Questions tagged [lie-algebras]
For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.
6,841
questions
-3
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11
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Epimorphism of Poisson algebras
Let P be a Poisson algebra over K. How can we construct an epimorphism $\phi: P \to K$?
- 1,310
0
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31
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Lie bialgebras and the Jacoci identity
Let $\mathfrak{g}$ be an $n$-dimensional real Lie algebra and let $\xi : \mathfrak{g}\to\wedge^2\mathfrak{g}$ be a linear map which is a $1$-cocycle, i.e. $\xi([x,y])=\mathrm{ad}_x\xi(y)-\mathrm{ad}_y\...
- 229
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38
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Mapping between vectors in irreducible Sp-representation
Let $V$ be the standard Sp-representation with symplectic basis $\{ a_i, b_i \}$. I believe the vector $(b_1 \wedge b_2) \otimes (b_1 \wedge b_2 \wedge a_3 \wedge a_4)$ lies inside the irreducible ...
- 326
2
votes
1
answer
44
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Does the formal character determine the representation?
Suppose $V,W$ are two finite-dimensional representations of a Lie algebra $\mathfrak{g}$.
Is it true that if their formal characters coincide, $$\mathrm{ch}_V=\mathrm{ch}_W ,$$ then the ...
- 1,562
1
vote
1
answer
65
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"Linear independency" of Lie Brackets
I was watching this eigenchris video. At 21:49, he says:
$$[g_i, g_j]=\sum_k {f_{ij}}^{k}g_k$$
for $\mathfrak{so}(3)$.
Does this mean $[g_i, g_j]$ and $g_i, g_j$ can be linear independent? What about ...
- 113
0
votes
0
answers
23
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Relation between the enveloping algebra $\mathcal{U}(\mathfrak{g})$ and the group von Neumann algebra $W^*(G)$
Let $G$ be a Lie group. Is it true that the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ of the associated Lie algebra $\mathfrak{g}$ generates the group von Neumann algebra $W^*(G) := \...
- 101
0
votes
0
answers
31
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Complexification of complex Lie algebras like $\mathfrak{su}(2)$
I'm reading Brian Hall's book on Lie theory.
He defines the complexification $V_{\mathbb{C}}$ of a real vector space $V$ as the linear combinations $v_1+iv_2$, with $v_1,v_2\in V$. Next, he proceeds ...
1
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0
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46
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Velocity vector vs Matrix Differential
I am having trouble understanding the equivalence of taking the derivative of the matrix and taking the velocity vector. I came across a proof of the Lie Algebra of $O(n)$ as follows:
Let $\gamma(t)$ ...
- 59
1
vote
1
answer
40
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uniqueness of generators of Lie groups
Are the generators of any particular Lie group always unique? Let's take $SU(3)$ group as an example. It does have 8 generators which are explicitly written as eight $3\times3$ matrices in the ...
- 111
0
votes
1
answer
23
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Lemma in proof of Cartan's Criterion (Humphreys' 4.3)
In Humphreys proof of Cartan's criterion there is a step I do not understand: $\text{ad y}=r(\text{ad s})$.
I tried to check this for $\{e_{ij}\}$:
$$
r(\text{ad s})(e_{ij}) \stackrel{r(0)=0}{=} r(\...
- 117
0
votes
0
answers
18
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Antisymmetric Structure Constants, $f_{ijk}$ of su(N) for generalised Gell-man/Pauli Matrices, $k$ is unique for a given $i,j$
I want to prove that for a fixed $(i,j)$ there exists only a single $k$ such that $f_{ijk} \neq 0$.
I did this by considering the Generalized Pauli matrices:
$$
\hat{T}_{\alpha_{nm}}=\frac{\hbar}{2}(|...
2
votes
1
answer
53
views
Real Commutant Algebra of a Set of Matrices
Suppose I have a collection of $N\times N$ real, symmetric matrices $R_1, R_2, \dots$ and I want to find their orthogonal commutant---that is, the group of real, orthogonal matrices that commute with ...
1
vote
0
answers
69
views
Simple Lie subalgebra of a semisimple Lie algebra
For a semisimple Lie algebra with a nondegenrate trace form, it's well-known that it can be decomposed as the direct sum of simple Lie ideals, and hence has a simple Lie ideal. But in general, how to ...
0
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0
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31
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Invariant Solutions of PDEs-Linear fokker-planck equation with an odd drift
I am currently writing my master thesis on Lie group analysis and recently I came across this infinitesimal generator:
I am trying to obtain group invariant solutions in their implicit form and so ...
1
vote
2
answers
91
views
The action of $SL(n,\Bbb{R})$ on the tangent space of $SL(n,\Bbb{R})/SO(n)$.
SETUP. It is a standard result that $\text{GL}(n,\Bbb{R})/O(n)$ is isomorphic to the set $P'$ of positive definite $n\times n$ matrices, as manifolds: the basic idea is that $\text{GL}(n,\Bbb{R})$ ...
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