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Results tagged with lie-algebras
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Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
5
votes
Reductive Lie algebra
As remarked by Jim Humphreys in a comment to my answer to a previous question, the notion of reductive for a Lie algebra (in characteristic zero) has no intrinsic interest, which means that the answer …
- 33.3k
3
votes
Algorithm to determine whether there is an injective homomorphism between two Lie algebras
Assuming that the question is whether there exists a Lie algebra monomorphism $\mathfrak{h} \to \mathfrak{g}$, then I am not aware of any algorithm, but with a little "native wit" (to quote one of my …
- 33.3k
11
votes
Exceptional Lie algebras
I think your first question has been answered by Jim and Richard Borcherds, so perhaps I can add the following.
As Robin points out in his answer, exceptional Lie algebras are not classical and hence …
- 33.3k
4
votes
Figure out the roots from the Dynkin diagram
There is also a succinct description of the algorithm in he very last paragraph of §11.1 in page 56 of Humphrey's Introduction to Lie algebras and representation theory.
Here's the oblgatory Google B …
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32
votes
How to interpret the Sugawara construction from a physical or mathematical viewpoint?
The origin of the Sugawara construction in Physics is, not surprisingly, the 1968 paper A field theory of currents by Hirotaka Sugawara. (There was also work of Sommerfield at about the same time.) …
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27
votes
Accepted
The Jacobi Identity for the Poisson Bracket
The Jacobi identity for the Poisson bracket does indeed follow from the fact that $d\Omega =0$.
I claim that (twice) the Jacobi identity for functions $f,g,h$ is precisely
$$d\Omega(X_f,X_g,X_h) = 0. …
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12
votes
Lie algebras and complements
Take $\mathfrak{g}$ to be the Lie algebra of the Heisenberg Lie group (i.e., the nilpotent Lie algebra of strictly upper triangular $3\times 3$ matrices) and let $\mathfrak{g}_1$ be the centre. This …
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5
votes
Accepted
how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6?
As a follow-up to Jim's answer (which came in as I was typing an inferior answer), let me add that the 7 possible embeddings are given in the $C_3$ entry of Table VI in the paper: Classification of se …
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2
votes
Radical of projection equals projection of radical?
I am not sure about infinite dimensions and/or positive characteristic, but the answer is Yes in the finite-dimensional, zero characteristic case.
In your situation we have an exact sequence of Lie a …
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3
votes
On the full reducibility of representations of reductive Lie algebras
In many applications, a (real) reductive Lie algebra arises as the Lie algebra of a compact Lie group. In this case, and if the representation integrates to one of the group, then it is fully reducib …
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5
votes
Low dimensional nilpotent Lie algebras
The paper Invariants of real low dimension Lie algebras (journal link) lists all real Lie algebras of dimension $\leq 5$ and all nilpotent of dimension $\leq 6$ along with its invariants. It also con …
- 33.3k
16
votes
Accepted
A terminology issue with the Killing form
Let me add a few comments to the answers by Mariano and Theo.
There is a one-to-one correspondence between bi-invariant metrics (of any signature) in a Lie group and ad-invariant nondegenerate symmet …
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2
votes
Permutable (Lie) subgroups
Let $i_A : A \to G$ and $i_B : B \to G$ be the embeddings and let $\mu : G \times G \to G$ be the group multiplication. Then the composition
$$
A \times B \stackrel{i_A\times i_B}{\longrightarrow} G …
- 33.3k
1
vote
Accepted
Casimir of a three dimensional solvable lie algebra
Casimirs for low-dimensional Lie algebras are given explicitly in Invariants of real low-dimensional Lie algebras by J Patera, RT Sharp, P Winternitz and H Zassenhaus.
As you point out, the Lie algeb …
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6
votes
Accepted
Decomposition of Lorentz-like groups
Provided that $n,p>0$, $O(n,p)$ has four connected components as well. There are many ways to see this. $O(n,p)$ is a matrix subgroup of the general linear group of $\mathbb{R}^{n+p}$:
$$ O(n,p) = \ …
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