Questions tagged [hochschild-cohomology]
The hochschild-cohomology tag has no usage guidance.
6
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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 ...
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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 $CRing$ $...
