All Questions
Tagged with deformation-theory ra.rings-and-algebras
19 questions
5
votes
0
answers
154
views
One-parameter family of algebra structures: characterizing trivial deformation as adjoint 2-cocycle being a 2-coboundary
$\DeclareMathOperator\C{\mathbf{C}}$Motivation: this post discusses a simple criterion for a 1-parameter family of $n$-dimensional complex algebras to be a "trivial deformation", i.e., be ...
- 63.9k
3
votes
0
answers
214
views
formal smoothness and cotangent complex
If $k$ is a field and $A$ is a formally smooth $k$-algebra, then we know that $\Omega^{1}_{A/k}$ is projective. What about its cotangent complex $L_{A/k}$? When is it quasi-isomorphic to $\Omega^{1}_{...
- 3,472
1
vote
0
answers
105
views
formal smoothness and McQuillan formal schemes
Let $k$ be an algebraically closed field, $A\rightarrow B$ be a continuous map of weakly admissible topological $k$-local algebras.
We assume that it is formally smooth and topologically of finite ...
- 3,472
7
votes
2
answers
386
views
Non-associative deformation quantization
Several physicists consider non-Poisson bivectors but still apply Kontsevich formality in order to get deformation quantization type results: see e.g. Szabos's review An introduction to nonassociative ...
- 3,880
6
votes
1
answer
214
views
Weak associativity
Let $(V,*)$ be an algebra and denote $A_*\in \text{Hom}(V^{\otimes 3},V)$ the associator of the binary product $*\in \text{Hom}(V^{\otimes 2},V)$ defined as $A_*(a,b,c):=(a*b)*c-a*(b*c)$.
The ...
2
votes
1
answer
175
views
Finite generation of flat deformations of algebras
Let $R=\mathbb C[q^{\pm 1}]$ and let $A$ be a graded (possibly non-commutative) $R$-algebra, $A=\oplus_{n=0}^\infty A_n,$ where $A_0=R$ and all $A_n$'s are free $R$-modules.
Then $A'=A/(q-1)$ is a ...
- 2,390
1
vote
0
answers
78
views
Compute action of the gauge group in deformation theory of an algebra
I am working on Balazs Szendroi's introduction to deformation theory, but I got stuck on the exercise on the bottom of page 6.
Consider a vector space $A$ with a multiplication $m$ that makes it into ...
- 1,509
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 ...
8
votes
1
answer
241
views
Classification of formality morphisms for chains and Drinfel'd associators
In his 1997 preprint q-alg/9709040, M. Kontsevich proved constructively the existence of a $L_\infty$-quasi-isomorphism between the differential graded algebra structure on the deformation complex of ...
4
votes
1
answer
269
views
3-Gerstenhaber algebra structure on the cohomology of deformation complexes?
In a seminal paper "On the Deformation of Rings and Algebras", M. Gerstenhaber showed that the deformation complex of any associative algebra (known as the Hochschild complex) is naturally endowed ...
3
votes
0
answers
157
views
Equivalence of deformations of non-associative algebras
Let $(\mathcal A,\mu)$ be an associative algebra. According to usual deformation theory, deformations of $(\mathcal A,\mu)$ as an associative algebra are controlled by a differential graded algebra (...
1
vote
0
answers
53
views
Generic properties of families of algebras over an infinite dimensional base space
Let $\mathbb{k}$ be an algebraically closed field and let $A$ be a $\mathbb{k}$-algebra which is a free module of rank $r$ over some central subalgebra $Z_0$. If $Z_0$ is affine and $r$ is a finite ...
- 599
4
votes
1
answer
291
views
Intrinsic formality versus rigidity of a differential graded Lie algebra
Let $\mathfrak g:=(V,d,[\cdot,\cdot])$ be a differential graded Lie algebra (DGLA) where $d$ is the zero differential.
Intrinsic formality: The DGLA $\mathfrak g$ will be said intrinsically formal ...
6
votes
0
answers
317
views
What is the technical difference between a deformation and a perturbation?
What is the technical difference between a deformation and a perturbation? Do they exist in somewhat different categories?
- 3,880
4
votes
1
answer
381
views
Classical deformation of algebras
Given a complex manifold (or a smooth scheme) $X$, the classical (infinitesimal) deformation theory is parametrized by the first cohomology with coefficients in the tangent sheaf $H^1 (X, T_X)$.
...
- 1,211
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
18
votes
1
answer
4k
views
Deformations of the punctured affine plane
Let $k$ be some field, algebraically closed and of characteristic $0$, if you like.
Let $U= \mathbb{A}^2_k \setminus \{ (0,0) \}$ be the punctured affine plane over $k$. Write $U$ as the union of $...
- 21.3k
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
31
votes
11
answers
10k
views
Introduction to deformation theory (of algebras)?
So I know that the idea of deformation theory underlies the concept of quantum groups; I haven't found any single introduction to quantum groups that makes me fully satisfied that I have some kind of ...
- 12.6k