All Questions
Tagged with qa.quantum-algebra deformation-theory
22 questions
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 ...
5
votes
1
answer
525
views
Motivating quantum groups from knot invariants
Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\...
- 2,283
1
vote
1
answer
158
views
Is the map in Kontsevich Formality Theorem $\mathcal{O}$-linear?
$X$ is smooth Poisson. Kontsevich formality theorem says that there is a $L_\infty$ quasi-isomorphism $$T_{\text{poly}}\xrightarrow{L_\infty}D_{\text{poly}},$$ where $T_{\text{poly}}:=(\bigwedge^\...
- 367
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
5
votes
0
answers
198
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Analogue of Kontsevich's formality theorem for quantization of Courant algebroids
In his 1997 preprint, M. Kontsevich proved the formality of the differential graded algebra controlling deformations of the associative and commutative algebra of functions on a manifold, seen as an ...
1
vote
1
answer
199
views
How to compute the deformation quantizations of a polynomial Poisson algebra?
As a specialist told me, given a smooth affine Poisson algebra $S$ over $\mathbb{C}$, up to a choice of certain characteristic class, one can find one of the deformation quantizations of $S$, say $A$, ...
- 57
4
votes
1
answer
423
views
Formality of the little $n$-disks operad and deformation theory
In [Another proof of M. Kontsevich formality theorem], Tamarkin provides a proof of the formality of the differential graded Lie algebra controlling the deformation of a polynomial associative algebra....
3
votes
1
answer
365
views
Brace algebra structure on the Hochschild complex of an associative algebra
As shown by Gerstenhaber and Voronov [Higher operations on the Hochschild complex], the Hochschild complex of an associative algebra is endowed with a natural structure of brace algebra. The first ...
4
votes
0
answers
220
views
What does "control of a deformation problem" mean?
Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...
- 3,880
8
votes
0
answers
463
views
On the cohomology of Kontsevich graph complex
Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...
- 1,609
4
votes
1
answer
147
views
Equivalence of star products on two differents Poisson algebras?
Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...
- 53
7
votes
2
answers
688
views
Alternative to Kontsevich formality
Has anyone considered an alternative approach to Kontsevich formality in which the DGLA of poly-vector fields is deformed to an $L_\infty$-algebra?
Some vocabulary:
DGLA = Differential Graded Lie ...
- 3,880
7
votes
1
answer
1k
views
What's the relation between the heat kernel proof of the index theorem and deformation quantization?
In the book "Heat kernels and Dirac operators", I found the slogan by Quillen in the Introduction: "Dirac operators are a quantization of the theory of connections, and the supertrace of the heat ...
- 9,019
5
votes
4
answers
532
views
Non-Drinfeld–Jimbo deformations and finite quantum groups
As is well-known, for the compact semi-simple Lie groups $G$, there exist non-commutative Hopf algebra deformations ${\cal O}_q[G]$ of their coordinate algebras ${\cal O}[G]$, the so-called Drinfeld--...
- 263
3
votes
1
answer
193
views
Koszul algebras deformations
Do we know the maximal class of Koszul algebras for which any deformation is Koszul?
- 3,880
1
vote
1
answer
119
views
Zero Sums in a $q$-Deformation Remain Zero for $q=1$
Looking at previous question, I begun to think and came upon the following more solid question: Let $SU_q(N)$ be the usual quantized coordinated algebra of $SU(N)$. Now consider, for some fixed value ...
- 389
0
votes
1
answer
126
views
Deformations and Dimensions: $q$-Deform Finite to Infinite?
Let $A$ be a (complex) (finitely generated) algebra with some type of $q$-deformation $A_q$, where $A_1 = A$. Moreover, let $V_q$ be a vector subspace of $A_q$, such that $V_1$ is a finite-...
- 1,479
7
votes
0
answers
182
views
Deformation of Noether's first theorem
Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...
- 3,880
10
votes
3
answers
2k
views
In the dictionary between Poisson and Quantum, what corresponds to Coisotropic?
I work entirely over a field of characteristic $0$, in case it matters.
Recall that a Poisson algebra is a commutative algebra $A$ with a bracket $\lbrace,\rbrace : A^{\wedge 2} \to A$ which is (1) a ...
- 54.6k
11
votes
1
answer
1k
views
Kontsevich's formality theorem from an explicit homotopy
Suppose that $X$ is a smooth manifold, whose $C^{\infty}$-functions we denote by $A$. Let $D_{poly}^*(A):=\bigoplus_{n\geq -1}Hom(A^{\otimes n+1},A)$ be the Lie algebra of polydifferential operators ...
- 386
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
18
votes
2
answers
2k
views
Why do my quantum group books avoid homotopical language?
I am sitting on my carpet surrounded by books about quantum groups, and the only categorical concept they discuss are the representation categories of quantum groups.
Many notes closer to "Kontsevich ...
- 904