# 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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### Relation between Lie algebra of conformal Killing vector fields and conformal algebra

I'm new to conformal transformations and I have a question. Following the book of Barrett O'Neill "Semi-Riemannian geometry with applications to relativity", there is a Lie anti-isomorphism between ...
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### $\mathfrak{su}(3)$ structure constants

The $\mathfrak{su}(3)$ structure constants $f^{abc}$ are defined by $$[T^a,T^b] = i f^{abc} T^c,$$ with $T^a$ being the generators of the group $\mathrm{SU}(3)$. They are usually written out in a very ...
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### How to prove the translation generator commutes with the spinors in SUSY algebra?

I was reading Modern Supersymmetry by John Terning, the book starts with SUSY algebra and says $$\left[ P_{\mu} , Q_{\alpha} \right] = \left[ P_{\mu} , Q_{\alpha}^{\dagger} \right] = 0$$ I am ...
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### Trace of 4 Gell-Mann matrices

Does any one know what would be $tr(t^a t^b t^c t^d)$, where $t^a$ etc are Gell-Mann matrices? This came about when analyzing the color factor for the compton effect for QCD. So, must be pretty ...
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### Unitary Representations in Conformal Field Theory

So I am currently studying conformal field theory from the perspective of the representation theory of Lie algebras. I am trying to understand exactly why we care about unitarizable Verma modules. For ...
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### Link between dynamical algebra and symmetry group

I was wondering if there is a known link between dynamical algebra and symmetry group. In particular: Do all Hamiltonians belonging to certain dynamical algebra share the same symmetry group? Do ...
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### Generators of conformal transformations

I'm currently reading about the Witt algebra, and I'm trying to understand in what sense the Witt algebra basis $L_n = -z^{n+1}\partial _z$ generates conformal maps in dimension $2$. From what I've ...
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### Noether's theorem for fields and infinitesimal transformations [duplicate]

I'm starting to learn QFT by myself using many references, mostly (QFT for the Gifted Amateur, and Tong's lectures) and both present a proof of Noether's theorem using infinitesimal tranformations, ...
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### Updating link variables in lattice $SU(N)$ gauge theory

I'm currently writing a basic program in python to simulate a 1 + 1 dimensional yang mills gauge theory with symmetry group $SU(2)$. On the lattice you work with link variables, which are $SU(N)$ ...
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### How to prove a set of matrices form a representation of Lie algebra?

When reading Paul Langacker's The Standard Model and Beyond, I am quite confused on equation 3.29, which says with a set of fields $\Phi _a$, where $a$ goes from 1 to $n$, are chosen to be transformed ...
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### For what angles (and why) does the equation for finite rotation fail to work?

I am learning rotations and group theory/representations and my lecturer's note mentioned that: "The group is considered connected, but not simply connected [...] As a result, the formula for a ...
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### Modified Lie bracket

In a paper by Barnich, they use a different definition of the Lie bracket for vector fields at null infinity. Can somebody please give me the intuition behind using this Lie bracket in Equation 4.12: ...
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### Does the sedenion algebra offer a grand unification theory?

Stephane Bronoff in The Standard Model of Particle Physics from Sedenions claims that studying the left-multiplication map of unit doubly-pure sedenions solves several mysteries of the standard model. ...
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### Time dependent Hamiltonian operator and $SU(1,1)$ generator method

In this screenshot of a paper I am reading, I have the following question: 1.What is a $SU(1,1)$ group and how do we find its generators? 2.From the expression for the Hamiltonian $\hat{H}$, how do ...
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### Integrating over a symmetry vector field by exponential map

Take some constant-of-motion, $H$, and the Poisson bracket, $\{\cdot, \cdot\}$. Then, we recover the symmetry vector field of $H$ by $$S_H = \{\cdot, H\}$$ so take for ...
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### Proposal of the Virasoro modes and algebra

Hi I am wondering what the first published paper on Virasoro modes was? And what about Virasoro algebra?
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### Basis-free, non-power series definition of the exponential of linear operator?

Given an arbitrary linear operator $A$ (be it real, complex or whatever), how can the exponential of it ($e^A$) be defined naturally, without stuff like power series? The exponential for regular ...
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### Angular momentum and Noether's theorem

Studying the lagrangian formulation of Noether's theorem and came upon how the invariance under rotations gives conservation of angular momentum. Whilst setting up the problem the notes state that if ...
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### Lie algebra valued potential vector [closed]

Maybe it is a simple question but I have some difficulty to understand the explicit matrix form of this usual relation: $$A_\mu=A^a_\mu \tau_a$$ where $A^a_\mu$ is the Lie algebra valued potential ...
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### How can the algebra of infinitesimal conformal transformations be infinite dimensional (in 2D)?

In Blumenhagen's book "Introduction to Conformal Field Theory", I found the statement The algebra of infinitesimal conformal transformations in an Euclidean 2-dimensional space is infinite ...
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### Composition of Lorentz transformations using generators and the Wigner rotation

I solved this problem by painful calculations of Lorentz matrices. However, I heard that there is a much easier solution using the generators of boosts and rotations and their commutation relations, ...
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### A question about a commutation relation

Here $a$, $b$,$c$, $d$ $u$, $v$ range from 0 to 4 and the metric $g^{ab}=\text{diag}(-1,1,1,1)$. 16 $4 \times 4$ matrices $M^{ab}$ are defined as follows: \begin{equation*} (M^{ab})_{uv} = -i(\delta^...
### Explicit Quadratic Casimir for $sp(2N)$
We know that $so(3)$ has the explicit quadratic Casimir $$L^2=\sum L_{i}^2.$$ Are there analogs to this in other simple lie algebras? I know that for a simple lie algebra I can always use the ...