10
votes
2answers
180 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
0
votes
0answers
32 views

Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
2
votes
1answer
48 views

What is the explicit formula for classical r-matrices?

It is said that classical r-matrices are those satisfy the classical Yang-Baxter equation $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0$, where $r \in \mathfrak{g} \otimes \mathfrak{g}$. ...
0
votes
0answers
33 views

Root space of a Semi simple group an LVS?

A semi-simple Lie group has a Cartan Subalgebra ($H$) (CSA) -an LVS, Dual to this CSA LVS is root space($H^*$), which is set funtionals that map elements of CSA to real numbers and hence useful in ...
3
votes
1answer
82 views

Representation theory and particle physics

Are there good books which explain clearly explain the connections between modern particle physics and representation theory of groups and lie algebras?
0
votes
1answer
55 views

Vector Space of Lie Algebra

Lie algebra $ \mathfrak{g} $ for a Lie group $ \mathcal{G}$ is closed under commutation. Also, the elements of Lie Algebra form a Linear Vector Space(LVS). Firstly, when is it allowed to define an ...
1
vote
1answer
163 views

SO(2) group generator Lie Algebra

For the $2 \times 2$ orthogonal group of matrices which for the $SO(2)$ group, there is only one free parameter in the group element and hence only one generator for the group. Which is, $$ X_g = ...
2
votes
1answer
124 views

Generators of Translation - Lie Algebra [duplicate]

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
2
votes
0answers
77 views

Status of a question from Freeman Dyson's 1972 article

In a famous article, Freeman Dyson mentions an interesting relationship between the $\tau$ functions of number theory and the dimensions of finite-dimensional simple Lie algebras (section 2). He ...
4
votes
1answer
89 views

What does this dynamic system represent for?

I know systems like $$\frac{dx}{dt}=Sx$$ where $S$ is a symmetric matrix admit a solution that dialates along eigendirection of $S$. And systems like $$\frac{dx}{dt}=Ax$$ where $A$ is a ...
2
votes
1answer
288 views

Lie algebras and physics

I often hear physicists talk about Lie algebras and their representation theory, but most of the time hardly understand them because my knowledge of physics is very limited. Does anyone know of any ...
4
votes
0answers
149 views

How to write $SO(2n)$ characters in terms of rotation angles?

Say one is working in a representation of $SO(2n)$ such that it has the highest weights $(h_1,...,h_n)$. And let $\{H_i\}_{i=1}^{n}$ be a basis in the Cartan of $so(2n) = Lie(SO(2n))$. Now one says ...
7
votes
1answer
203 views

What are spinors mathematically?

In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing. There are essentially two frameworks for viewing the notion of a ...
5
votes
1answer
395 views

What are defining & fundamental representations?

In physics terminology, one hears of the fundamental & defining representations of lie algebras or groups - are these the same as irreducible representations?
0
votes
1answer
38 views

relation between the Poincaré and Euclidean algebra

Take $d$ a strictly positive integer, and consider the (proper) Euclidean group $E^d$ (the symmetry group of $\mathbf{R}^d$ with the conventional inner product), and the (proper, ortochronous) ...
2
votes
1answer
142 views

Conjugate Representations of Lie Algebra of Lorentz Group

I'm trying to understand the Lie algebra of the Lorentz group and am almost there, but am stuck at the final hurdle! It's easy to prove that $$\frak ...
1
vote
1answer
267 views

Lie Algebra of the Lorentz Group $SO(1,3)^{\uparrow}$

I'm trying to get my head around the Lie algebra of the Lorentz group once and for all, but have got tied up in knots. Where is my error in the following? The universal covering group of the Lorentz ...
2
votes
1answer
74 views

Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
6
votes
2answers
282 views

Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
2
votes
1answer
319 views

Finding All Irreducible Representations of $SO(3)$

I've read that one may prove that all irreducible representations of $SO(3)$ are tensor product representations of the fundamental representation (or tensor product representations of the spin 1/2 ...
5
votes
1answer
658 views

Some questions about representations of $SO(6)$

I would like to know the proof/explanation for the following three properties of the representation of $SO(6)$, What is the importance of symmetric traceless tensors of arbitrary rank w.r.t $SO(6)$ ...
4
votes
1answer
418 views

Fundamental and the anti-fundamental representation of $U(n)$

I guess that conventionally one thinks of the fundamental representation and the anti-fundamental representation of $U(n)$ as the complex $n-$dimensional representation and its complex conjugate. ...
1
vote
1answer
57 views

Question about Lie superalgebra.

What are the generators and relations for the Lie superalgebra $\mathfrak{psu}(2, 2 | 4)$? Thank you very much.
2
votes
0answers
196 views

Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...