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We study finite group actions on Hom-Leibniz algebras. We introduce equivariant cohomology of Hom-Leibniz algebras with coefficients in $$O_G$$ -Hom-algebras. We show that there is a cup-product operation in the graded equivariant Hom-Leibniz cohomology, which satisfies the graded Hom-Zinbiel relation.

For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not restrict to systems with short-range entanglement, stability against stacking with trivial systems nor assume the existence of quasi-particles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the $P$-equivariant cohomology of a $d$-dimensional torus gives rise to different interacting phases, where $P$ denotes the point group of the crystalline structure. We compare our results to bosonic symmorphic crystallographic SPT phases and to non-interacting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases., Comment: Stronger results. Included examples in section 6. Accepted for publication in JHEP

In this paper, motivated by Chen–Ruan’s stringy orbifold theory on almost complex orbifolds, we construct a new equivariant cohomology ring $${\mathscr {H}}^*_G(X)$$ for an equivariant almost complex pair (X, G), where X is a compact connected almost complex manifold, G is a connected compact Lie group which acts on X and preserves the almost complex structure.

J. De Loera & T. McAllister and K. D. Mulmuley & H. Narayanan & M. Sohoni independently proved that determining the vanishing of Littlewood–Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood–Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas, A. Yong. Our proof then combines a saturation theorem of D. Anderson, E. Richmond, A. Yong, a reading order independence property, and E. Tardos’ algorithm for combinatorial linear programming.

We describe the torus-equivariant cohomology of weighted partial flag orbifolds $${\mathrm {w}}\Sigma $$ of type A. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as “Schubert Calculus on $${\mathrm {w}}\Sigma $$ ”. For the weighed Schubert classes in $${\mathrm {w}}\Sigma $$, we give the Chevalley’s formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley–Monk’s formula.

We compute the rational Borel equivariant cohomology ring of a cohomogeneity-one action of a compact Lie group., Comment: 20 pages, one figure. Minor corrections and referee suggestions and additional references included. To appear in Geometriae Dedicata

This paper is dedicated to introducing and studying a cohomology theory for monoids enriched with the action by endomorphisms of a fixed monoid of operators. This equivariant cohomology theory extends both Whitehead’s cohomology for groups with operators and Leech’s cohomology for monoids.

We investigate the topological nilpotence degree, in the sense of Henn-Lannes-Schwartz, of a connected Noetherian unstable algebra $R$. When $R$ is the mod $p$ cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian $p$-subgroups. By replacing centralizers of elementary abelian $p$-subgroups with components of Lannes' $T$-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn's result to a large class of connected Noetherian unstable algebras. We show how this generalizes Kuhn's result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac-Moody groups. In fact, our results apply much more generally, for example, we establish results for $p$-local compact groups in the sense of Broto-Levi-Oliver, for connected $H$-spaces with Noetherian mod $p$ cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson's depth conjecture in the case of a Noetherian unstable algebra of minimal depth., 42 pages; comments welcome v2; Significantly revised version, accepted for publication in Selecta Mathematica

For any connected reductive group $G$ over $\mathbb{C}$, we revisit Goresky-Kottwitz-MacPherson's description of the torus equivariant Borel-Moore homology of affine Springer fibers $\mathrm{Sp}_\gamma\subset \mathrm{Gr}_G$, where $\gamma=at^d$, and $a$ is a regular semisimple element in the Lie algebra of $G$. In the case $G = GL_n$, we relate the equivariant cohomology of $\mathrm{Sp}_\gamma$ to Haiman's work on the isospectral Hilbert scheme of points on the plane. We also explain the connection to the HOMFLY homology of $(n, dn)$-torus links, and formulate a conjecture describing the homology of the Hilbert scheme of points on the curve $\{x^n=y^{dn}\}$., Comment: 42 pages, section 3 rewritten due to an erroneous lemma in the previous version

We find an explicit formula for the elliptic stable envelope in the case of the Hilbert scheme of points on a complex plane. The formula has a structure of a sum over trees in Young diagrams. In the limit we obtain the formulas for the stable envelope in equivariant $K$-theory (with arbitrary slope) and equivariant cohomology., 65 pages

This paper mainly deals with the question of equivalence between equivariant cohomology Chern numbers and equivariant K-theoretic Chern numbers when the transformation group is a torus. By using the equivariant Riemann-Roch relation of Atiyah-Hirzebruch type, it is proved that the vanishing of equivariant cohomology Chern numbers is equivalent to the vanishing of equivariant K-theoretic Chern numbers.

We relate the old and new cohomology monoids of an arbitrary monoid $M$ with coefficients in semimodules over $M$, introduced in the author's previous papers, to monoid and group extensions. More precisely, the old and new second cohomology monoids describe Schreier extensions of semimodules by monoids, and the new third cohomology monoid is related to a certain group extension problem.

We compare relative cohomology theories arising from using different proper resolutions of modules. Criteria for the vanishing of such distinctions are given in certain cases, and we show that this is related to the generalized Tate cohomology theory. We also demonstrate that the two balance properties admitted by the two different cohomology theories are actually equivalent in some cases. As applications, we recover many results obtained earlier in various contexts. At last we investigate derived functors with respect to the Auslander and Bass classes.

In this paper, we investigate the properties of right orthogonal modules of $${{\mathscr {C}}}$$ , where $${{\mathscr {C}}}$$ is a class of left R-modules. As an application, we investigate the properties of right orthogonal modules of Ding injective left R-modules, and present various characterizations of semisimple and von Neumann regular rings and so on. Moreover, we also consider another cohomology, strong Tate cohomology, which connects the usual cohomology with the Ding cohomology.

We recall some unusual features of the cohomology of arithmetic groups, and propose that they are explained by a hidden action of certain motivic cohomology groups.

Complex analytic invariants of hypersurface isolated singularities are considered in the context of symbolic computation. The motivations for this paper are computer calculations of μ ∗-sequences that introduced by B. Teissier to study the Whitney equisingularity of deformations of complex hypersurfaces. A new algorithm that utilizes parametric local cohomology systems is proposed to compute μ ∗-sequences. Lists of μ ∗-sequences of some typical cases are also given.

The Hochschild cohomology groups are computed for algebras of dihedral type, which form the family D(3K) (from the famous K. Erdmann’s classification) over an algebraically closed field of characteristic different from 2. In the calculation, the beforehand constructed bimodule resolution for algebras from the family under discussion is used. Bibliography: 28 titles.

This paper is a sequel of the joint paper by the author with S. O. Ivanov, Yu. Volkov, and G. Zhou. In the present paper, the BV -structure, and therefore, the Gerstenhaber algebra structure on the Hochschild cohomology of local algebras of generalized quaternion type is completely described over a field of characteristic 2. The family of algebras under investigation contains group algebras of generalized quaternion groups for which the case of characteristic 2 is the only one where the calculation of Hochschild cohomology and structures on it is a highly nontrivial problem. Also the group algebras of generalized quaternion groups represent classes of Morita-equivalence of tame group blocks from K. Erdmann’s classification. In particular, the BV -structure on the Hochschild cohomology of group algebras of some noncommutative groups is described.

Let C be a triangulated category with a proper class E of triangles. We prove that there exists an Avramov–Martsinkovsky type exact sequence in C, which connects E-cohomology, E-Tate cohomology and E-Gorenstein cohomology.

A group action on the input ring or category induces an action on the algebraic $K$-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic $K$-theory is, for example, that the map of spectra with $G$-action induced by a $G$-map of $G$-rings is not equivariant. We define a version of equivariant algebraic $K$-theory which encodes a group action on the input in a functorial way to produce a $genuine$ algebraic $K$-theory $G$-spectrum for a finite group $G$. The main technical work lies in studying coherent actions on the input category. A payoff of our approach is that it builds a unifying framework for equivariant topological $K$-theory, Atiyah's Real $K$-theory, and existing statements about algebraic $K$-theory spectra with $G$-action. We recover the map from the Quillen-Lichtenbaum conjecture and the representational assembly map studied by Carlsson and interpret them from the perspective of equivariant stable homotopy theory., Comment: Final version to appear in Mathematische Zeitschrift. The last section about Waldhausen G-categories has been removed from this paper

We present a survey of results concerning the use of Hochschild cohomology in representation theory of algebras.

For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $\Gamma$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/\Gamma$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$ of $G$. In this paper we construct, up to a finite group, a new Lie algebra $\tilde{\mathfrak g}$ whose cohomology is isomorphic to the de Rham cohomology of $G/\Gamma$ by using a modification of $G$ associated with a algebraic sub-torus of the Zariski-closure of the image of the adjoint representation. This technique includes the construction due to Guan and developed by the first two authors. In this paper, we also give a Dolbeault version of such technique for complex solvmanifolds, i.e. for solvmanifolds endowed with an invariant complex structure. We construct a finite dimensional cochain complex which computes the Dolbeault cohomology of a complex solvmanifold $G/\Gamma$ with holomorphic Mostow bundle and we give a construction of a new Lie algebra $\breve {\mathfrak g}$ with a complex structure whose cohomology is isomorphic to the Dolbeault cohomology of $G/\Gamma$., Comment: 23 pages; to appear in Transformation Groups

We give a classification of the 2-groupoid of singular extensions of categorical groups by means of a second cohomology categorical group, whose definition uses the notion of nerve of a categorical group. This second cohomology is, then, shown to be involved in a five term 2-exact sequence \(\grave{\text {a}}\) la Hochschild–Serre which is associated with any essentially surjective homomorphism of categorical groups.

Algebraic local cohomology classes associated with parametric semi-quasihomogeneous hypersurface isolated singularities are considered in the context of symbolic computation. The motivations for this paper are computer calculations of complete lists of Tjurina numbers of semi-quasihomogeneous polynomials with isolated singularity. A new algorithm, that utilizes parametric local cohomology systems, is proposed to compute Tjurina stratifications associated with $$\mu $$μ-constant deformations of weighted homogeneous isolated singularities. The resulting algorithm gives in particular a suitable decomposition of the parameter space depending on the structure of the parametric local cohomology systems. An efficient algorithm of computing parametric standard bases of relevant ideals is also given as an application of parametric local cohomology systems.

Let $X_1$ and $X_2$ be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann (CR) manifolds of dimensions $2m-1$ and $2n-1$ in $\mathbb{C}^{m+1}$ and $\mathbb{C}^{n+1}$, respectively. We introduce the Thom-Sebastiani sum $X=X_1\oplus X_2$ which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension $2m+2n+1$ in $\mathbb{C}^{m+n+2}$. Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in $\mathbb{C}^{n+1}$ for all $n\geq 2$ forms a semigroup. $X$ is said to be an irreducible element in this semigroup if $X$ cannot be written in the form $X_1\oplus X_2$. It is a natural question to determine when $X$ is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly, we show that if $X=X_1\oplus X_2$, then the Kohn-Rossi cohomology of the $X$ is the product of those Kohn-Rossi cohomology coming from $X_1$ and $X_2$ provided that $X_2$ admits a transversal holomorphic $S^1$-action.

We modify the definition of Dirac cohomology in such a way that the standard properties of the usual Dirac cohomology, valid for modules with infinitesimal character, become valid also for modules with only generalized infinitesimal character.

The realizations of the basic representation of \(\widehat{\mathfrak {gl}}_r\) are known to be parametrized by partitions of r and have an explicit description in terms of vertex operators on the bosonic/fermionic Fock space. In this paper, we give a geometric interpretation of these realizations in terms of geometric operators acting on the equivariant cohomology of certain Nakajima quiver varieties.

For a B-algebra A we introduce a Hochschild-like cohomology and use it to describe simultaneous deformations of the product and of the B-algebra structure on A[[t]]. These deformations have the property that the natural projection A[[t]]→A is a morphism of B-algebras.

The Hochschild cohomology ring for one of the two series of self-injective algebras of tree class E6 is described in terms of generators and relations.

We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups. These embeddings are the projectivizations of reductive monoids. Our main result describes their equivariant cohomology in terms of roots, idempotents, and underlying monoid data. Also, we characterize those embeddings whose equivariant cohomology ring is obtained via restriction to the associated toric variety. Such characterization is given in terms of the closed orbits., 25 pages. Final version. To appear in Transformation Groups

On a smoothly stratified space, we identify intersection cohomology of any given perversity with an associated weighted \(L^2\) cohomology for iterated fibred cusp metrics on the smooth stratum.

The authors first construct an explicit minimal projective bimodule resolution (ℙ, δ) of the Temperley-Lieb algebra A, and then apply it to calculate the Hochschild cohomology groups and the cup product of the Hochschild cohomology ring of A based on a comultiplicative map Δ: ℙ → ℙ ⊗ A ℙ. As a consequence, the authors determine the multiplicative structure of Hochschild cohomology rings of both Temperley-Lieb algebras and representation-finite q-Schur algebras under the cup product by giving an explicit presentation by generators and relations.

Expanding a result of Serre on finite CW-complexes, we show that the Brauer group coincides with the cohomological Brauer group for arbitrary compact spaces. Using results from the homotopy theory of classifying spaces for Lie groups, we give another proof of the result of Antieau and Williams that equality does not hold for Eilenberg-MacLane spaces of type K(ℤ/nℤ, 2). Employing a result of Dwyer and Zabrodsky, we show the same for the classifying spaces BG where G is an infinite-dimensional F p -vector space. In this context, we also give a formula expressing phantom cohomology in terms of homology.

We use a differential form cohomology theory on transitive digraphs to give a new proof of a theorem of Gerstenhaber and Schack about isomorphism between simplicial cohomology and Hochschild cohomology of a certain algebra associated with the simplicial complex.

We study the Bredon-Illman cohomology with local coefficients for a G-space X in the case of G being a totally disconnected, locally compact group. We prove that any short exact sequence of equivariant local coefficients systems on X gives a long exact sequence of the associated Bredon-Illman cohomology groups with local coefficients.

We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Chern character form, a loop space refinement of the twisted Chern character form, which represent classes in the completed periodic exotic twisted $S^1$-equivariant cohomology of $LZ$. We establish a localisation theorem for the completed periodic exotic twisted $S^1$-equivariant cohomology for loop spaces and apply it to establish T-duality in a background flux in type II String Theory from a loop space perspective., 23 pages. To appear in CMP

In this paper, by using the de Rham model of Chen-Ruan cohomology, we define the relative Chen-Ruan cohomology ring for a pair of almost complex orbifold (G,H) with H being an almost sub-orbifold of G. Then we use the Gromov-Witten invariants of Ĝ, the blow-up of G along H, to give a quantum modification of the relative Chen-Ruan cohomology ring H*CR(G,H) when H is a compact symplectic sub-orbifold of the compact symplectic orbifold G.

We explicitly compute the first and second cohomology groups of the Schrodinger algebra S(1) with coefficients in the trivial module and the finite-dimensional irreducible modules. We also show that the first and second cohomology groups of S(1) with coefficients in the universal enveloping algebras U(S(1)) (under the adjoint action) are infinite dimensional.

In this paper we study K-cosymplectic manifolds, i.e., smooth cosymplectic manifolds for which the Reeb field is Killing with respect to some Riemannian metric. These structures generalize coK\"ahler structures, in the same way as K-contact structures generalize Sasakian structures. In analogy to the contact case, we distinguish between (quasi-)regular and irregular structures; in the regular case, the K-cosymplectic manifold turns out to be a flat circle bundle over an almost K\"ahler manifold. We investigate de Rham and basic cohomology of K-cosymplectic manifolds, as well as cosymplectic and Hamiltonian vector fields and group actions on such manifolds. The deformations of type I and II in the contact setting have natural analogues for cosymplectic manifolds; those of type I can be used to show that compact K-cosymplectic manifolds always carry quasi-regular structures. We consider Hamiltonian group actions and use the momentum map to study the equivariant cohomology of the canonical torus action on a compact K-cosymplectic manifold, resulting in relations between the basic cohomology of the characteristic foliation and the number of closed Reeb orbits on an irregular K-cosymplectic manifold., Comment: 33 pages, no figures. Comments are welcome! V2 with more references and some corrections (Proposition 3.5 in V1 deleted)

In this paper the authors investigate the structure of the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose E 2-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen–Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifying as a ring with the E 2-term. We present criteria for the collapsing of this spectral sequence and provide some examples where the ring isomorphism fails. Furthermore, we show that there are instances when the spectral sequence does not collapse and yields cohomology rings which are not Cohen-Macaulay.

We describe the multiplicative structure of the Hochschild cohomology ring H H ∗(Λ) of the generalized preprojective algebra \(\Lambda =\mathbb {B}_{n}\). This is done by giving the structure of the cohomology groups as modules over the center of Λ and by giving a presentation of H H ∗(Λ), as a bigraded algebra, by means of generators and relations.

The cohomology with trivial coefficients of the Lie algebra ℌ of Hamiltonian vector fields in the plane and of its maximal nilpotent subalgebra L 1ℌ is considered. The cohomology H 2(L 1ℌ) is calculated, and some far-reaching conjectures concerning the cohomology of the Lie algebras mentioned above and based on an extensive experimental material are formulated.

We consider the conormal bundle of a Schubert variety \(S_I\) in the cotangent bundle \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) of the Grassmannian \({{\mathrm{\mathrm {Gr}}}}\) of \(k\)-planes in \({{\mathrm{\mathbb {C}}}}^n\). This conormal bundle has a fundamental class \({\kappa _I}\) in the equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\). Here \({{\mathrm{\mathbb T}}}=({{\mathrm{\mathbb {C}}}}^*)^n\times {{\mathrm{\mathbb {C}}}}^*\). The torus \(({{\mathrm{\mathbb {C}}}}^*)^n\) acts on \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) in the standard way and the last factor \({{\mathrm{\mathbb {C}}}}^*\) acts by multiplication on fibers of the bundle. We express this fundamental class as a sum \(Y_I\) of the Yangian \(Y(\mathfrak {gl}_2)\) weight functions \((W_J)_J\). We describe a relation of \(Y_I\) with the double Schur polynomial \([S_I]\). A modified version of the \(\kappa _I\) classes, named \(\kappa '_I\), satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold \(T^*\!{{\mathrm{\mathrm {Gr}}}}\). This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on \({{\mathrm{\mathrm {Gr}}}}\). The classes \((\kappa '_I)_I\) form a basis in the suitably localized equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\). This basis depends on the choice of the coordinate flag in \({{\mathrm{\mathbb {C}}}}^n\). We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.

We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting group. We show that the algebra structure of the equivariant cohomology can be read off from this graph. In comparison with ordinary abelian GKM theory, there are some special features due to the more complicated structure of the non-abelian 1-skeleton., Comment: 25 pages

In this paper we apply the theory of finitely generated FI-modules developed by Church et al. to certain sequences of rational cohomology groups. An FI-module over \({\mathbb {Q}}\) is a functor from the category of finite sets and injections to the category of vector spaces over \({\mathbb {Q}}\). Our main examples are the cohomology of the moduli space of \(n\)-pointed curves, the cohomology of the pure mapping class group of surfaces and some manifolds of higher dimension, and the cohomology of classifying spaces of some diffeomorphism groups. We introduce the notion of FI\([G]\)-module and use it to strengthen and give new context to results on representation stability discussed by the author in a previous paper. Moreover, we prove that the Betti numbers of these spaces and groups are polynomial and find bounds on their degree. Finally, we obtain rational homological stability of certain wreath products.

For a semistable family of varieties over a curve in characteristic \(p\), we prove the existence of a “Clemens–Schmid type” long exact sequence for the \(p\)-adic cohomology. The cohomology groups appearing in such a long exact sequence are defined locally.

Given an arithmetic lattice of the unitary group U(3,1) arising from a hermitian form over a CM-field, we show that all unitary representations of U(3,1) with nonzero cohomology contribute to the cohomology of the attached arithmetic complex 3-manifold, at least when we pass to a finite-index subgroup of the given arithmetic lattice.

We present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex $G$-variety $X$ by its associated arc space $J_{\infty} X$, with its induced $G$-action. This not only allows us to obtain geometric classes in equivariant cohomology of arbitrarily high degree, but also provides more flexibility for equivariantly deforming classes and geometrically interpreting multiplication in the equivariant cohomology ring. Under appropriate hypotheses, we obtain explicit bijections between $\mathbb{Z}$-bases for the equivariant cohomology rings of smooth varieties related by an equivariant, proper birational map. We also show that self-intersection classes can be represented as classes of contact loci, under certain restrictions on singularities of subvarieties. We give several applications. Motivated by the relation between self-intersection and contact loci, we define higher-order equivariant multiplicities, generalizing the equivariant multiplicities of Brion and Rossmann; these are shown to be local singularity invariants, and computed in some cases. We also present geometric $\mathbb{Z}$-bases for the equivariant cohomology rings of a smooth toric variety (with respect to the dense torus) and a partial flag variety (with respect to the general linear group)., 35 pages; v2 contains new material introducing generalized equivariant multiplicities

In this paper, we describe cohomology theories constructed on the category of pairs of topological spaces and continuous maps and based on finite-valued cochains with coefficients in abelian group.