Gerstenhaber Bracket Research Articles

Polysymplectic formulation for topologically massive Yang–Mills field theory

We analyze the De Donder–Weyl covariant field equations for the topologically massive Yang–Mills theory. These equations are obtained through the Poisson–Gerstenhaber bracket described within the polysymplectic framework. Even though the Lagrangian defining the system of our interest is singular, we show that by appropriately choosing the polymomenta one may obtain an equivalent regular Lagrangian, thus avoiding the standard analysis of constraints. Further, our simple treatment allows us to only consider the privileged [Formula: see text]-forms in order to obtain the correct field equations, in opposition to certain examples found in the literature.

BV structure on the Hochschild cohomology of Sullivan algebras

Abstract Let X be a closed, simply connected manifold of dimension m and LX the space of free loops on X . If (∧ V, d ) is the minimal Sullivan model of X where V is finite dimensional, then there is a Gerstenhaber algebra ( ∧ V ⊗ ∧ s − 1 V # , d 0 ) , where V # is the graded dual of V , and its homology is isomorphic to the loop space homology H * ( L X ) . In this paper we define a BV structure on ( ∧ V ⊗ ∧ s − 1 V # , d 0 ) which extends the Gerstenhaber bracket.

The derived non-commutative Poisson bracket on Koszul Calabi–Yau algebras

Let A be a Koszul (or more generally, N -Koszul) Calabi–Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on A , which induces a graded Lie algebra structure on the cyclic homology of A ; moreover, we show that the Hochschild homology of A is a Lie module over the cyclic homology and the Connes long exact sequence is in fact a sequence of Lie modules. Finally, we show that the Leibniz–Loday bracket associated to the derived non-commutative Poisson structure on A is naturally mapped to the Gerstenhaber bracket on the Hochschild cohomology of its Koszul dual algebra and hence on that of A itself. Relations with some other brackets in literature are also discussed and several examples are given in detail.

Brackets in the Pontryagin Algebras of Manifolds

Given a smooth oriented manifold $M$ with non-empty boundary, we study the Pontryagin algebra $A=H_*(\Omega)$ where $\Omega$ is the space of loops in $M$ based at a distinguished point of $\partial M$. Using the ideas of string topology of Chas-Sullivan, we define a linear map $\{\{-,-\}\} : A \otimes A \to A \otimes A$ which is a double bracket in the sense of Van den Bergh satisfying a version of the Jacobi identity. For $\dim(M)>2$, the double bracket $\{\{-,-\}\}$ induces Gerstenhaber brackets in the representation algebras associated with $A$. This extends our previous work on the case $\dim(M)=2$ where $A=H_0(\Omega)$ is the group algebra of $\pi_1(M)$ and $\{\{-,-\}\}$ induces the classical Poisson brackets on the moduli spaces of representations of $\pi_1(M)$.

A little bit of extra functoriality for Ext and the computation of the Gerstenhaber bracket

We show that the action of the Lie algebra HH1(A) of outer derivations of an associative algebra A on the Hochschild cohomology HH•(A) of A given by the Gerstenhaber bracket can be computed in terms of an arbitrary projective resolution of A as an A-bimodule, without having recourse to comparison maps between the resolution and the bar resolution.

A Comparison of Products in Hochschild Cohomology

In this paper we transport Steenrod's cup-i products, i ≥ 0, from the singular cochains on the free loop space Maps(S1, BG) to Hochschild's original cochain complex Homk(k[G]⊗*, k[G]) defining Hochschild cohomology. Here G is a discrete group, k an arbitrary (commutative) coefficient ring, and BG the classifying space of G. This induces a natural action of the (mod 2) Steenrod algebra on the Hochschild cohomology of a group ring. For cochains supported on BG, we prove that Gerstenhaber's cup product agrees with the simplicial cup product and Gerstenhaber's pre-Lie product agrees with Steenrod's cup-one product. As a consequence, for cocycles f and g supported on BG, the Gerstenhaber bracket [f, g] = 0 in HH*(k[G]; k[G]). This is interpreted in terms of the Batalin–Vilkovisky structure on HH*(k[G]; k[G]).

Gerstenhaber brackets on Hochschild cohomology of quantum symmetric algebras and their group extensions

We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute brackets also in some cases for skew group algebras arising as group extensions of quantum symmetric algebras.

PBW deformations of quantum symmetric algebras and their group extensions

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

An alternate approach to the Lie bracket on Hochschild cohomology

We define Gerstenhaber's graded Lie bracket directly on complexes other than the bar complex, under some conditions. The Koszul complex of a Koszul algebra in particular satisfies our conditions. As examples we recover the Schouten-Nijenhuis bracket for a polynomial ring and the Gerstenhaber bracket for a group algebra of a cyclic group of prime order.

The odd origin of Gerstenhaber brackets, Batalin-Vilkovisky operators, and master equations

Using five basic principles, we treat Gerstenhaber/Lie brackets, Batalin-Vilkovisky (BV) operators, and master equations appearing in mathematical and physical contexts in a unified way. The different contexts for this are given by the different types of (Feynman) graphs that underlie the particular situation. Two of the maxims we bring forth are (1) that extending to the non-connected graphs gives a commutative multiplication forming a part of the BV structure and (2) that there is a universal odd twist that unifies and explains seemingly ad hoc choices of signs and is responsible for the BV operator being a differential. Our treatment results in uniform, general theorems. These allow us to prove new results and recover and connect many constructions that have appeared independently throughout the literature. The more general point of view also allows us to disentangle the necessary from the circumstantial.

The Gerstenhaber bracket of Hochschild cohomology of triangular quadratic monomial algebra

In this paper, we give a combinatorial description for the Gerstenhaber bracket on the Hochschild cohomology of triangular quadratic monomial algebras. Moreover, we describe explicitly the Gerstenhaber bracket of the cohomology of Fibonacci algebras in terms of parallel paths, and obtain that its Lie algebra structure is nontrivial while the ring structure is trivial.

Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology

In this monograph, we extend S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish an explicit description of an isomorphism by A. Neeman and V. Retakh, which links $\mathrm{Ext}$-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, we show that our construction behaves well with respect to structure preserving functors between exact monoidal categories. We use our main result to conclude, that both the Lie bracket and the squaring map in Hochschild cohomology are invariants under Morita equivalence. For quasi-triangular bialgebras, we further determine a significant part of the Lie bracket's kernel, and thereby prove a conjecture by L. Menichi. Along the way, we introduce $n$-extension closed and entirely extension closed subcategories of abelian categories, and study some of their properties.

Models for classifying spaces and derived deformation theory

Using the theory of extensions of L ∞ algebras, we construct rational homotopy models for classifying spaces of fibrations, giving answers in terms of classical homological functors, namely the Chevalley–Eilenberg and Harrison cohomology. We also investigate the algebraic structure of the Chevalley–Eilenberg complexes of L ∞ algebras and show that they possess, along with the Gerstenhaber bracket, an L ∞ structure that is homotopy abelian.

Poisson–Gerstenhaber brackets in representation algebras

We introduce cyclic bilinear forms on coalgebras and use them to generalize Van den Berghʼs Poisson brackets in representation algebras.

Drinfeld orbifold algebras

We define Drinfeld orbifold algebras as filtered algebras deforming the skew group algebra (semi-direct product) arising from the action of a finite group on a polynomial ring. They simultaneously generalize Weyl algebras, graded (or Drinfeld) Hecke algebras, rational Cherednik algebras, symplectic reflection algebras, and universal enveloping algebras of Lie algebras with group actions. We give necessary and sufficient conditions on defining parameters to obtain Drinfeld orbifold algebras in two general formats, both algebraic and homological. Our algebraic conditions hold over any field of characteristic other than two, including fields whose characteristic divides the order of the acting group. We explain the connection between Hochschild cohomology and a Poincare-Birkhoff-Witt property explicitly (using Gerstenhaber brackets). We also classify those deformations of skew group algebras which arise as Drinfeld orbifold algebras and give applications for abelian groups.

Group actions on algebras and the graded Lie structure of Hochschild cohomology

Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms. We examine the Gerstenhaber bracket with a view toward deformations and developing bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We give some general results describing when brackets are zero for polynomial skew group algebras, which allow us in particular to find noncommutative Poisson structures. For abelian groups, we express the bracket using inner products of group characters. Lastly, we interpret results for graded Hecke algebras.

The Lie structure on the Hochschild cohomology of a modular group algebra

We prove that the Gerstenhaber bracket on the Hochschild cohomology of the group algebra of a cyclic group over a field of positive characteristic is not trivial. In this case, we relate the Lie algebra structure on the odd degrees of the Hochschild cohomology with a Witt-type algebra.

截面箭图代数的Gerstenhaber 括号积

Based on the minimal projective bimodule resolution, the Gerstenhaber bracket products on the Hochschild cohomology spaces for truncated quiver algebras are described explicitly in terms of parallel paths. As a consequence, for truncated basic cycle algebra Λ, each element in HH2(Λ) defines a noncommutative Poisson structure on Λ, and in turn defines the first multiplication of a one-parameter deformation of Λ.

Noncommutative Poisson structures on orbifolds

In this paper, we compute the Gerstenhaber bracket on the Hoch-schild cohomology of C ∞ ( M ) ⋊ G C^\infty (M)\rtimes G for a finite group G G acting on a compact manifold M M . Using this computation, we obtain geometric descriptions for all noncommutative Poisson structures on C ∞ ( M ) ⋊ G C^\infty (M)\rtimes G when M M is a symplectic manifold. We also discuss examples of deformation quantizations of these noncommutative Poisson structures.

The Lie module structure on the Hochschild cohomology groups of monomial algebras with radical square zero

We study the Lie module structure given by the Gerstenhaber bracket on the Hochschild cohomology groups of a monomial algebra with radical square zero. The description of such Lie module structure will be given in terms of the combinatorics of the quiver. The Lie module structure will be related to the classification of finite dimensional modules over simple Lie algebras when the quiver is given by the two loops and the ground field is the complex numbers.