# 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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### Infinitesimal Poincare transformations , Taylor expansion

Let $(\Lambda,a)\in\text{ ISO}_o(3,1)$ be a finite (proper) Poincare transformation and Let $U(\Lambda,b)$ be the corresponding unitary operator implementing this transformation on the Hilbert space ...
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### Cartan Killing metric and Casimir operators

I'm a little confused about Casimir operators and Cartan-Killing metric. The Lorentz group is a semi-simple group and its Cartan-Killing metric is non-degenerate, say $g_{ab}$; it is invertible and ...
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### Reference recommendation for Projective representation, group cohomology, Schur's multiplier and central extension

Recently I read the chapter 2 of Weinberg's QFT vol1. I learned that in QM we need to study the projective representation of symmetry group instead of representation. It says that a Lie group can have ...
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### How to show that an $N$-dimensional SHO's dynamics symmetry is $SU(N)$?

From Wikipedia: The dynamical symmetry group of the $n$-dimensional quantum harmonic oscillator is the special unitary group $SU(n)$. As an example, the number of infinitesimal generators of ...
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### Interpretation of Dynkin diagrams

I am having trouble in understanding the physics represented by dynkin diagrams. Say I have the following diagram: What is the difference between the square nodes and circular nodes? What does the ...
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### Invariance under boosts but not rotations?

I am aware that there are 6 independent infinitesimal Lorentz transformations that can be separated into 3 rotations and 3 boosts. Is it possible for a quantum field theory to be invariant under the ...
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### What is “broken symmetry”?

For reference, I come from a mathematics background (mostly differential geometry). I have a very limited understanding of upper-level physics (I'm currently trying to fix this). This is my current ...
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### Where in fundamental physics are Lie groups actually important (and not just Lie algebras)?

I was wondering where in fundamental physics the global structure of a Lie group actually makes a difference. Most of the time physicists are sloppy and don't distinguish groups and algebras ...
### How is the invariant speed of light encoded in $SL(2, \mathbb C)$?
In quantum field theory, we use the universal cover of the Lorentz group: $SL(2, \mathbb C)$, instead of $SO(3,1)$. The reason for this is, of course, that $SO(3,1)$ representations aren't able to ...
### Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$?
The number of generators of Lie algebra $sl(2,\mathbb{C})$ is 6, and $sl(2,\mathbb{R})$ has 3 generators, Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$? Say \...