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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34 views

Borel subalgebras contain solvable radical

Let $L$ be a Lie algebra and let $B$ be a Borel subalgebra (a maximal solvable subalgebra) of $L$. I want to understand why $\operatorname{Rad} L \subseteq B$. In his proof, Humphreys ...
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1answer
50 views

Humphreys proof check

Theorem: Let $L$ be a subalgebra of $\mathfrak{gl}(V)$ with $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V\ne 0$, then $\exists v\in V. v\ne 0$, such that $Lv = 0$. ...
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1answer
64 views

Characterisation of Cartan subalgebras as maximal toral; redundant “abelian” in the definition of “toral”

Edit: The reasoning on which this question is based is wrong. See my own answer for a counterexample. Let $L$ be a (finite dimensional) semisimple Lie algebra over a field $k$ with $char(k) = 0$. ...
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1answer
81 views

Problem 9.7 - Lie Algebras - Humphreys

Let $\alpha,\beta\in\Phi$ span a subspace $E'$ of $E$. Prove that $E'\cap\Phi$ is a root system in $E'$. Prove similarly that $\Phi\cap(\mathbb{Z}\alpha+\mathbb{Z}\beta)$ is a root system in E' (must ...
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2answers
43 views

How do I show that there are $q^2$ solutions $M$ to $MX-XM=0$ where $X$ is non central in $\mathop{GL}(2,q),$ $M\in M(2,q)$ and $q$ is an odd prime?

Equivalently, in the Lie Algebra $M(2,q)$, how can I show that there are precisely $q^2$ solutions M to $[M,X]=0,$ where $X$ is a non central element of $\mathop{GL}(2,q)$, where $q$ is an odd prime? ...
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103 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 ...
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46 views

Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
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1answer
39 views

When is the Lie algebra of the center of Lie group the center of its Lie algebra

Suppose that $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and the center of $G$ is denoted by $Z(G)$ with its Lie algebra denoted $Z(\mathfrak{g})$. It's easy to show that ...
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19 views

Maximal noncompact forms in classical Lie algebra?

In this short note on Lie algebra, discussing about classical Lie algebra A,B,C,D class, in page 4 after Eq.(7), on the part of B,D class of O(2n,F) and O(2n+1,F) group (or algebra?), there is a ...
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39 views

Quaternion, Dihedral groups and A-D-E classification

$\bullet$ What is the role of Quaternion group $H$ and dihedral groups $D_n$ in A-D-E classification? $\bullet$ Is Quaternion group $H$ in $A$ (special linear Lie algebra of traceless operators) or ...
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28 views

References request about exponentials in Lie algebras.

I saw two formulas about Lie algebras. Let $G$ be an algebraic group over $k$ and $\mathfrak{g}$ its Lie algebra. For any $x \in \mathfrak{g}$, $a \in k$ and $g \in G$, we have $$ g \exp(ax) g^{-1} = ...
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71 views

Lie algebra of Derivations as a functor?

To an associative algebra $A$ one can associate a Lie algebra $\operatorname{Der} A$ of all derivations $D:A\to A$. To any morphism of associative algebras $\alpha:A\to B$, how can one associate a ...
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1answer
32 views

Decomposition into weights of semisimple Lie algebra

Let $H$ be an abelian subalgebra, in a complex semisimple Lie algebra $L\subset {\rm gl}\ ({\bf C}^n)$, whose elements are semisimple. Assume that $H$ is abelian Lie subalgebra. Define $$H^\ast = ...
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34 views

Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
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24 views

Lie algebra homomorphism preserves Jordan form

Fact : $\phi : L_1\rightarrow L_2$ is $surjective$ Lie algebra homomorphism. If $h\in L_1$ and ${\rm ad}_h$ is diagonalizable then ${\rm ad}_{\phi(h)}$ is diagonalizable ...
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48 views

Proof of Jordan Theorem on Lie Algebra

$ L$ is semisimple Lie algebra in ${\rm gl}\ V$ where $V$ is a complex vector space. Then we have two Jordan forms $$ x=d+n,\ {\rm ad}_x = {\rm ad}_d + {\rm ad}_n\ (x\in L) $$ (cf. Theorem 9.15 in ...
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46 views

Lie groups with structure constant $f_{abc} \neq f_{bca}$.

The structure constant $f_{abc}$ of Lie group is defined by the commutators of generators, $$[T^a,T^b]=i f_{abc}T_c$$ automatically $f_{abc}=-f_{bac}$. Can someone give a list of explicit examples ...
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1answer
32 views

Size of Derivation of Lie algebra

If $L$ is a Lie algebra in ${\rm gl}\ (n,{\bf C})$ then $$ {\rm Der}\ (L)\subset {\rm gl}\ (n,{\bf C})$$ (If $L$ is semisimple then $ L = {\rm ad}\ L ={\rm Der}\ L$) Is this true ? Thank you.
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36 views

Extend to a homomorphism.

My question is regarding a step in "p-Automorphisms of Finite p-Groups" by Evgenii I. Khukhro (p. 117 line 7) and would like some response to my argumentation/understanding of it. p-Automorphisms of ...
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1answer
30 views

Killing form on ${\rm ad}\ L$

What is Killing form on ${\rm ad}\ L$ ? Note that $L$ has a Killing : $$ \kappa(x,y) = {\rm tr}\ ({\rm ad}_x{\rm ad}_y) $$
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62 views

Weyl orbits of integral dominant weights and convex polytopes

Let $\xi$ be an integral dominant weight of a root system $\Delta$, and let $\mathcal{O}_{\xi}$ be its orbit under the action of the Weyl group. The elements of the orbit are the vertices of an ...
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1answer
47 views

ODE system and single PDE “equivalence”, reference request

The answers to this question Replacing large-dimensional ODE systems with one PDE suggest that, in general, one can not hope for "replacing" an ODE system with a single PDE. On the other hand, this ...
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2answers
89 views

A left invariant vector field on a Lie group

Let $G$ be a matrix Lie group. Let $v$ be a left invariant vector field on $G$ and $v_1 \in \frak g$, where $\frak g$ is a Lie group of $G$. Let $v_1$ be its value at the identity. We define $\phi_t ...
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1answer
24 views

Simple Lie algebra over ${\bf R}$

As far as I know classification of simple Lie algebra over ${\bf C}$ in ${\rm gl}\ (n, {\bf C})$ is done. And note that $$ {\bf R}^3_\wedge \otimes {\bf C} = {\rm sl}\ (2,{\bf C})$$ (all of ...
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14 views

A property on $SU(2^N)$

Is there a simple formula for the following: $\frac{d}{d \epsilon} \exp(\lambda^k G_k + \epsilon G_j) |_{\epsilon = 0}$ where ${G_k}$ are the standard generators of $SU(2^N)$ given by N-fold tensor ...
2
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1answer
82 views

Action of the Weyl group on the symmetric algebra $ S\mathfrak{h} $

Let $\mathfrak{g}$ be a complex semi-simple Lie algebra. Let $\mathfrak{h}$ be a cartan subalgebra. Let $ \Delta $ be the resulting root system. Denote by $ V $ the real span of the roots. Let $ ...
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1answer
45 views

The decomposition of the exterior of the symmetric square over Lie algebra sl(3)

I am studying the representation theory of finite dimensional modules over the simple Lie algebra $\operatorname{sl}(3)$. I know some basics facts about the decomposition of some construction of ...
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1answer
56 views

how can I show $H^1(g , Hom_C(g,M))=0$?

For a simple Lie algebra $g$ and a finite dimensional vector space $M$ with a trivial $g-$action, how can I show $H^1(g , Hom_C(g,M))=0$?
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1answer
43 views

An existence of exponential function for a Lie algebra.

Let $G$ be a Lie group (given by a matrix). Let $\frak g$ be its Lie algebra. I would like to know if the following is true. "Let $X$ be a matrix in $\frak g$. Then $\gamma(t)=\exp(tX)$ is a curve ...
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0answers
37 views

Example of $3$-dimensional Lie algebra

I have a question on $3$-dimensional Lie algebra $L$ over ${\bf C}$ (cf. Erdmann and Wildon's book) Assume that $$ L=(x,y,z),\ L'=(y,z)$$ Then the book states that there exits two kinds of $L$ : ...
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62 views

sl2(C) does not have a nontrivial 1-dimensional central extension?

How can I show lie algebra sl2(C) does not have a nontrivial 1-dimensional central extension?
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43 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
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1answer
16 views

${\rm tr}\ {\rm ad}\ z =0$ for $z$ in commutator ideal $L'$

If $L$ is a Lie algebra in ${\rm gl}\ (n,{\bf R})$ then $$\ast\ {\rm tr}\ {\rm ad}\ z =0$$ for $z$ in commutator ideal $L'$ This is followed from matrix expression. But in 2.5 exercise in Erdmann and ...
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32 views

Correspondence between unipotent and nilpotent elements

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...
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1answer
31 views

Problem about structure of Lie algebra

This is 2.13 exercise in Erdmann and Wildon's book. Define a center $$ Z(L) = \{ z\in L |\ [z,x]=0\ \forall \ x\in L \} $$ If $I$ is ideal of $L$ then let $$ B = C_L(I) = \{ z\in L|\ [z,x]=0\ ...
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1answer
34 views

Isomorphism between Lie algebras

This is an exercise (cf. exe 2.11 in the Erdmann and Wildon's book) Define $$ gl_S(n, F) = \{ x\in gl(n,F)|\ x^t S = -Sx \} $$ Here $t$ is transpose. Then let $T=P^tSP$ and show that $$ ...
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1answer
40 views

Invertible derivation of $[L, \operatorname{Rad}(L)]$

Suppose $L$ is a finite-dimensional Lie algebra over the field of characteristic $0$. Let $\operatorname{Rad}(L)$ denote the radical of $L$. My question is: Does there always exist an invertible ...
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1answer
12 views

product of ideals in decending central series

Let $L$ be a Lie algebra. Let $C^n(L)$ be defined by: $C^0(L) = L$, $C^k(L) = [L,C^{k-1}(L)]$ for $k \geq 1$. Then how can I show that $[C^r(L),C^s(L)] \subseteq C^{r+s}(L)$ for all $r,s \in ...
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0answers
19 views

Base of a root system

Let $R \subset V$ be a reduced root system, and $R' \subset R$. Assume that: (i) $\alpha \in R' \ \to \ - \alpha \notin R'$, (ii) $ \alpha, \beta \in R'$ and $\alpha + \beta \in R$ implies $\alpha ...
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20 views

irreducible representations of lie algebras

We have the following criterion for the irreducibility of a Lie algebra representation (we work with $L$-modules here). Let $L$ be a Lie algebra, $V$ a finite dimensional vector space, and let $L ...
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1answer
155 views

How to determine the matrix of adjoint representation of Lie algebra?

My questions will concern two pages: http://mathworld.wolfram.com/AdjointRepresentation.html and http://mathworld.wolfram.com/KillingForm.html In the first page, we know the basis of four matrix ...
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1answer
28 views

Are Lie algebras $u_n$ and $su_n$ simple?

I think, that $u_n$ isn't simple, because, for example, any matrix $(\begin{matrix} ia & 0 \\ 0 & ia \end{matrix}) \in Z(u_n)$, and hence $u_n$ has non-trivial ideal. But i don't know ...
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1answer
53 views

On the Lie bracket of the Lie algebra of the group of invertible elements of an algebra

Assume $A$ is a finite dimensional associative $\mathbb{R}-$algebra, with identity $1( \neq 0)$, let $A^{\times}$ be the set of all invertible elements of $A$, then it's easy to see that ...
3
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2answers
70 views

Lie algebras ${\rm sl}(2,{\bf R})$ and $({\bf R}^3,\wedge)$ are not isomorphic

As I said in the title, I want distinguish algebras between ${\rm sl}(2,{\bf R})$ and $({\bf R}^3,\wedge)$ : On ${\rm sl}(2,{\bf R})$ $$ e=\begin{bmatrix} 0 & 1 \\ 0 & 0\end{bmatrix},\ f= ...
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2answers
71 views

Lie algebra adjoint representation

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Then there are representations $$Ad : G \rightarrow GL(\mathfrak{g}), \; \; ad : \mathfrak{g} \rightarrow GL(\mathfrak{g}).$$ Subrepresentations ...
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1answer
96 views

Lie-brackets and solution space of PDE

I have a linear, first-order homogeneous PDE system with polynomial coefficients $$L_j\, f =0,\text{ for } j=1,..,J\quad \text{ where } L_j \text{ is a first order, diff. operator with polynomial ...
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0answers
25 views

parabolic subalgebra

Let $G$ be a semisimple lie group, a parabolic subgroup of $P$ is a connected subgroup that contains a conjugate of $B$, (which $B$ is Borel subgroup of $G$) then I can not see why lie algebra of $P$ ...
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2answers
38 views

Conjugate of Lie subalgebra

What does it mean that "all Cartan subalgebras of a semisimple Lie algebra are conjugates"? I know this refers to adjoint action but I don't know exactly what it means. The most obvious definition to ...
5
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0answers
52 views

Corestriction map in lie algebra cohomology

Given a lie algebra $\mathfrak{g}$ over a field $k$, we can define the cohomology groups of $\mathfrak{g}$ as follows: $$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$ where ...
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62 views

How do I find the Cartan subalgebra?

I know the definition of a Cartan subalgebra, but how do I actually find it explicitly for a particular Lie algebra over the complex numbers?