All Questions
Tagged with deformation-theory hochschild-cohomology
14 questions
23
votes
4
answers
2k
views
A matrix algebra has no deformations?
I have often heard the slogan that "a matrix algebra has no deformations," and I am trying to understand precisely what that means. While I would be happy with more general statements about finite-...
- 8,559
14
votes
2
answers
724
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 ...
- 185
14
votes
2
answers
2k
views
"Spec" of graded rings?
From the discussion at Hochschild cohomology and A-infinity deformations, it seems that general Hochschild cohomology classes correspond to deformations where the deformation parameter can have ...
- 21k
11
votes
1
answer
1k
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 ...
- 595
10
votes
2
answers
2k
views
A simple proof of the Weyl algebra's rigidity.
I am wondering if there is a nice presentation of the Hochschild cohomology of $A_n$ the Weyl algebra. It is known that $H^m(A_n,A_n)=0$ for $m>0$, and thus it is rigid. A proof can be found in ...
- 4,842
7
votes
3
answers
3k
views
Hochschild cohomology and A-infinity deformations
When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.
...
- 21k
7
votes
1
answer
679
views
Relation between symplectic (co)homology and Hochschild (co)homology and deformations
A very fluffy question in which I'm ignorant of homology/cohomology, grading etc:
The open-closed and closed-open string maps relating the symplectic (co)homology and Hochschild (co)homology of the ...
- 73
5
votes
3
answers
2k
views
Differential Hochschild Cohomology, general tools?
Background: in deformation quantization one wants to construct formal associative deformations (the star products $\star$) of the algebra of smooth complex-valued functions on a Poisson manifold $M$ ...
- 8,098
5
votes
0
answers
134
views
Transferred $L_\infty$-structure from Hochschild dgLA
Let $D_{poly}$ be the differential graded Lie algebra (dgLA) of differentiable Hochschild cochains on a manifold $\mathscr M$, endowed with the usual Gerstenhaber bracket $[-,-]_G$ and Hochschild ...
4
votes
0
answers
93
views
Drinfeld centres and formal moduli problems
If $\mathcal{P}$ is a sufficiently nice operad, then by [Higher Algebra, 5.3] you can form its centre:
$$\mathcal{Z}_{\mathcal{P}}\ :\ \mathcal{P}\text{-Alg}\ \to\ \mathbf{E}_1\text{-Alg}(\mathcal{P}\...
- 5,701
4
votes
0
answers
196
views
divided powers of a deformation class
Let $A$ be a (unital, associative) $k$-algebra where $k$ is a field. Given a flat deformation of $A$ one gets the deformation class $h$ in the second Hochschild cohomology $HH^2(A)$. Suppose $k$ has ...
- 1,526
4
votes
0
answers
863
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 $ ...
- 595
3
votes
1
answer
249
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 ...
3
votes
0
answers
76
views
Natural transformation and Hochschild cohomology
I am reading the lecture note by Căldăraru:https://arxiv.org/pdf/math/0501094, in the last chapter of this note, he said that we should consider dg category instead of the derived category of coherent ...
- 395