# Tagged Questions

For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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### $\operatorname{span}\{AB-BA : A, B \in M_n(\Bbb R)\}$ is the set of all matrices with trace $0$

Let $W$ be the space $$\operatorname{span}\{AB-BA\} ,$$ where A and B are square matrices, and let $H$ be the space of all square matrices of trace $0$. Then prove that $W=H$. The fact that $W$ is a ...
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### Dimension of maximal tori

Let $G$ be a compact Lie group. $T$ $-$ its maximal torus. Is there a simple reasoning to show that dimensions of $T$ and $G$ have the same parity? I am sorry if this quesion is for children, but ...
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### Is the exponential map of GL(n,C) holomorphic?

Let $GL(n, \mathbb{C})$ be the complex general linear (Lie) group consisting of all invertible complex $n\times n$ matrices, and $gl(n,\mathbb{C})\cong C^{n^2}$ be its Lie algebra. The exponential map ...
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### Existence of product $\ast$ in a Lie algebra so that $[X,Y]=X*Y-Y*X$

I've been studying particle physics, and studying Lie algebra using physics text book doesn't give me enough information, so I'm asking my question here. Given a Lie algeba $\mathcal{A}$ where ...
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### How to find the Lie algebra of a specific subgroup of a product Lie group

My question is about finding the Lie algebra of a specific Lie group. Start with a Lie group $G$, with normal Lie subgroup $C \unlhd G$. Then define the following subgroup $\hat{G} \leq G \times G$: \...
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### Indecomposable representations of Lie algebra

Let $\mathfrak{g}$ be the nonabelian $2$-dimensional complex Lie algebra. It can be generated by two independent vectors $e_1,e_2$ such that $[e_1,e_2]=e_1$. Thus, $\mathfrak{g}$ is solvable and it ...
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### Direct sum decomposition of weight spaces and relation to Tensor products.

There are 3 parts to the question that I am trying to understand, and while it is not homework it seems instrumental in decomposition modules into weight spaces and their relation to tensor products. ...
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### Some examples for Lie algebras

I need some small examples for Lie algebras over finite fields ( GF(2) or GF(3)) including some simple Lie algebras and some others which are not simple. And I would be thankful if anyone could give ...
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### Convolution and Characters

I am confused about the purpose of the Formal Character, character functions, and the convolution in representation theory of Lie algebras. Is the Character function different than just the Character? ...
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### A criteria for a subalgebra of M(n,C) being M(n,C)

Suppose $S$ is a subalgebra of the matrix algebra $M_n(\mathbb{C})$. If for any vector $v$ and $w$ in $\mathbb{C}$, there always exists a matrix $A$ in $S$, depending on $v$ and $w$ of course, which ...
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### Isometries of S^3 and some Lie algebras

By considering $S^3$ as the group of unit quaternions, and letting it act on itself from both the left and right, one can get an isomorphism $SO(4)\cong (S^3\times S^3)/C_2$, where the $C_2$ subgroup ...
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### How can we compute a Lie bracket for powers of elements of given lie algebra?

Let $L$ be a lie algebra over finite field, for $x,y$ in $L$ I want to solve the following bracket: $[yx^k,x]=?$ How can we describe that in the format of $[...[y,x],x],...,x]=[y,x]_i$ ($i-times$)
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### Considerations for moving a function inside or outside of an integral

Excluding the possibility that $A(t)$ is the limit of a sequence, are there any special considerations I should be concerned with regarding the following assertion: Let $A(t)$ be an $n\times n$ ...
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### Complex conjugation of positive roots

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...
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### Multiple Cartan sub-algebras

How is it that for a Semi-simple Lie Algebra there is not one Cartan Sub-Algebra? I assume since there are multiple CSA's of a SS Lie algebra that must mean some of the ss elements of the Lie ...
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let $\mathsf{g}$ be a finite dimensional lie algebra and $\xi\in\mathsf{g}$. Under which conditions the adjoint map $ad_\xi :\mathsf{g}\longrightarrow \mathsf{g}$ is diagonalizable? what about ...
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### Closed Connected Subgroup of $SO(5)$

I was reading a paper in which a part of it they want to classify the closed connect subgroups of $SO(5)$. What they write is this: Let $G^0$ be a closed connected subgroup of $SO(5)$. Let $T$ be a ...
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### Don't understand Levi decomposition theorem

Levi decomposition theorem states that any finite-dimensional real Lie algebra $L$ is the semidirect product of a solvable ideal and a semisimple subalgebra. I don't understand this since to me it ...