# Questions tagged [lie-algebra]

A vector space $\mathfrak{g}$ over some field $F$ and kitted with a bilinear, antisymmetric and Jacobi-identity-fulfilling product ("Lie Bracket" or "commutator"). In physics, most often arises as the Lie algebra (tangent space to the identity) of a Lie group; in gauge theories, basis vectors of the gauge group's Lie algebra correspond to Noether currents and conserved quantities.

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### Weight $h$ as a function of the central charge $c$ in a Unitary Highest Weight Irrep of the Virasoro algebra

A highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by $(c,h)$, with $c$ the central charge and $h$ the "Witt weight" (the weight of the Witt algebra part of ...
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### Fierz identity for symplectic group

For the fundamental representation of $SU(N)$, there is a Fierz identity: $$\sum_iT^i_{ab}T^i_{cd}=\frac{1}{2}\left(\delta_{ad}\delta_{bc}-\frac{1}{N}\delta_{ab}\delta_{cd}\right)$$ where $T^i$ is ...
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### Why are all transformations of quantum operators inner automorphisms?

Operators in quantum mechanics are basically related to each other through their Lie algebra i.e. the commutator $\times \frac{1}{i\hbar}$. This is then connected to the state space i.e. the Hilbert ...
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### Is there any feature which distinguishes the Hamiltonian in the Poincare algebra?

The Poincare algebra is defined as \begin{align*} i[J^{\mu\nu},J^{\rho\sigma}]&=\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}+\eta^{\sigma\nu}J^{\rho\mu}-\eta^{\sigma\mu}J^{\rho\nu}\\ ...
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### Can we parameterise $SU(3)$ in such a way that there are clearly 2 parameters corresponding to the cartan torus?

We can parameterise the lie algebra of $SU(3)$ using the Gell-Mann matrices, so that a general element of LA is $\theta_i T_i$, where $T_i=\lambda_i/2$ and $\lambda_i$ are the Gell-Mann matrices. ...
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### Bosonic commutation relations for force carriers?

Why are force carriers bosons? The easiest answer that I can give myself is that the gauge field $A_\mu$ is introduced like this: $$\partial_\mu \rightarrow D_\mu = \partial_\mu+ieA_\mu,$$ so it ...
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### Poincaré invariance of linearly polarised plane wave

I am reading a book that just quotes the Lie group generators and the discrete subgroups that leave a linearly polarised plane wave unchanged. And I have no idea how to derive them. Context The ...
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### There is a classification of matrix representations of Euclidean group $\mathrm{E}(n)$ (also called $\mathrm{ISO}(n)$)?

I'm wondering if someone has already done a classification of matrix representations of the Euclidean group $\mathrm{E}(n)$, more specifically I'm trying to classify $\mathrm{E}(2)$. I watched a ...
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### Commutation relation Lorentz Algebra

Related question, which I don't understand either. I think is easier to get the Lorentz group algebra as defined by Maggiore,  [J^{\mu\nu},J^{\rho\sigma}] = i(\eta^{\nu\rho}J^{\mu\sigma} - \eta^{\...
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### Representations of the rotation group

(I have already done a similar question, but I did not express myself very well and the question was a bit confusing, so let me try again. If you find the question repetitive, I apologize and you can ...
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### Is it always possible to move to the “Cartan Gauge”?

Forgive me for potentially coming up with a new name for what I am about to describe. Let's say we have a scalar field $\phi^a$ which transforms with respect to the adjoint representation of some Lie ...
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### Hypercharge normalization for $SU(5)$ GUT

Reading about $SU(5)$ unification, texts says that they use the renormalization factor $\sqrt{3/5}$ for weak hypercharges in order to embed SM into a $SU(5)$ group. This implies a new $U(1)_Y$ ...
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### Connection between Classical and Quantum symmetries

I am an advanced undergraduate student.I would like to know about the importance of symmetry in classical and quantum mechanics.Also a good book concerning the connection between symmetries of ...
I am currently doing a math research project on the representation theory of Cherednik (double affine Hecke) algebras, specifically the algebra $\mathcal{H}_{t,c}(\mathfrak{S}_n,\mathfrak{h})$, which ...
I am following Classical and Quantum Mechanics via Lie Algebras by Neumaier and Westra. Setup I am stuck at part of Thm 2.3.1. Consider the matrix group $\mathbb{G}$. The set of $\mathbb{G}$-motions ...