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

learn more… | top users | synonyms (1)

3
votes
0answers
82 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 ...
1
vote
1answer
35 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.
1
vote
1answer
42 views

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

What is Killing form on ${\rm ad}\ L$ ? Note that $L$ has a Killing : $$ \kappa(x,y) = {\rm tr}\ ({\rm ad}_x{\rm ad}_y) $$
4
votes
0answers
100 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 ...
2
votes
1answer
61 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 ...
1
vote
2answers
321 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 ...
1
vote
1answer
34 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 ...
3
votes
1answer
118 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 $ ...
1
vote
1answer
120 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 ...
0
votes
1answer
65 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 $H^1(g , Hom_C(g,M))=0$?
1
vote
1answer
53 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 ...
1
vote
0answers
43 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 $L$ : ...
2
votes
0answers
136 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 ...
0
votes
1answer
22 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 ...
3
votes
0answers
58 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 ...
1
vote
1answer
47 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\ ...
0
votes
1answer
53 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 $$ ...
1
vote
1answer
46 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 ...
1
vote
1answer
18 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 ...
1
vote
0answers
62 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 ...
0
votes
0answers
37 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 ...
3
votes
1answer
707 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 ...
0
votes
1answer
36 views

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

I think, that $u_n$ isn't simple, because, for example, any matrix $\begin{pmatrix} ia & 0 \\ 0 & ia \end{pmatrix} \in Z(u_n)$, and hence $u_n$ has non-trivial ideal. But i don't know ...
2
votes
1answer
70 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
votes
2answers
140 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= ...
0
votes
2answers
131 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 ...
5
votes
1answer
144 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 ...
0
votes
2answers
113 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 ...
6
votes
1answer
132 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 ...
1
vote
0answers
105 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?
2
votes
1answer
130 views

Jacobi identity and Leibniz rule - the same thing?

Is there any formal connection between the Jacobi identity $$[[a,b],c] = [a,[b,c]] + [b,[c,a]]$$ and the Leibniz rule $$d(a \cdot b) \cdot c = a \cdot d(b) \cdot c + b \cdot c \cdot d(a) ~\text{?}$$
1
vote
0answers
53 views

The Lie subgroup of the compact Lie group

$G$ is a compact connected Lie group with Lie algebra $g$ whose center is $h$. Let $h^{\bot}$ be the orthogonal complement of $h$ where the inner product is chosen to be invariant under the adjoint ...
3
votes
1answer
116 views

With the branching rules of subalgebra, how can I write down explicit matrix elements for a representation?

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
4
votes
1answer
83 views

Linear structure on the category of formal groups

Let $R$ be a commutative ring. If $R$ is a $\mathbb{Q}$-algebra, then the category of formal groups over $R$ (or the category of formal group laws) carries the structure of an $R$-linear category; ...
2
votes
0answers
84 views

Status of a question from Freeman Dyson's 1972 article

In a famous article, Freeman Dyson mentions an interesting relationship between the $\tau$ functions of number theory and the dimensions of finite-dimensional simple Lie algebras (section 2). He ...
1
vote
2answers
107 views

If Lie algebra is equal to its commutant, is it semisimple?

I am searching for a prove or a counterexample for this statement: If finite-dimensional complex Lie algebra is equal to its commutant, then it is semisimple. I suppose it is not true, because ...
0
votes
0answers
12 views

Is the matrix of every set of base vectors of $\Bbb{C}^n$ symmetric?

The book "Theory of Lie Groups" by Chevalley says A linear endomorphism $\alpha$ of $C^n$ is determined when the elements $\alpha e_i=\sum\limits_{j=1}^n a_{ji}e_j$ are given. There corresponds ...
1
vote
1answer
26 views

Possible use of the rank of a nilpotent Lie algebra to construct a maximal dimensional solvable Lie algebra

Let $\mathfrak{g}$ be a nilpotent Lie algebra. It is possible to find the Lie algebra of derivations of $\mathfrak{g}$ denoted $Der\mathfrak{g}.$ Then we could consider the maximal abelian subalgebra ...
1
vote
1answer
129 views

Lie algebra homomorphism and action on a manifold

In Introduction to smooth manifolds Lee says on page 527: If $\mathfrak{g}$ is an arbitrary finite-dimensional Lie algebra, any Lie algebra homomorphism ...
1
vote
1answer
52 views

Is there a name for this Lie algebra?

Consider the three dimensional, complex Lie algebra with basis $\{a,a^\dagger, I\}$ and the following structure relations: \begin{align} [a,a^\dagger] = I, \qquad [a,I] = 0, \qquad [a^\dagger, I] = ...
2
votes
1answer
75 views

Definition of (co)homology of groups and Lie algebras: actions and augmentations

In the Chevalley-Eilenberg chain complex, what is $ux_i$? What does "trivial $\frak{g}$-module $k$" mean? Below I denote $R=k$ (any commutative unital ring). How is the augmentation (last map in the ...
2
votes
1answer
69 views

What is $\mathfrak{gl}(\infty)$

As title says, I know what is $\mathfrak{gl}(n,\mathbb{C})$, but what is $\mathfrak{gl}(\infty)$? Where can I find good reference for this?
3
votes
1answer
51 views

Commutativity and Maximal Tori in Connected, Compact Lie Groups

Let $G$ be a path-connected, compact Lie Group. Let $x \in G$ and let $T_x \subset G$ denote the union of all the maximal tori in $G$ that contain $x$. Question: Is it true that if $y \notin T_x$, ...
1
vote
1answer
50 views

$[[x,y],z]=[x,[y,z]] \Rightarrow [x,y]=0$?

I got the next problem: Let $A$ be a Lie algebra, prove that if the bracket associates $([[x,y],z]=[x,[y,z]]$) then the bracket is zero $([x,y]=0)$. Can't get the result using the properties ...
3
votes
0answers
69 views

Computing the fundamental groups of simple algebraic groups of type $A$

I'm interested in seeing the computation for the fundamental groups of the simple algebraic groups of type $A$. Below is the definition of the fundamental group for a simple algebraic group $G$. Let ...
0
votes
1answer
71 views

Real representations of SL(2,C)

Is there a classification of real-linear (rather than complex-linear) finite-dimensional representations of SL(2,C)?
1
vote
1answer
89 views

Zero-product property for enveloping algebras

Let $L$ be a finite-dimensional Lie algebra $L$ over a field $k$. Let $(U(L), i)$ be a universal enveloping algebra of $L$. If $x,y \in U(L) - \{0\}$ is there something contradictory about the ...
1
vote
0answers
49 views

Intertwiner for $U(n-1) \subset U(n)$

I'm using the notation of Vilenkin and Klimyk, ''Part3: Representations of Lie Groups and Special Functions''', chapter 18. Given an irreducible representation $T_m$ of the complex Lie algebra $U(n)$ ...
5
votes
3answers
888 views

How should I show that the Lie algebra so(6) of SO(6) is isomorphic to the Lie algebra su(4) of SU(4)?

As far as I can see, an isomorphism of Lie algebras is a bijective map which preserves the Lie bracket. I need to show that $\mathfrak{so}(6)$ (the Lie algebra of SO(6)) is isomorphic to the ...
4
votes
1answer
399 views

What is the correspondence between structure constants and a Lie group?

Let $T^a$ (with $a = 1,2,\ldots,n$) be a set of generators of a Lie group that satisfy the commutation relations: \begin{equation} [T^a,T^b] = i \sum_{c=1}^n f^{abc} T^c \,, \end{equation} where ...