All Questions
Tagged with ra.rings-and-algebras mp.mathematical-physics
20 questions
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 ...
- 43
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-...
- 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, ...
- 21.8k
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 ...
- 3,515
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 ...
- 3,515
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, ...
- 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 ...
- 3,880
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 ...
- 327
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 $\...
- 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 ...
- 31
12
votes
1
answer
3k
views
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 ...
- 2,680
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 ...
- 235
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) ...
- 971
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 ...
- 133
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 ...
- 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 $\star$-...
- 1,368
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 ...
- 21.5k
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 ...
- 379
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 ...
- 111
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 ...
- 13.2k