2
votes
1answer
82 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 ...
6
votes
1answer
199 views

alternative to Kontsevich formality

Has anyone considered an alternative approach to Kontsevich formality in which the DGCA of poly-vector fields is deformed to an $L_\infty$-algebra?
6
votes
1answer
280 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 ...
4
votes
4answers
288 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 ...
1
vote
1answer
110 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 ...
0
votes
1answer
110 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 ...
7
votes
3answers
782 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 ...
10
votes
1answer
806 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 ...
4
votes
3answers
1k 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 $M$ ...
16
votes
2answers
1k 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 ...