Laure Helme-Guizon, Yongwu Rong
For each commutative, graded algebra with finite dimension in each degree, we construct a graded cohomology theory for graphs whose graded Euler characteristic is the chromatic polynomial of the graph. This extends our previous work which was based on the algebra $\mathbb {Z}[x]/(x^2)$.
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We study algebras constructed by quantum Hamiltonian reduction associated with symplectic quotients of symplectic vector spaces, including deformed preprojective algebras, symplectic reflection algebras (rational Cherednik algebras), and quantization of hypertoric varieties introduced by Musson and Van den Bergh. We determine BRST cohomologies associated with these quantum Hamiltonian reductions. To compute these BRST cohomologies, we make use of method of deformation quantization (DQ-algebras) and F-action studied in [Kashiwara-Rouquier], and in [Gordon-Losev].
We present an unified construction for algebras and modules homologies and cohomologies, in the case of associative, commuttaive, Lie and Gerstenhaber algebras. We make a distinction between the linear part of the construction of algebras and cogebras, characterized by the symmetries of the defining relations and the structure itself which appears as a differential on these algebras and cogebras.
We apply discrete algebraic Morse theory to the computation of Hochschild cohomologies of associative conformal algebras. As an example, we evaluate the dimensions of the universal associative conformal envelope $U(3)$ of the Virasoro Lie conformal algebra relative to the associative locality $N=3$ on the generator with scalar coefficients.
For a class of pointed Hopf algebras including the quantized enveloping algebras, we discuss cleft extensions, cocycle deformations and the second cohomology. We present such a non-standard method of computing the abelian second cohomology that derives information from the non-abelian second cohomology classifying cleft extensions. As a sample computation, a quantum analogue of Whitehead's second lemma for Lie-algebra cohomology is proved.
In this paper we continue to study Belavin-Drinfeld cohomology introduced in arXiv:1303.4046 [math.QA] and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra. Here we compute Belavin-Drinfeld cohomology for all non-skewsymmetric $r$-matrices from the Belavin-Drinfeld list for simple Lie algebras of type $B$, $C$, and $D$.
We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_p$ for any integer $p>2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of $T_p$, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild co…
A pre-Lie-Rinehart algebra is an algebraic generalization of the notion of a left-symmetric algebroid. We construct pre-Lie-Rinehart algebras from r-matrices through Lie algebra actions. We study cohomologies of pre-Lie-Rinehart algebras and show that abelian extensions of pre-Lie-Rinehart algebras are classified by the second cohomology groups. We introduce the notion of crossed modules for pre-Lie-Rinehart algebras and show that they are classified by the third cohomology groups of pre-Lie-Rinehart algebras. At last, we use (pre-)Lie-Rinehart 2-algebras to characterize the crossed modules fo…
Associative conformal algebras of conformal endomorphisms are of essential importance for the study of finite representations of conformal Lie algebras (Lie vertex algebras). We describe all semisimple algebras of conformal endomorphisms which have the trivial second Hochschild cohomology group with coefficients in every conformal bimodule. As a consequence, we state a complete solution of the radical splitting problem in the class of associative conformal algebras with a finite faithful representation.
We study Hochschild homology and cohomology for some polynomial algebras mixing both ``classical'' relations ($XY-YX=1$) and ``quantum'' relations ($XY=łYX$). More specifically, we prove that the algebra of differential operators on any quantum affine space (quantum Weyl algebra) have the same Hochschild homology, and satisfy the same duality relation, as the classical Weyl algebra does.
In the present note we show that the Duflo isomorphism extends to an isomorphism of associative algebras of tangential cohomologies. This result confirms the B.Shoikhet's conjecture.
In this paper, we review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras, which have attracted quite much interest recently. Using Voronov's higher derived brackets, one can obtain an $L_\infty$-algebra whose Maurer-Cartan elements are relative Rota-Baxter Lie algebras. Then using the twisting method, one can obtain the $L_\infty$-algebra that controls deformations of a relative \RB Lie algebra. Meanwhile, the cohomologies of relative Rota-Baxter Lie algebras can also be defined with the help of the twisted $L_\infty$-algebra. Using the controlling algebra ap…
The fundamental example of Gerstenhaber algebra is the space $T_{poly}({\mathbb R}^d)$ of polyvector fields on $\mathbb{R}^d$, equipped with the wedge product and the Schouten bracket. In this paper, we explicitely describe what is the enveloping $G_\infty$ algebra of a Gerstenhaber algebra $\mathcal{G}$. This structure gives us a definition of the Chevalley-Harrison cohomology operator for $\mathcal{G}$. We finally show the nontriviality of a Chevalley-Harrison cohomology group for a natural Gerstenhaber subalgebra in $T_{poly}({\mathbb R}^d)$.
In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $\N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For…
We show that the Gerstenhaber-Schack cohomology of a Hopf algebra determines its Hochschild cohomology, and in particular its Gerstenhaber-Schack cohomological dimension bounds its Hochschild cohomological dimension, with equality of the dimensions when the Hopf algebra is cosemisimple of Kac type. Together with some general considerations on free Yetter-Drinfeld modules over adjoint Hopf subalgebras and the monoidal invariance of Gerstenhaber-Schack cohomology, this is used to show that both Gerstenhaber-Schack and Hochschild cohomological dimensions of the coordinate algebra of the quantum p…
It is well known that anomaly cancellations for D_16 Lie algebra are at the root of the first string revolution. For E_8 Lie algebra, cancellation of anomalies is the principal fact leading to the existence of heterotic string. They are in fact nothing but the 6th order cohomologies of corresponding Lie algebras. Beyond 6th order, the calculations seem to require special care and it could be that their study will be worthwhile in the light of developments of the second string revolution. As we have shown in a recent article, for A_N Lie algebras, there is a method which are based on the calcul…
The space $T_{poly}(\mathbb R^d)$ of all tensor fields on $\mathbb R^d$, equipped with the Schouten bracket is a Lie algebra. The subspace of ascending tensors is a Lie subalgebra of $T_{poly}(\mathbb R^d)$. In this paper, we compute the cohomology of the adjoint representations of this algebra (in itself and $T_{poly}(\mathbb R^d)$), when we restrict ourselves to cochains defined by aerial Kontsevitch's graphs like in our previous work (Pacific J of Math, vol 229, no 2, (2007) 257-292). As in the vectorial graphs case, the cohomology is freely generated by all the products of odd wheels.
Let $V$ be a grading-restricted vertex algebra and $W$ a $V$-module. We show that for any $m\in \mathbb{Z}_{+}$, the first cohomology $H^{1}_{m}(V, W)$ of $V$ with coefficients in $W$ introduced by the author is linearly isomorphic to the space of derivations from $V$ to $W$. In particular, $H^{1}_{m}(V, W)$ for $m\in \mathbb{N}$ are equal (and can be denoted using the same notation $H^{1}(V, W)$). We also show that the second cohomology $H^{2}_{\frac{1}{2}}(V, W)$ of $V$ with coefficients in $W$ introduced by the author corresponds bijectively to the set of equivalence classes of square-zero …
Given a mechanical system $(M, \mathcal{F}(M))$, where $M$ is a Poisson manifold and $\mathcal{F}(M)$ the algebra of regular functions on $M$, it is important to be able to quantize it, in order to obtain more precise results than through classical mechanics. An available method is the deformation quantization, which consists in constructing a star-product on the algebra of formal power series $\mathcal{F}(M)[[\hbar]]$. A first step toward study of star-products is the calculation of Hochschild cohomology of $\mathcal{F}(M)$. The aim of this article is to determine this Hochschild cohomology i…
We study finite dimensional representations of the quantum affine algebra, using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.
We compute the Hochschild and Gerstenhaber-Schack cohomological dimensions of the universal cosovereign Hopf algebras, when the matrix of parameters is a generic asymmetry. Our main tools are considerations on the cohomologies of free product of Hopf algebras, and on the invariance of the cohomological dimensions under graded twisting by a finite abelian group.
$\mathcal{O}$-operators are important in broad areas in mathematics and physics, such as integrable systems, the classical Yang-Baxter equation, pre-Lie algebras and splitting of operads. In this paper, a deformation theory of $\mathcal{O}$-operators is established in consistence with the general principles of deformation theories. On the one hand, $\mathcal{O}$-operators are shown to be characterized as the Maurer-Cartan elements in a suitable graded Lie algebra. A given $\mathcal{O}$-operator gives rise to a differential graded Lie algebra whose Maurer-Cartan elements characterize deformatio…
Let $C$ be a differential graded coalgebra, $ \barΩC$ the Adams cobar construction and $C^\vee$ the dual algebra. We prove that for a large class of coalgebras $C$ there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies $HH^\ast (C^\vee, C ^\vee)$ and $HH^\ast (\barΩC ; \barΩC)$. This result permits to describe a Hodge decomposition of the loop space homology of a closed oriented manifold, in the sense of Chas-Sullivan, when the field of coefficients is of characteristic zero.
It is clarified how cohomologies and Gerstenhaber algebras can be associated with linear pre-operads (comp algebras). Their relation to mechanics and operadic physics is concisely discussed.
It is well known that a foliation F of a smooth manifold M gives rise to a rich cohomological theory, its characteristic (i.e., leafwise) cohomology. Characteristic cohomologies of F may be interpreted, to some extent, as functions on the space P of integral manifolds (of any dimension) of the characteristic distribution C of F. Similarly, characteristic cohomologies with local coefficients in the normal bundle TM/C of F may be interpreted as vector fields on P. In particular, they possess a (graded) Lie bracket and act on characteristic cohomology H. In this paper, I discuss how both the Lie …
We compute the low dimensional cohomologies $\tilde H^q(gc_N,C)$, $H^q(gc_N,\C)$ of the infinite rank general Lie conformal algebras $gc_N$ with trivial coefficients for $q\le3, N=1$ or $q\le2, N\ge2$. We also prove that the cohomology of $gc_N$ with coefficients in its natural module is trivial, i.e., $H^*(gc_N,\C[\ptl]^N)=0$; thus partially solve an open problem of Bakalov-Kac-Voronov in [{\it Comm. Math. Phys.,} {\bf200} (1999), 561-598].
Bialgebra cohomology and exact sequences
本文研究正合序列中两个 Hopf 代数的双代数上同调之间的关系，在第三个因子为有限维余半单的条件下建立联系，并据此给出泛余主权 Hopf 代数在一般情形下双代数上同调计算的简短新证明。
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