Tagged Questions

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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 ...
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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 ...
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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?
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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 ...
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How to decompose a representation of $so(n)$ into representations of a subalgebra

In some cases, it is possible. For instance the representation $16$ of $so(9)$ decomposes as $8_c+8_s$ of $so(8)$. Now I would like to do the same with representations of $so(8)$ into a sum of ...
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$\mathbb{RP}^3$ is homeomorphic to the solid ball with antipodal points identified

I am reading the book Application of Path integrals by Schulman, which has a chapter on applications of homotopy theory to path integrals. In that he says we can geometrically describe $SO(3)$ by a ...
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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 ...
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Quotients of $S^{2n+1}$

Any sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$ Now say one wants to quotient it ...
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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 ...
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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?
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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) ...
Can anyone help me with the following? Let $\rho$ be the right-invariant Maurer-Cartan 1-form $$\rho = dg\ g^{-1}$$ I want to show that the MC equation $$d\rho - \rho \wedge\rho = 0$$ holds. So ...