# 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.

607 questions
Filter by
Sorted by
Tagged with
143 views

### Construct an SO(3) rotation inside the two SU(2) fundamental rotations

We know that two SU(2) fundamentals have multiplication decompositions, such that $$2 \otimes 2= 1 \oplus 3.$$ In particular, the 3 has a vector representation of SO(3). While the 1 is the trivial ...
320 views

### Relation between projective representations, connectivity of a group manifold and number of equivalence classes of paths

The projective unitary representations of a multiply-connected group $G$ is defined as $$U(g_1)U(g_2)=c(g_1,g_2)U(g_1g_2)$$ where $c(g_1,g_2)$ is phase. Reading various articles, and this old post of ...
40 views

### 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}\\ ...
113 views

87 views

### Infinitesimal generator flux of Lorentz trasformations in spacetime

I'm considering the following matrixs which I know that they form a flux of Lorentz trasformation in spacetime. I want to know how to calculate the infinitesimal generator of this flux. ...
780 views

### Casimir Operators and the Poincare Group

Following along in QFT (Kaku) he introduces the Casimir Operators (Momentum squared and Pauli-Lubanski) and claims that the eigenvalues of the operators characterize the irreducible representations of ...
589 views

58 views

### Evaluating $n \otimes_A n^*$ in $SU(n)$

In "Quantum Field Theory in a Nutshell" pg424 the author (Zee) writes: $$(n\oplus n^*)\otimes_A(n \oplus n^*)\quad\cong\quad(n^2-1)\oplus 1 \oplus n(n-1)/2 \oplus ((n(n-1))/2)^*$$ From what I ...
52 views

529 views

### Infinitesimal transformation

I came across this statement in the book "Quantum Field theory and the Standard Model" by Schwartz. "We would now like to find all the representations of the Lorentz group. The Lorentz group ...
700 views

104 views

### What are the matrix representations of super Poincaré algebras?

I have seen that Lie superalgebras are classified by some algebras like $\mathfrak{osp}(m|2n)$, but I don't know how to fit super Poincar\'e algebras into this. Especially what are the fundamental ...
### How to derive an $E_8$ algebra?
What is the simplest way to derive an $E_8$ algebra? I am not interested in $E_8$ itself but what would compel one to think about it. I know for example why you would want to think about $SU(2)$ and ...
### The Lie algebra of the Lorentz group is $su(2) \oplus su(2)$. Is there a similar relation for the algebra of the Poincare group?
It can be shown easily, by introducing new generators from the usual ones that we can think of the Lie algebra of the Lorentz group as being built up by two copies of the $SU(2)$ Lie algebra:  \...