# Tagged Questions

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### a question about definition m-tuples [on hold]

Let $m>0$ and we have an $m$-tuple of natural numbers. For example we know that $\varepsilon_2=(0,1,0,\ldots,0)$. Now question is that : Does it make sense for $\varepsilon_0$ ?
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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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### Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
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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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### 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 ...
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Suppose $V=V_0\oplus V_1$ be a $Z_2$-graded semi-simple lie algebra and, $\xi\in V_1$. The maps $ad_\xi \circ ad_\xi :V_0\longrightarrow V_0$ and $ad_\xi \circ ad_\xi :V_1\longrightarrow V_1$ are ...
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### Convergence of Baker-Cambpbell-Hausdorf for compact groups

It is well known that the Baker-Campbell-Hausdorf formula doesn't need to converge for general elements of a Lie algebra, resp. for matrices with norms larger then 1. On the other side, if $G$ is a ...
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### killing form and the dot product

When going from talking about roots as functionals to talking about roots as vectors in a Euclidian space (root system), does the killing form become the dot product? Are the killing form and dot ...
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### Killing form and Roots

I know that the roots of a Lie Algebra are functionals such that if $\alpha$ is a root and $h \in \mathfrak h$ is an element of the Cartan subalgebra, then $\alpha(h)$ is an eigenvalue. I'm looking ...
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### Root space decomposition

Regarding the direct sum of vector spaces/algebras, the dimensions of the parts of the sum should equal the whole. With the root decomp, the cartan sub algebra seems to have a dimension as high as the ...
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### Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to$End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ ...
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### An extension of an algebraic question from my test

Let $A$ and $B$ be two real $n\times n$ matrices s.t. $AB=BA$. We now that $\det(A^2+B^2) \geq 0$. Is the similar question true for $n$ matrices which commute with each other? If not, how do I ...
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### Invariant tensors in adjoint representation

Suppose we have a simple Lie group $G$ with algebra $\mathfrak{g}=\{X_a\}$, where the generators $X_a$ are in some matrix representation. Is it true that the only invariant rank $n$ tensor in the ...
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### Derived series of a Lie algebra

I've been studying semisimple Lie algebras and solvability and was wondering if someone could explain to me the meaning of the derived series of a Lie algebra L and this part: $$L^{(1)}=[LL]$$ I don't ...
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### Show that $\exp: \mathfrak{sl}(n,\mathbb R)\to \operatorname{SL}(n,\mathbb R)$ is not surjective

It is well known that for $n=2$, this holds. The polar decomposition provides the topology of $\operatorname{SL}(n,\mathbb R)$ as the product of symmetric matrices and orthogonal matrices, which can ...
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### Is algebraic closure required in Weyl's theorem on complete reducibility? (Lie algebras)

Weyl's theorem states that finite-dimensional representations of finite dimensional semisimple Lie algebras are completely reducible (expressible as a direct sum of irreducible submodules), with some ...
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### Why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?

Let $G=SO(n,R)$ be a Lie group and $\mathbb{g}$ its lie algebra. Take $X\in \mathbb{g}$. Then why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?
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### Finding the Lie Algebra of a Lie Group

I am having a hard time finding the set of all $X \in M(n, \mathbb{R})$ such that $e^{tX^T}Be^{tX} = B$ for all $t \in \mathbb{R}$. where $b$ is any matrix in $M(n, \mathbb{R})$.