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12 votes
1 answer
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Is there a way to embed Clifford algebras into the corresponding tensor algebra?

$\newcommand{\talg}{\mathcal{T}(V)}$$\newcommand{\clalg}{\mathcal{Cl}_q(V)}$$\newcommand{\qalg}{\mathcal{I}_q(V)}$Is there a way to embed Clifford algebras into the corresponding tensor algebra? There ...
Chill2Macht's user avatar
  • 2,680
12 votes
1 answer
653 views

What are the simple Lie superalgebras of type E?

Background Simple finite dimensional Lie superalgebras over $\Bbb C$ have been classified. There are the Cartan type superalgebras (algebras of purely odd vector fields), two strange families P(n) ...
AndreA's user avatar
  • 971
11 votes
2 answers
1k views

What are important examples of filtered/graded rings in physics?

Hi, what comes to the mind of a physicist, when he hears words like filtered ring and associated graded? What do these guys describe? What are basic/typical/illuminating examples in physics? Of ...
Jan Weidner's user avatar
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10 votes
0 answers
227 views

What is the meaning of the coefficients of the Alekseev-Torossian associator

Drinfeld associators became a central object in mathematics and mathematical physics. They appear in deformation quantization, quantum groups, in the proof of the formality of the little disks operad, ...
DamienC's user avatar
  • 8,385
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 ...
Jim Stasheff's user avatar
  • 3,880
5 votes
1 answer
254 views

Well defined Tensoring of spectral triples

Hi, I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it. Question: In connes standard model he takes ...
SMF's user avatar
  • 133
5 votes
1 answer
256 views

Definition of a Dirac operator

So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as: $D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...
CristinaSardon's user avatar
4 votes
1 answer
917 views

How does Constructive Quantum Field Theory work?

Please correct me if I'm wrong, but it seems to me that two and three dimensional axiomatic quantum field theory were constructed as follow: the wightman axioms were formulated in euclidean space via ...
Jimbo's user avatar
  • 235
4 votes
3 answers
1k views

Fractional Quantum Hall Effect - Mathematics

Just to include something that starts to answer my own question Topological Quantum Computation Lecture notes covers a lot of the Mathematics of the Fractional Quantum Hall effect, or topological ...
Peadar Coyle's user avatar
4 votes
1 answer
674 views

What is the relationship between the Dirac algebra and the Clifford algebra?

While I'm still trying to understand the issues raised on my previous question, I decided to first address the Clifford algebra used on formulating the famous Dirac equation. In this context, what is ...
JustWannaKnow's user avatar
4 votes
1 answer
251 views

Connected Frobenius algebras non-semisimple as an object

A Frobenius algebra object $A$ in a tensor category $\mathcal C$ is said to be connected if $\text{Hom}_{\mathcal C}(\mathbb{1}, A)$ is a one dimensional vector space, where $\mathbb {1} $ denotes the ...
Mainak's user avatar
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4 votes
0 answers
154 views

decomposition of paraunitary matrices

Let $A(X,Y)$ be a matrix with coefficients in $\mathbb{C}[X,X^{-1},Y,Y^{-1}]$, where the conjugation operation is defined on this ring by ${(z X^a Y^b)}^* =\bar{z}X^{-a}Y^{-b}$. So $A$ is said to be ...
Vincent Nesme's user avatar
3 votes
3 answers
430 views

Open symplectic embeddings and deformation quantization

I would like to know whether there is a specific relationship between the deformation quantizations of the Poisson algebras of the functions on a symplectic manifold, say $M$, and of the functions on ...
Igor Khavkine's user avatar
3 votes
3 answers
552 views

Graph of a Lie super algebra

Let $A$ be a generalized Cartan matrix and let $\mathfrak{g}$ be the Kac-Moody Lie algebra associated to $A$. There is an associated graph of $\mathfrak{g}$ which is known as the Dynkin diagram of $\...
GA316's user avatar
  • 1,269
3 votes
1 answer
277 views

What is the relation between cobar duality and Feynman transform

If $O$ is a cyclic operad, it can be regared as a modular operad $P$ with $P(g,n)=0$, for $g >0$. So we have cobar dual $BO$ and Feynman transform $FP$(with trivial cocycle). Is there any ...
Hao Yu's user avatar
  • 31
2 votes
1 answer
586 views

Gamma matrices are irreducible

For $\mu=0,1,2,3$, let $\gamma^{\mu}$ the set of Dirac gamma matrices. What does it mean to say that $\{\gamma^{\mu}: \hspace{0.1cm} \mu=0,1,2,3\}$ is irreducible? From my previous question, I know ...
JustWannaKnow's user avatar
2 votes
1 answer
438 views

Reference request (or otherwise): Adjoint action

I am interested in clearing up a confusion of mine. I will try to make my question as clear as I can but I apologize in advance if this is not the case. Given a unitary group of some unital ...
SMF's user avatar
  • 133
0 votes
1 answer
268 views

Mappings between states on *-algebras

Consider finite-dimensional (for-simplicity) $\star$-algebras, that is, unital associative algebras over the complex numbers equipped with an antilinear antiautomorphism $\star$. A state on a $\star$-...
Vanessa's user avatar
  • 1,368
0 votes
0 answers
121 views

Representation of anti-commuting matrices in $\mathbb{C}^{2}$

This is a cross posting updated question from MSE. I have not got any answers there yet and I really want to understand this problem. The basic question is the following. Let $V$ be a finite-...
MathMath's user avatar
  • 1,305
-1 votes
1 answer
159 views

Classification of real Clifford algebras

$\DeclareMathOperator\Cl{Cl}$Let $V$ be a real vector space of dimension $p+q$. Let $Q$ be a non-degenerate quadratic form on $V$ of signature $(p,q)$ where $p$ is the number of positive eigenvalues, ...
asv's user avatar
  • 21.8k