Questions tagged [hochschild-cohomology]

The tag has no usage guidance.

6
votes
0answers
99 views

Hochschild cohomology of the $A_\infty$-category of paths

I would like to describe the Hochschild cohomology (in the sense of $A_\infty$-categories) of the following $A_\infty$-category associated to a topological space $X$: It has points of $X$ as objects. ...
1
vote
0answers
55 views

Two definitions of minimal models

Is there any relationship between both definitions of minimal models? (the couple of definitions I know are the one mentioned in Lefèvre's thesis, in the sense that the differential is zero, and the ...
4
votes
0answers
128 views

A relation between Hochschild cohomology of a $C^*$ algebra and its bidual

Let $A$ be a $C^*$ algebra and $A''$ be its bidual with the Arens product. Is there any relation between the Hochschild cohomolgy of $A$ with complex coefficients and the Hochschild cohomology of $A''$...
0
votes
1answer
71 views

Compute the cohomology of $\mathrm{Hom} (\Omega^*(M),\Omega^*(M))$

Let $M$ be a compact smooth manifold. And particularly I am interested in the case the torus $M=T^n$. Consider the de Rham complex $(\Omega^*(M), d)$ and the cochain complex $$ C:=\mathrm{Hom} (\...
1
vote
0answers
94 views

Question on vanishing Hochschild cohomology

Recall that for an $K$-algebra $A$ with $A^e:=A^{op} \otimes_K A$ the Hochschild cohomology is defined as $HH^n(A,M):=Ext_{A^e}^n(A,M)$. Question: Is there a finite dimensional selfinjective ...
1
vote
0answers
53 views

A 2- cocycle $\tau$ which is not cyclic but it still satisfies the stability of $\tau(e,e,e)$ for idempotent $e$

I learned the following statement from page $20$ of the book Noncommutative Geometry by Alain Connes: Let $\tau$ be a $2$-cyclic cocyle on a $C^*$ algebra. Then for every smooth curve $e(t)$ of ...
2
votes
0answers
45 views

Units in the (stable) center of a Frobenius algebra [duplicate]

Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is ...
12
votes
2answers
282 views

Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?

I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why ...
10
votes
0answers
420 views

Higher traces in Hochschild cohomology

Let $A$ be an associative algebra over a field $k$. Let $\rho:A \rightarrow \mathrm{End}(M)$ a left module, finite dimensional over $k$. Then the map $a \mapsto \mathrm{tr}_M \rho(a)$ is a well ...
1
vote
0answers
136 views

The comparison of certain modules arising from the Cauchy-Riemann differential operator

Let $\Gamma=C^{\infty}(\mathbb{R}^2)$ be the space of all smooth complex valued functions on the plane. We define the following Cauchy Riemann differential operator $D$ on $\Gamma$: $$D:\Gamma \...
4
votes
1answer
232 views

A precise definition of contractible Banach algebras

I asked this question at MSE but I did not received any answer. So I ask it here at MO I am sorry if this question is elementary: What is a precise definition of a contractible Banach ...
2
votes
0answers
56 views

Notion of “strict $A_\infty$ centre”

There is definition of "$A_\infty$ Centre" in article The A_\infty-Centre of the Yoneda Algebra and the Characteristic Action of Hochschild Cohomology on the Derived Category at p.28. It can be ...
3
votes
1answer
376 views

Relation between the Hochschild cohomology of group algebras and groupoids

Is there a known relation between the Hochschild cohomology of group algebras and cohomology of groupoids? Clarification: It is known that 1-dimensional Hochschild cohomology of the Group algebra C[...
7
votes
0answers
283 views

Hochschild cohomology of a universal enveloping algebra of a Lie algebra

I was told that the following equation is true: Given a finitely generated Lie algebra $\mathfrak g$, there is a Gerstenhaber algebra isomorphism $$ HH(U\mathfrak g) \cong HH(\wedge^* \mathfrak g^\vee,...
19
votes
2answers
2k views

intuition for hochschild homology

According to this post Intuition for group homology, I wonder what is the intuition for Hochschild homology. The Hochschild homology is defined as the homology of this complex chain. Given a ...
3
votes
1answer
119 views

A differential graded Lie algebra with the Hochschild differential

Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following ...
6
votes
1answer
236 views

Morita equivalence and isomorphisms in cohomology theories

Let $A,B$ be two unital algebras. We say that $A,B$ are Morita equivalent if there are $A-B$ and $B-A$ bimodules $P,Q$ such that $$P \otimes_{B} Q \cong A, Q \otimes_A P \cong B$$ (as $A-A$ and $B-B$ ...
2
votes
1answer
190 views

First Hochschild cohomology of $A=K[x]/(x^n)$

Given the algebra $A=K[x]/(x^n)$ for some field $K$ and natural number $n \geq 2$ with enveloping algebra $A^e=A \otimes_K A$. It is easy to see that the 1. Hochschild cohomology of $A$ is nonzero ...
6
votes
1answer
271 views

Hochschild cohomology of certain local algebras

Let $K$ be a field and $A$ the algebra $K\langle x_1,...,x_n\rangle/J^m$ for $n \geq 1$ and $m \geq 2$, where $K\langle x_1,...,x_n\rangle$ is the non-commutative polynomial ring in $n$ variables over ...
6
votes
1answer
233 views

Lie algebra of a p-group

Given a p-group P, the first hochschild cohomology of the group algebra (over a field of characteristic p) of P is a nonzero Lie algebra. Is it known what Lie algebra results depending on P? I have no ...
2
votes
0answers
104 views

Another question on Hochschild cohomology

All algebras are assumed to be connected over a field. Given an algebra $A$ with minimal faithful projective-injective left module $Af$ for some idempotent f. Let $M$ be an $A$-module of infinite ...
3
votes
1answer
176 views

A question on Hochschild cohomology

Given a nonsemisimple symmetric algebra B and a non-selfinjective algebra A (all algebras are finite dimensional over a field and connected). Can A and B have isomorphic Hochschild-cohomology rings?
1
vote
0answers
92 views

Hochschild coboundary on the space of alternative forms

Assume that $A$ is a complex algebra. By $C^{n}(A)$ we mean the space of all $n-$linear map $\phi:A^n \to \mathbb{C}$. An alternative $k-$ form is an element $\phi \in C^{k}(A)$ ...
5
votes
0answers
323 views

Definitions of Hochschild Cohomology $HH^{\bullet}(A)$

Let $A$ be an associative unital $k$-algebra, and let $M$ be a bimodule of $A$. The Hochschild cohomology of $A$ with coefficients in $M$ can be defined as $$HH^{n}(A,\,M)=\mathrm{Ext}^{n}_{A^{e}}(A,\,...
9
votes
0answers
497 views

From classical to quantum mechanics

Let ($X,\omega$) be a symplectic manifold (phase space of some physical system). Consider the algebra $\mathcal{C}^{\infty}(X,\mathbb{R})$ of smooth functions on $X$ and $[\omega]\in \textrm{H}^{2}_{\...
10
votes
0answers
513 views

Is “Determinant” a Hochschild coboundary?

Assume that $n>2$. Is there an associative unital algebra structure on $\mathbb{C}^{n}$ such that $D$, the determinant as a $n-\text{form} $ on $\mathbb{C}^{n}$, would be a ...
7
votes
1answer
930 views

What is the negative cyclic homology of a smooth projective variety?

Let X be a smooth and projective variety. Hochschild homology and cohomology have a very simple definition in terms of Ext groups of the diagonal of X. The Hochschild-Kostant-Rosenberg (HKR) theorem ...
14
votes
1answer
425 views

Multiplicativity twisted Hochschild Kostant Rosenberg isomorphism

Let $X$ be a smooth projective variety over $\mathbb{C}$. I call (following Swan) Hochschild cohomology of $X$ the graded algebra: $$ \mathrm{HH}^{\bullet}(X) := \mathrm{Ext}^{\bullet}_{X \times X}(\...
4
votes
2answers
371 views

Normalization of Hochschild cocycles

Let $A$ be a unital algebra over $\mathbb{C}$. Let $C^n(A)$ be the space of all $n+1$-linear maps $f:A^{n+1} \to \mathbb{C}$ (to be called $n$-cochains). Define $b:C^n(A) \to C^{n+1}(A)$ by the ...
4
votes
0answers
187 views

Inner automorphisms acts as identity on Hochschild homology

Let $A$ be a unital algebra and $u \in A$ be an invertible element. Let us consider $u_n(a_0 \otimes a_1 \otimes ... \otimes a_n):=ua_0u^{-1} \otimes ua_1u^{-1} \otimes ... \otimes ua_nu^{-1}$. Then $(...
9
votes
1answer
421 views

Isomorphism in cyclic cohomology vs isomorphism in Hochschild cohomology

Let $A$ be a unital algebra over a field $K$, $C^n(A)$ a space of all $n+1$ linear maps into scalar field $k$ (I'm interested in case $k=\mathbb{C}$) and $$(bf)(a_0,...,a_{n+1})=\sum_{i=0}^n(-1)^if(...
7
votes
1answer
366 views

Can the Hochschild cochain complex be given the structure of a “homotopy BV algebra”?

In a 1993 letter, Deligne posed the following (paraphrased from a paper of Gerstenhaber and Voronov's): Conjecture (Deligne). The Hochschild cochain complex $CC^*(A)$ of an associative algebra ...
5
votes
0answers
218 views

Hochschild cohomology of SU(2)

I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth. Let's ...
12
votes
2answers
420 views

Gerstenhaber conjecture for free loop space

I- Is the following statement still a conjecture see this article ? Conjecture (?) Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild ...
16
votes
0answers
764 views

What is the Hochschild cohomology of the Fukaya-Seidel category?

Let $(Y, \omega)$ be a compact symplectic manifold and let $Fuk(X,\omega)$ be its Fukaya category. The Hochschild cohomology of this category should be given by $HH^\bullet(Fuk(Y,\omega))=H^\bullet(Y, ...
5
votes
0answers
254 views

Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...
5
votes
1answer
139 views

Antisymmetrization of the Hochschild cocycle

Let $A$ be a commutative (unital, complex) algebra and let $\varphi$ be a $n+1$-linear functional on $A$ (we will call it cochain). Define $$(b\varphi)(a_0,a_1,...,a_{n+1}):=\sum_{j=0}^{n}(-1)^j\...
1
vote
0answers
358 views

Contraction with a vector field and pullback bundle

While trying to understand the proof of "smooth" version of Kostant-Hochschild-Rosenberg theorem (which is due to Connes for compact smooth manifolds) I found the following argumentation: one is ...
5
votes
0answers
166 views

Hochschild cohains-type models for smooth functions on shifted cotangent bundles

Let $M$ be a smooth manifold. Then, by the well known Hochschild-Kostant-Rosenberg theorem, the cochain complex $C^*(C^\infty(M))$ of Hochshild cochains on the algebra $C^\infty(M)$ of smooth ...
11
votes
2answers
1k views

Relationship between Hochschild cohomology and Drinfeld centers

Let $HH_*(A,N)$ (or $HH^*(A,N)$) be the Hochschild homology (or cohomology) of an associative algebra $A$ with coefficients in an $A$-bimodule $N$. I was reading nlab's entry on Hochschild cohomology ...
2
votes
1answer
187 views

Hochschild cohomology of the skew group ring D(X)#G in the complex analytic case

Let $X$ be a n-dimensional complex (not necessarily compact) manifold, let $G$ be a finite subgroup of $Aut(X)$ acting by biholomorphic maps on $X$ and let $D(X)$ be the ring of differential operators ...
2
votes
1answer
221 views

Hochschild cohomology of commutative quotients

Notation: Let $k$ be a commutative local ring and let $HH^{i}(A,N)$ denote the $i^{th}$ Hochschild cohomology $k$-module of a $k$-algebra A with coefficients in an $(A,A)$-bi-module $N$. If $x:=\{...
2
votes
1answer
386 views

Interpretation of Hochschild Homology groups

In all the literature I've come across there are many concrete interpretations of the first few Hochschild Cohomology groups. For example $HH^1(A,M)\cong Derivation/Inner Derivations$ etc.... In ...
10
votes
1answer
734 views

Motivation behind the definition of hochschild cohomology

For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as ...
3
votes
0answers
554 views

Hochschild cohomology and bar resolutions

I asked the following question on mathstack but didn't receive any comments, so I thought I'd try my luck here. Let $A$ be an associative algebra over a field $k$. One can define $HH^n (A,A)$ as $ ...
1
vote
1answer
434 views

Hochschild cohomology and formal smoothness

Hochschild cohomology can be used to characterise formal smoothness of unital associative algebras; in that such an algebra $A$ is formally smooth if and only if it is of Hochschild cohomological ...
0
votes
0answers
372 views

A noncommutative vector bundle

We know that a noncommutative vector bundle is a finitely generated projective $A$-module where $A$ is a non commutative $C^{*}$ algebra. In this question we introduce a particular non commutative ...
4
votes
0answers
388 views

Soft Question: What does periodic cyclic theory measure?

Ex1) The cyclic homology of $\mathbb{C}[X,Y]$ and that of the algebra of functions on the sphere $S^2$ have the same periodic cyclic homology. Clearly, however, these objects are topologically very ...
2
votes
0answers
81 views

Poincaré Duality of a quasi-free algebra

I'm completely stumped on this one (yet I feel it is obviously true or obviously false) If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality? All i need to find is a protective ...
1
vote
1answer
187 views

Jacobi-Zariski exact sequence question

Denote by $HC(A,M)$ the Hochschild homological complex of an algebra $A$ with coefficients in an $A$-bimodule $M$, and let $B\rightarrow A$ be an $R$-flat extension of $R$-algebras, for some $CRing$ $...