# 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.

1,445 questions
504 views

### Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
536 views

### Abelian Cartan subalgebras

If a Lie algebra is semisimple or reductive, its Cartan subalgebras are Abelian, and their elements semisimple. Are there non-reductive algebras with Abelian Cartan subalgebras all of whose ...
358 views

### Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
235 views

### Why are parabolic subgroups called “parabolic” subgroups?

I used to think that things called "parabolic" must have something to do with parabolas or their defining quadratic equations. In fact, terms like parabolic coordinate, parabolic partial differential ...
205 views

338 views

### Non-degenerate bilinear forms of Lie algebra with a degenerate Killing form

Definition: A Lie algebra is defined by: $$[e_a,e_b]={f_{ab}}^ce_c$$ The Killing form is $$g_{ab}=-{f_{ac}}^d {f_{bd}}^c$$ Set-Up: The type of Lie algebra of our interests (found out during a ...
181 views

### Application of SU(2) in physics

How can we interpret the representations of SU(2) and $\mathfrak{su}(2)$ in physics? I have studied a lot of mathematics including representation theory and differential geometry, so I understand SU(...
203 views

### Generators of so(7)

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is isomorphic to $\mathfrak{so}(7)$. Can ...
192 views

### A subspace is invariant by the Lie group if it is invariant by the Lie algebra

Let $G$ be a connected Lie group and $$\varphi:G\to \mathrm{GL}(V)$$ a representation on a finite dimensional real vector space $V$. Let $$\psi:\mathfrak{g}\to\mathrm{End}(V)$$ be the associated Lie ...
386 views

### Coproduct of Lie algebras

Fix a commutative ring $k$ and look at the category of Lie algebras over $k$. How do coproducts in that category look like? Notice that what is usually called the "direct sum" of Lie algebras is not ...
139 views

290 views

### Irreducible representations of $\mathfrak{sl}_3\mathbb{C}$

I am working through the exercises in Fulton and Harris's Representation Theory, and am stuck on two on page 189. Let $\text{Sym}^2V$ denote the second symmetric power of the standard 3-dimensional ...
439 views

131 views

### When are the roots of a Lie algebra the differences of the weights?

In books about group theory written for physicists, there's a strange procedure used to find the roots of a Lie algebra. The steps are: Write down the fundamental representation. This is the '...
151 views

### Beilinson-Bernstein localization, equivariant modules

I have a question regarding the equivariance in the Beilinson-Bernstein localization. Let $G$ be an simply connected algebraic group over a field of charateristic $0$ and $K$ a closed subgroup of $G$ ...
339 views

### Direct sum of injective modules is injective.

By the Bass-Papp Theorem, for a unital ring $R$, any direct sum of injective left $R$-modules is injective if and only if $R$ is left Noetherian. I would like to restrict my consideration to an ...
106 views

### Is there a relationship between the trace and the Clifford/geometric product?

In what follows, let $V=\mathbb{R}^n$ (although the following probably applies also to a larger number of finite-dimensional spaces). We assume throughout that we have made a choice for an inner ...
96 views

### Real representations of Lie groups

I am looking for a reference that discusses the representation theory of Lie groups, especially the classical groups, on real vector spaces. It seems that almost every text I've looked at restricts ...
309 views

### Orthosymplectic Lie Superalgebra

I am trying to work out a presentation for the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2n)$. I am following Musson's book "Lie Superalgebras and Enveloping Algebras". From what I ...