# 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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### Torsion form and exterior covariant derivative

The torsion form can be defined as the exterior covariant derivative of a solder form, $\Theta=d_\omega\theta$. This derivative is always in the fundamental representation of the algebra $\mathfrak g$ ...
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### Why use $SU(3)$ and not $SL(3, \mathbb{R})$ for color charge? [duplicate]

Why do we use the group $SU(3)$ and not $SL(3, \mathbb{R})$ for color charge? As far as I can tell, the $SL(3, \mathbb{R})$ is volume and orientation preserving, by the fact that it has unit ...
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### Regarding notation used for infintesimal parameters of the Lorentz algebra and generators of the Lorentz group

I have a confusion regarding the notation that is used for infintesimal Lorentz transformations and the parameters that define the Lorentz transformation (used in various books such as Srednicki's and ...
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### Levi-Civita tensor and the Lorentz group generators in the vector representation

In the vector representation of the Lorentz group its generators are given by - $$(J^{\mu\nu})_{\alpha\beta} = i(\delta^\mu_\alpha\delta^\nu_\beta-\delta^\mu_\beta\delta^\nu_\alpha)$$ It can be ...
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### Why the full conformal symmetry is $Vir\otimes \overline{Vir}$ instead of $Vir\oplus \overline{Vir}$

In 2D CFT, we have the Virasoro generators $L_m$ and the generators $\bar L_m$, which are such that $[L_m,\bar L_n]=0$. Hence I thought that the full conformal algebra was $Vir\oplus \overline{Vir}$. ...
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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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### Generators of conformal transformations change of basis

I recently started going through Introduction to Conformal Field Theory by Blumenhagen and Plauschinn ( springer link ). On page 11, they glue together the generators of conformal transformations as ...
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### $\mathrm{SU}(2)$ as a representation of the rotation group

I have read in a book that the group $\mathrm{SU}(2)$ is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators $J_{1}, J_{2}$ and ...
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### How to prove $(α·σ)(β·σ) = α·β +iα×β·σ$ (where, $α$ and $β$ are 3 dimensional vectors and $σ$ represents Pauli matrices)?

I tried to evaluate the LHS first and obtained the first term of RHS easily. Then i tried to use the commutation relations of $\mathrm{SU}(2)$ group to proceed further to obtain the second term of the ...
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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 ...
I am doing a question that asks me to identify the gauge groups of a Lagrangian with the field strength tensors $$\bf{F}_{\mu \nu} = \partial_{\mu}\bf{W}_{\nu} - \partial_{\nu} \bf{W}_{\mu} - g\bf{... 3answers 2k views ### Why do we need complex representations in Grand Unified Theories? EDIT4: I think I was now able to track down where this dogma originally came from. Howard Georgi wrote in TOWARDS A GRAND UNIFIED THEORY OF FLAVOR There is a deeper reason to require ... 0answers 44 views ### Error with generators of Lorentz group (basis of Lorentz Lie algebra) [closed] Can someone help me figure out why my J_y is incorrect? :/ It's supposed to be \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & -... 2answers 79 views ### Symplectic group Sp(2N) in Srednicki's book There is a question in Mark Srednicki's Book (Problem 24.4, p.160) about Sp(2N), but I am not sure I understand the significance (application?) of this group. In that chapter, he talks about SO(N) ... 4answers 214 views ### Why in QFT what really matters is \exp(\mathfrak{so}(1,3)) instead of O(1,3)? In QFT fields are classified according to representations of the Lorentz group O(1,3). Now, most books when getting into this say that in order to understand the representations of O(1,3) we need ... 2answers 103 views ### Hypercharge normalization for SU(5) GUT Reading about SU(5) unification, texts says that they use the renormalization factor \sqrt{3/5} for weak hypercharges in order to embed SM into a SU(5) group. This implies a new U(1)_Y ... 1answer 37 views ### Product of generators in fundamental representation of SU(N) I'm trying to prove equation 25.20 in Schwartz:$$T^a T^b=\frac{1}{2N}\delta ^{ab}+\frac{1}{2}d^{abc}T^c + \frac{1}{2}if^{abc}T^c,\tag{25.20}$$where T^a are the fundamental representation ... 0answers 23 views ### Connection between Classical and Quantum symmetries I am an advanced undergraduate student.I would like to know about the importance of symmetry in classical and quantum mechanics.Also a good book concerning the connection between symmetries of ... 1answer 69 views ### Reference request for Lie algebras My future adviser just published a beautiful paper, https://arxiv.org/abs/1904.08304, and I am looking for some references/textbooks to look into the following concepts: Lie algebra (central) ... 1answer 131 views ### Quantum spin, tensor product: a long time relationship Anyone who has studied quantum mechanics know the following relation:  2 \otimes 2 = 3 \oplus 1  But how did a man woke up and said "Hell yeah, I'll use tensor product of two spin 1/2 to simulate ... 1answer 191 views ### What Lie group structure is used for infinite-dimensional Unitary Groups in Quantum Mechanics? Given an infinite-dimensional Hilbert space H, the set U(H) of all unitary operators on H forms a group, known as the unitary group. Now several subgroups of this group play an important role ... 4answers 491 views ### Why does the Lie algebra corresponding to the unitary group contain Hermitian operators? I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on: We also know that when t' = t, we must have the unitary time evolution operator U(t, t) = 1. Therefore, ... 2answers 53 views ### Lie algebra vs. position and momentum commutators Most theoretical texts on high energy physics make statements like below:$$[A_i , A_j] = i C^k_{i,j} A_k  (I suppose $\hbar$ may or may not be needed) and of course they describe this as being ...
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 ...