be isomorphic to the Poisson chain complex with values in the corresponding twisted Poisson module. Then a version of twisted Poincaré duality is proved between the Poisson homologies and cohomologies. Furthermore, a notion of pseudo-unimodular Poisson structure is defined. It is proved that the Poisson cohomology as a Gerstenhaber algebra admits a Batalin-Vilkovisky operator inherited from some one of its Poisson cochain complex if and only if the Poisson structure is pseudo-unimodular. This generalizes the geometric version due to P. Xu. The modular derivation and Batalin-Vilkovisky operator are also described by using the dual basis of the Kähler differential module. [ABSTRACT FROM AUTHOR]

In this paper, we define a cohomology theory of a modified λ -differential left-symmetric algebra. Moreover, we introduce the notion of modified λ -differential left-symmetric 2-algebras, which is the categorization of a modified λ -differential left-symmetric algebra. As applications of cohomology, we classify linear deformations and abelian extensions of modified λ -differential left-symmetric algebras using the second cohomology group and classify skeletal modified λ -differential left-symmetric 2-algebra using the third cohomology group. Finally, we show that strict modified λ -differential left-symmetric 2-algebras are equivalent to crossed modules of modified λ -differential left-symmetric algebras. [ABSTRACT FROM AUTHOR]

To a strongly 퐺-graded algebra 퐴 with 1-component 퐵, we associate the group Picent gr ⁢ (A) of isomorphism classes of invertible 퐺-graded (A , A) -bimodules over the centralizer of 퐵 in 퐴. Our main result is a Picent version of the Beattie–del Río exact sequence, involving Dade's group G ⁢ [ B ] , which relates Picent gr ⁢ (A) , Picent ⁢ (B) , and group cohomology. [ABSTRACT FROM AUTHOR]

We give an explicit description of a diagonal map on the Bardzell resolution for any monomial algebra, and we use this diagonal map to describe the cup product on Hochschild cohomology. Then, we prove that the cup product is zero in positive degrees for triangular monomial algebras. Our proof uses the graded-commutativity of the cup product on Hochschild cohomology and does not rely on explicit computation of the Hochschild cohomology modules. [ABSTRACT FROM AUTHOR]

The notion of O -operator is a generalization of the Rota–Baxter operator in the presence of a bimodule over an associative algebra. A compatible O -operator is a pair consisting of two O -operators satisfying a compatibility relation. A compatible O -operator algebra is an algebra together with a bimodule and a compatible O -operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O -operators and compatible O -operator algebras as Maurer–Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible O -operators and compatible O -operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible O -operators. [ABSTRACT FROM AUTHOR]

We study the cohomology of families of |$(\varphi ,\Gamma)$| -modules with coefficients in pseudoaffinoid algebras. We prove that they have finite cohomology, and we deduce an Euler characteristic formula and Tate local duality. We classify rank- |$1$| |$(\varphi ,\Gamma)$| -modules and deduce that triangulations of pseudorigid families of |$(\varphi ,\Gamma)$| -modules can be interpolated, extending a result of [ 29 ]. We then apply this to study extended eigenvarieties at the boundary of weight space, proving in particular that the eigencurve is proper at the boundary and that Galois representations attached to certain characteristic |$p$| points are trianguline. [ABSTRACT FROM AUTHOR]

We provide a cohomology of n-Hom–Lie color algebras, in particular, a cohomology governing oneparameter formal deformations. Then we also study formal deformations of the n-Hom–Lie color algebras and introduce the notion of Nijenhuis operator on an n-Hom–Lie color algebra, which may give rise to infinitesimally trivial (n − 1)th-order deformations. Furthermore, in connection with Nijenhuis operators, we introduce and discuss the notion of product structure on n-Hom–Lie color algebras. [ABSTRACT FROM AUTHOR]

For n a power of two, we give a complete description of the cohomology algebra H*(G͠n,3; ℤ2) of the Grassmann manifold G͠n,3 of oriented 3-planes in ℝn. We do this by finding a reduced Gröbner basis for an ideal closely related to this cohomology algebra. Using this Gröbner basis we also present an additive basis for H*(G͠n,3; ℤ2). [ABSTRACT FROM AUTHOR]

Based on Nijenhuis-Richardson bracket and bidegree on the cohomology complex for a Lie conformal algebra, we develop a twisting theory of Lie conformal algebras. By using derived bracket constructions, we construct L ∞ -algebras from (quasi-)twilled Lie conformal algebras. And we show that the result of the twisting by a C [ ∂ ] -module homomorphism on a (quasi-)twilled Lie conformal algebra is also a (quasi-)twilled Lie conformal algebra if and only if the C [ ∂ ] -module homomorphism is a Maurer-Cartan element of the L ∞ -algebra. In particular, we show that relative Rota-Baxter type operators on Lie conformal algebras are Maurer-Cartan elements. Besides, we propose a new algebraic structure, called NS-Lie conformal algebras, that is closely related to twisted relative Rota-Baxter operators and Nijenhuis operators on Lie conformal algebras. As an application of twisting theory, we give the cohomology of twisted relative Rota-Baxter operators and study their deformations. [ABSTRACT FROM AUTHOR]

The main goal of this paper was to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, and α k -derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from the one given by Guan and Chen. This definition is inspired by the process of the queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we developed a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we established a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology. [ABSTRACT FROM AUTHOR]

Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra A$A$ forms a Lie algebra, and a restricted Lie algebra if A$A$ contains a field of characteristic p$p$. We deduce that the space of integrable classes in HH1(A)${\operatorname{HH}}^1(A)$ forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self‐injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos. Along the way, we compute the first Hochschild cohomology of the group algebra of any symmetric group. [ABSTRACT FROM AUTHOR]

Historically, it was known by the work of Artin and Mazur that the ℓ -adic homotopy type of a smooth complex variety with good reduction mod p can be recovered from the reduction mod p, where ℓ is not p. This short note removes this last constraint, with an observation about the recent theory of prismatic cohomology developed by Bhatt and Scholze. In particular, by applying a functor of Mandell, we see that the étale comparison theorem in the prismatic theory reproduces the p-adic homotopy type for a smooth complex variety with good reduction mod p. [ABSTRACT FROM AUTHOR]

We introduce the notion of a 푝-Cartier smooth algebra. It generalises that of a smooth algebra and includes valuation rings over a perfectoid base. We give several characterisations of 푝-Cartier smoothness in terms of prismatic cohomology and deduce a comparison theorem between syntomic and étale cohomologies under this hypothesis. [ABSTRACT FROM AUTHOR]

Recently, relative Rota–Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider relative Rota–Baxter Leibniz algebras (rRB Leibniz algebras) as the object of our study. We construct an $ L_\infty $ L ∞ -algebra that characterizes rRB Leibniz algebras as its Maurer–Cartan elements. Then we define representations of an rRB Leibniz algebra and introduce cohomology with coefficients in a representation. As applications of cohomology, we study deformations and abelian extensions of rRB Leibniz algebras. [ABSTRACT FROM AUTHOR]

The cohomology algebra of the space H∗ (X) defines neither cohomology modules of the loop space H∗ (ΩX) nor cohomologies of the free loop space H∗ (ΛX). But by the author's minimality theorem, there exists a structure of A(∞)-algebra (H∗ (X), {mi}) on H∗ (X), which determines H∗ (ΩX). Here will be shown that the same A(∞)-algebra (H∗(X), {mi}) determines also cohomology modules H∗ (ΛX). [ABSTRACT FROM AUTHOR]

In this paper, the notion of generalized Reynolds operators on Lie-Yamaguti algebras is introduced, and the cohomology of a generalized Reynolds operator is established. The formal deformations of a generalized Reynolds operator are studied using the first cohomology group. Then, we show that a Nijenhuis operator on a Lie-Yamaguti algebra gives rise to a representation of the deformed Lie-Yamaguti algebra and a 2-cocycle. Consequently, the identity map will be a generalized Reynolds operator on the deformed Lie-Yamaguti algebra. We also introduce the notion of a Reynolds operator on a Lie-Yamaguti algebra, which can serve as a special case of generalized Reynolds operators on Lie-Yamaguti algebras. [ABSTRACT FROM AUTHOR]

In this paper, we first propose the concept of Rota–Baxter family Ω-associative conformal algebras, then we study the cohomology theory of Rota–Baxter family Ω-associative conformal algebras of any weight and justify it by interpreting the lower degree cohomology groups as formal deformations. [ABSTRACT FROM AUTHOR]

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including most quantum complete intersections. [ABSTRACT FROM AUTHOR]

We bound the Gelfand–Kirillov dimension of unitary Banach space representations of p-adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and p-adic Banach space representations of the group of units of a quaternion algebra over ℚ p appearing in the p-adic Jacquet–Langlands correspondence, deducing finiteness results in favorable cases. [ABSTRACT FROM AUTHOR]

In this note we review a construction of a BPS Hilbert space in an effective supersymmetric quiver theory with 4 supercharges. We argue abstractly that this space contains elements of an equivariant generalized cohomology theory E G ∗ − of the quiver representation moduli space giving concretely Dolbeault cohomology, K-theory or elliptic cohomology depending on the spacial slice is compactified to a point, a circle or a torus respectively, and something more amorphous in other cases. Furthermore BPS instantons — basic contributors to interface defects or a Berry connection — induce a BPS algebra on the BPS Hilbert spaces representing Fourier-Mukai transforms on the quiver representation moduli spaces descending to an algebra over E G ∗ − as its representation. In the cases when the quiver describes a toric Calabi-Yau three-fold (CY3) the algebra is a respective generalization of the quiver BPS Yangian algebra discussed in the literature, in more general cases it is given by an abstract generalized cohomological Hall algebra. [ABSTRACT FROM AUTHOR]

By applying Gröbner basis theory to spectral sequences algebras, we develop a new computational methodology applicable to any Leray-Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. We demonstrate the procedure by deducing the cohomology of the free loop space of flag manifolds, presenting a significant extension over previous knowledge of the topology of free loop spaces. A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank 2 complete flag manifolds are SU(3)/T 2, Sp(2)/T 2, Spin(4)/T², Spin(5)/T² and G2/T². In this paper we calculate the cohomology of the free loop space of the rank 2 complete flag manifolds. [ABSTRACT FROM AUTHOR]

Let X be a finitistic space with the mod 2 cohomology of the product space of a projective space and a 4-sphere. Assume that X admits a free involution. In this paper we study the mod 2 cohomology algebra of the quotient of X by the action of the free involution and derive some consequences regarding the existence of Z2-equivariant maps between such X and an n-sphere. [ABSTRACT FROM AUTHOR]

Let A be a Hopf algebra and B a graded twisting of A by a finite abelian group Γ. Then, categories of comodules over A and B are equivalent (but they are not necessarily monoidally equivalent). We show the relation between the Hochschild cohomology of A and B explicitly. This partially answer a question raised by Bichon. As an application, we prove that A is a twisted Calabi–Yau Hopf algebra if and only if B is a twisted Calabi–Yau algebra, and give the relation between their Nakayama automorphisms. [ABSTRACT FROM AUTHOR]

We show that the exterior algebra Λ R [ α 1 , ⋯ , α n ] , which is the cohomology of the torus T = (S 1) n , and the polynomial ring R [ t 1 , ... , t n ] , which is the cohomology of the classifying space B (S 1) n = (C P ∞) n , are S n -equivariantly log-concave. We do so by explicitly giving the S n -representation maps on the appropriate sequences of tensor products of polynomials or exterior powers and proving that these maps satisfy the hard Lefschetz theorem. Furthermore, we prove that the whole Kähler package, including the Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann bilinear relations, holds on the corresponding sequences in an equivariant setting. [ABSTRACT FROM AUTHOR]

We define generalized Koszul modules and rings and develop a generalized Koszul theory for N -graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings with degree zero part artinian semisimple developed by Beilinson-Ginzburg-Soergel and the ungraded Koszul theory for noetherian semiperfect rings developed by Green and Martinéz-Villa. Let A be a left finite N -graded ring generated in degree 1 with A 0 noetherian semiperfect, J be its graded Jacobson radical. By the Koszul dual of A we mean the Yoneda Ext ring Ext _ A • (A / J , A / J). If A is a generalized Koszul ring and M is a generalized Koszul module, then it is proved that the Koszul dual of the Koszul dual of A is the associated graded ring G r J A and the Koszul dual of the Koszul dual of M is the associated graded module G r J M. If A is a locally finite algebra, then the following statements are proved to be equivalent: A is generalized Koszul; the Koszul dual Ext _ A • (A / J , A / J) of A is (classically) Koszul; G r J A is (classically) Koszul; the opposite ring A o p of A is generalized Koszul. As an application, it is proved that if A is generalized Koszul with finite global dimension then A is generalized AS regular if and only if the Koszul dual of A is self-injective. [ABSTRACT FROM AUTHOR]

Let X be a smooth affine variety over a field \mathbf k of characteristic 0 and T(X) be the Lie algebra of regular vector fields on X. We compute the Lie algebra cohomology of T(X) with coefficients in \mathbf k. The answer is given in topological terms relative to any embedding \mathbf k\subset \mathbb {C} and is analogous to the classical Gel′fand-Fuchs computation for smooth vector fields on a C^\infty-manifold. Unlike the C^\infty-case, our setup is purely algebraic: no topology on T(X) is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts. [ABSTRACT FROM AUTHOR]

Let Gk,n(ℍ) for 2≤k

In this paper, first we introduce the notions of relative Rota-Baxter operators of nonzero weight on 3-Lie algebras and 3-post-Lie algebras. A 3-post-Lie algebra consists of a 3-Lie algebra structure and a ternary operation such that some compatibility conditions are satisfied. We show that a relative Rota-Baxter operator of nonzero weight induces a 3-post-Lie algebra naturally. Conversely, a 3-post-Lie algebra gives rise to a new 3-Lie algebra, which is called the subadjacent 3-Lie algebra, and an action on the original 3-Lie algebra. Then we construct an L ∞ -algebra whose Maurer-Cartan elements are relative Rota-Baxter operators of nonzero weight. Consequently, we obtain the twisted L ∞ -algebra that controls deformations of a given relative Rota-Baxter operator of nonzero weight on 3-Lie algebras. Finally, we introduce a cohomology theory for a relative Rota-Baxter operator of nonzero weight on 3-Lie algebras and use the second cohomology group to classify infinitesimal deformations. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic cohomology theory on a regular multiplier Hopf algebra. Given a modular pair consisting of a group-like element, a character and in involution for a regular multiplier Hopf algebra, we associate a cyclic module. The approach is based on the Connes-Moscovici cyclic module of Hopf algebras and we obtain the same cyclic modules when our algebra has an identity. [ABSTRACT FROM AUTHOR]

The deformation cohomology of a tensor category controls deformations of the constraint of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur–Weyl categories). Using this description we compute the deformation cohomology of free symmetric tensor categories generated by one object with an algebra of endomorphism free of zero-divisors. We compare the answers with the exterior invariants of the general linear Lie algebra. The results make precise an intriguing connection between the combinatorics of partitions and invariants of the exterior of the general linear algebra observed by Kostant. [ABSTRACT FROM AUTHOR]

We study the dg-algebra ... of algebraic de Rham forms of a real soft function algebra A, i.e., the algebra of global sections of a soft sub-sheaf of Cx, the sheaf of continuous functions on a space X. We obtain a canonical splitting ..., where V is some vector space. In particular, we consider the cases A = C(X) for X a compact Hausdorff space and A = C∞(XC) for X a compact smooth manifold. For the algebra PPolk(|K|) piecewise polynomial functions on a polyhedron K the above splitting reduces to a canonical isomorphism ... We also prove that the algebraic de Rham cohomology ... is non-trivial for each n ≥ 1 if X is an infinite compact Hausdorff space. [ABSTRACT FROM AUTHOR]

The unstable Adams spectral sequence is a spectral sequence that starts fromalgebraic information about the mod 2 cohomology H* (X) of a space X as an unstable algebra over the Steenrod algebra A, and converges, in good cases, to the 2-localized homotopy groups of X. Bousfield and Don Davis looked at the case when X was either of the infinite matrix groups SO or U. Bousfield and Davis created algebraic spectral sequences and conjectured that they agreed with the unstable Adams spectral sequences for SO and U. To this end the following algebraic decomposition must hold ... where M1 ∈ M2 ∈ ⋯ is the well known dyadic filtration of the A-module H* (ℝP∞, ℤ/2) ≄ F2 [u] given by the dyadic expansion of the powers of u. This paper aims at showing that this decomposition holds for numerous values of s and t. [ABSTRACT FROM AUTHOR]

The Rumin algebra of a contact manifold is a contact invariant C∞-algebra of differential forms which computes the de Rham cohomology algebra. We recover this fact by giving a simple and explicit construction of the Rumin algebra via Markl's formulation of the Homotopy Transfer Theorem. [ABSTRACT FROM AUTHOR]

In this work we show that for any topological space X having the homotopy type of a CW-complex and for any commutative ring R, the singular cohomology H*(X; R) is a Gerstenhaber algebra (see. [7]). In fact we prove that H*(X; R) satisfies the conditions of a generalization of the Gerstenhaber algebras. [ABSTRACT FROM AUTHOR]

A Lie bialgebra is a vector space endowed simultaneously with the structure of a Lie algebra and the structure of a Lie coalgebra, and some compatibility condition. Moreover, Lie brackets have skew symmetry. Because of the close relation between Lie bialgebras and quantum groups, it is interesting to consider the Lie bialgebra structures on the Lie algebra L related to the Virasoro algebra. In this paper, the Lie bialgebras on L are investigated by computing Der (L ,   L ⊗ L) . It is proved that all such Lie bialgebras are triangular coboundary, and the first cohomology group H 1 (L ,   L ⊗ L) is trivial. [ABSTRACT FROM AUTHOR]

In this paper, we review deformation, cohomology and homotopy theories of relative Rota–Baxter (RB$\mathsf {RB}$) Lie algebras, which have attracted quite much interest recently. Using Voronov's higher derived brackets, one can obtain an L∞$L_\infty$‐algebra whose Maurer–Cartan elements are relative RB$\mathsf {RB}$ Lie algebras. Then using the twisting method, one can obtain the L∞$L_\infty$‐algebra that controls deformations of a relative RB$\mathsf {RB}$ Lie algebra. Meanwhile, the cohomologies of relative RB$\mathsf {RB}$ Lie algebras can also be defined with the help of the twisted L∞$L_\infty$‐algebra. Using the controlling algebra approach, one can also introduce the notion of homotopy relative RB$\mathsf {RB}$ Lie algebras with close connection to pre‐Lie∞$_\infty$‐algebras. Finally, we briefly review deformation, cohomology and homotopy theories of relative RB$\mathsf {RB}$ Lie algebras of nonzero weights. [ABSTRACT FROM AUTHOR]

We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group Oq²(SL(2)) providing a topological interpretation for its structure morphisms. We also show that its stated skein algebra lifts in a suitable sense the Reshetikhin-Turaev functor and in particular we recover the dual R-matrix for Oq²(SL(2)) in a topological way. We deduce that the skein algebra of a surface with n boundary components is an algebra-comodule over Oq²(SL(2))⊗n and prove that cutting along an ideal arc corresponds to Hochshild cohomology of bicomodules. We give a topological interpretation of braided tensor product of stated skein algebras of surfaces as "glueing on a triangle"; then we recover topologically some braided bialgebras in the category of Oq²(SL(2))-comodules, among which the "transmutation" of Oq²(SL(2)). We also provide an operadic interpretation of stated skein algebras as an example of a "geometric non symmetric modular operad". In the last part of the paper we define a reduced version of stated skein algebras and prove that it allows to recover Bonahon-Wong's quantum trace map and interpret skein algebras in the classical limit when q→1 as regular functions over a suitable version of moduli spaces of twisted bundles. [ABSTRACT FROM AUTHOR]

In this work we find the groups of Hochschild cohomologies for the Chinese monoid algebra and derive its Hilbert and Poincaré series. In order to obtain this result we construct the Anick resolution via the algebraic discrete Morse theory and Gröbner–Shirshov basis for the Chinese monoid. [ABSTRACT FROM AUTHOR]

By a result of Gerstenhaber and Schack, the simplicial cohomology ring of a poset is isomorphic to the Hochschild cohomology ring of the category algebra , where the poset is viewed as a category and is a field. Extending results of Mishchenko, under certain assumptions on a category , we construct a category and a graded -linear isomorphism . Interpreting the degree one cohomology, we also show how the -space of derivations on graded by some semigroup corresponds to the -space of characters on . [ABSTRACT FROM AUTHOR]

In this work, we demonstrate that the integral formula for a generalised Sato-Levine invariant is consistent in certain situations with Evans and Berger's formula for the fourth-order winding number. Also, we found that, in principle, one can derive analogous high-order winding numbers from which one can calculate the entanglement of braids. The winding number for the Brunnian 4-braid is calculated algebraically using the cup product on the cohomology of a finite regular CW-space, which is the complement R³nB4. [ABSTRACT FROM AUTHOR]

We devote to the calculation of Batalin-Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi-Yau generalized Weyl algebras. We first establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras, and translate the homological information into cohomological one by virtue of the Van den Bergh duality, obtaining the desired Batalin-Vilkovisky algebra structures. Finally, we apply our results to quantum weighted projective lines and Podleś quantum spheres, and the Batalin-Vilkovisky algebra structures for them are described completely. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the cohomology theory of O -operators on Hom-associative algebras. This cohomology can also be viewed as the Hochschild cohomology of a certain Hom-associative algebra with coefficients in a suitable bimodule. Next, we study infinitesimal and formal deformations of an O -operator and show that they are governed by the above-defined cohomology. Furthermore, the notion of Nijenhuis elements associated with an O -operator is introduced to characterize trivial infinitesimal deformations. Moreover, we provide relevant constructions that twists objects on associative algebras to objects on Hom-associative algebras along morphisms. [ABSTRACT FROM AUTHOR]

We resolve the long-standing problem of constructing the action of the operad of framed (stable) genus-0 curves on Hamiltonian Floer theory; this operad is equivalent to the framed E 2 operad. We formulate the construction in the following general context: we associate to each compact subset of a closed symplectic manifold a new chain-level model for symplectic cohomology with support, which we show carries an action of a model for the chains on the moduli space of framed genus 0 curves. This construction turns out to be strictly functorial with respect to inclusions of subsets, and the action of the symplectomorphism group. In the general context, we appeal to virtual fundamental chain methods to construct the operations over fields of characteristic 0, and we give a separate account, over arbitrary rings, in the special settings where Floer's classical transversality approach can be applied. We perform all constructions over the Novikov ring, so that the algebraic structures we produce are compatible with the quantitative information that is contained in Floer theory. Over fields of characteristic 0, our construction can be combined with results in the theory of operads to produce explicit operations encoding the structure of a homotopy BV algebra. In an appendix, we explain how to extend the results of the paper from the class of closed symplectic manifolds to geometrically bounded ones. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of relative and absolute Ω-Rota-Baxter algebras of weight λ , which can be considered as a family algebraic generalization of relative and absolute Rota-Baxter algebras of weight λ. We study the deformations of relative and absolute Ω-Rota-Baxter algebras of arbitrary weight. Explicitly, we construct an L ∞ [ 1 ] -algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative Ω-Rota-Baxter algebra structures of weight λ. For a relative Ω-Rota-Baxter algebra of weight λ , the corresponding twisted L ∞ [ 1 ] -algebra controls its deformations, which leads to the cohomology theory of it, and this cohomology theory can interpret the formal deformations of the relative Ω-Rota-Baxter algebra. Moreover, we also obtain the corresponding results for absolute Ω-Rota-Baxter algebras of weight λ from the relative version. [ABSTRACT FROM AUTHOR]

We introduce the notion of quantum-symmetric equivalence of two connected graded algebras, based on Morita–Takeuchi equivalences of their universal quantum groups, in the sense of Manin. We study homological and algebraic invariants of quantum-symmetric equivalence classes, and prove that numerical Tor-regularity, Castelnuovo–Mumford regularity, Artin–Schelter regularity, and the Frobenius property are invariant under any Morita–Takeuchi equivalence. In particular, by combining our results with the work of Raedschelders and Van den Bergh, we prove that Koszul Artin–Schelter regular algebras of a fixed global dimension form a single quantum-symmetric equivalence class. Moreover, we characterize 2-cocycle twists (which arise as a special case of quantum-symmetric equivalence) of Koszul duals, of superpotentials, of superpotential algebras, of Nakayama automorphisms of twisted Frobenius algebras, and of Artin–Schelter regular algebras. We also show that finite generation of Hochschild cohomology rings is preserved under certain 2-cocycle twists. [ABSTRACT FROM AUTHOR]

The notion of a F -manifold algebras is an algebraic description of a F -manifold. In this paper, we introduce the notion of Hom-F -manifold algebras which is generalisation of F -manifold algebras and Hom-Poisson algebras. We develop the representation theory of Hom-F -manifold algebras and generalize the notion of Hom-pre-Poisson algebras by introducing the Hom-pre-F -manifold algebras which give rise to a Hom-F -manifold algebra through the subadjacent commutative Hom-associative algebra and the subadjacent Hom-Lie algebra. Using O-operators on a Hom-F -manifold algebras we construct a Hom-pre-F -manifold algebras on a module. Then, we study Hom-pre-Lie formal deformations of commutative Hom-associative algebra and we prove that Hom-F -manifold algebras are the corresponding semiclassical limits. Finally, we study Hom-Lie infinitesimal deformations and extension of Hom-pre-Lie n-deformation to Hom-pre-Lie (n + 1)-deformation of a commutative Hom-associative algebra via cohomology theory. [ABSTRACT FROM AUTHOR]

Both the basic cohomology groups and the reduced cohomology groups of the Schrödinger–Virasoro conformal algebra with trivial coefficients are completely determined. [ABSTRACT FROM AUTHOR]

In this paper we show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra. [ABSTRACT FROM AUTHOR]

We consider the finite generation property for cohomology of a finite tensor category \mathscr {C}, which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny∙}_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V in \mathscr {C}, the graded extension group \operatorname {Ext}^\text {\tiny∙}_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on \mathscr {C}. For example, the stated result holds when \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0, we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes. [ABSTRACT FROM AUTHOR]

The quiver Yangian, an infinite-dimensional algebra introduced recently in [1], is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional N = 2 supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories. [ABSTRACT FROM AUTHOR]