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Results tagged with deformation-theory
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user 12482
for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.
18
votes
Clifford algebras as deformations of exterior algebras
In addition to the answer of Bertram Arnold, let me point out that there is a very explicit formula for "fermionic" Weyl-Moyal product. Let us assume that your vector space $V$ (or module) is defined …
- 8,098
14
votes
Deformation Quantization
Unfortunately, there is no real textbook on DQ around. One has Fedosov's book on his construction of star products including a detailed exposition of his index theorem.
There is a chapter on DQ in t …
- 8,098
8
votes
Accepted
Some elementary questions about deformation quantization
a lot of questions, let me try on some of them :)
The bad news is that in most of the interesting situations the higher order terms of the star product, the $B_i$ will not vanish. Heuristically this …
- 8,098
8
votes
Accepted
In the dictionary between Poisson and Quantum, what corresponds to Coisotropic?
Concerning your first question I have a couple of suggestions: first coisotropic is in some sense the best we can have in a truely Poisson situation: there is nothing like lagrangian (unfortunately).
…
- 8,098
6
votes
Accepted
Formal series convergence in deformation quantization and $C^*$-condition
OK, let me give a try on this question. There are several problems hidden underneath which one has to address.
First, for physical reasons a formal deformation is not sufficient. $\hbar$ is a constan …
- 8,098
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 $M$ i …
- 8,098
3
votes
graded generalization of the Moyal–Weyl product
Yes, it's just putting signs correctly. Martin Bordemann has a preprint from the 90s where he adapted Fedosov's construction in the graded setting. If you are only interested in the flat situation thi …
- 8,098
3
votes
Equivalence of star products on two differents Poisson algebras?
to 1) A $\mathbb{k}[[\hbar]]$-linear map between $A[[\hbar]]$ and $B[[\hbar]]$ is necessarily of the form $T = T_0 + \hbar T_1 + \cdots$ with $T_r\colon A \longrightarrow B$ being $\mathbb{k}$-linear …
- 8,098
3
votes
Reverse Engineering to find deformation problem (from cohomology groups)?
In this generality, I would say that this is not possible: the same cohomology can be responsible for controlling quite different deformations problems.
Just an example: in formal deformation quantiz …
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3
votes
Open problems in deformation theory
Deformation theory is of course a very very wide field and one can take many different points of view on it. Working in deformation quantization, i.e. formal associative deformations of algebras and t …
2
votes
Tamarkin-Tsygan Formalism
Well, this is probably not the deep insight you are looking for, but if you consider the polyvector fields on a manifold, you can first define the Lie derivative $L_X$ of a polyvector field $X$ on dif …
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2
votes
Is the algebra of sections of a bundle of complex Clifford algebra over an oriented Riemanni...
Let's try this: assume that the complex vector bundle you start with has even fiber dimension such that the corresponding Clifford algebra bundle is a bundle of complex matrix algebras. Suppose furthe …
- 8,098
1
vote
How to compute the deformation quantizations of a polynomial Poisson algebra?
Not a complete answer but some observations. First it might be necessary not to take formal power series but formal Laurent series in order to get a reasonable behaviour of the Hochschild cohomology. …
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