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)

2
votes
0answers
26 views

Inner-product on skew-hermitian matrices

Let $$\mathfrak{u}(n)=\{X\in M(n,\Bbb C):X+X^*=0\}$$ where $X^*$ is the conjugate transpose. Then, $\mathfrak{u}(n)$ is a real vector space. Problem. Show that $\langle X,Y\rangle=\...
1
vote
0answers
31 views

Trace of the product of a Lie algebra and Lie group element

Take $U \in SU(n)$ and $X \in \mathfrak{su}(n)$. What can we learn about \begin{align} \text{Tr} (UX) \end{align} In particular Is there a closed form expression? When does $\text{Tr} (UX)$ vanish?...
1
vote
1answer
45 views

In what sense are complex representations of a real Lie algebra and complex representations of the complexified Lie algebra equivalent?

In this book I read Proposition A.1. The irreducible complex representations of a real Lie algebra $\mathfrak{g}$ are in one-to-one correspondence with the irreducible complex-linear ...
1
vote
0answers
32 views

Reference Request: Lie Theory For Quantum Field Theory

I have encountered the section on non-Abelian gauge theories in Peskin and Schroeder's QFT textbook, and although I am comfortable with the derivation of the Yang-Mills Lagrangian they present, the ...
1
vote
2answers
35 views

problem about nilpotent leibniz algebra

I want to prove this statement if $A$ is nilpotent leibniz algebra then $ H \subsetneq N_A(H)$. $H$ is a subalgebra of $A$ Can you help me how to show this.thanks
2
votes
0answers
44 views

Is the commutator subgroup $[G,G]$ isomorphic to $G/Z_G$?

Let $G$ be a connected reductive Lie group with Lie algebra $\mathfrak{g}$. That means that $\mathfrak{g}=Z_\mathfrak{g}\oplus[\mathfrak{g},\mathfrak{g}]$, and $[\mathfrak{g},\mathfrak{g}]$ is ...
6
votes
0answers
96 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 ...
1
vote
1answer
24 views

What is the root system and the Weyl group of the group spin$(2n)$?

Reading on root systems and Weyl groups, unfortunately I am highly confused when it comes to the spin-groups (the two-fold universal cover of SO$(2n, \mathbb{C})$, realizable as a quotiënt in a ...
0
votes
1answer
58 views

Cartan decomposition diffeomorphism at the level of (compact) groups

I am trying to understand the Cartan decomposition in the case of a compact group $G$ with subgroup $K$, such that their respective Lie algebras $(\mathfrak{g},\mathfrak{k})$ correspond to a Cartan ...
0
votes
0answers
12 views

About exceptional Lie group E6

How to show the group of determinant preserving linear transformations of $ z $ is isomorphic to $$ \{a \in Isom_\mathbb{C}(z^\mathbb{C},z^\mathbb{C})|det(aX)=det(X),<aX,aY>=<X,Y>\} $$ ...
-1
votes
0answers
34 views

homogeneous dimension of the Heisenberg group [closed]

How to compute the homogeneous dimension of the Heisenberg group $\mathbb C \times \mathbb R $ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ ...
1
vote
0answers
31 views

The relation between Weyl character formula and Frobenius characteristic map

Let $\mathfrak{gl}(n)$ be the general linear Lie algebra of rank $n$, and $\mathfrak{S}_d$ be the symmetric group of rank $d$. It is well-known that the Schur-Weyl duality provide a equivalence ...
0
votes
0answers
44 views

Is it possible to study Lie algebras without knowing too much of representation theory?

There's a course on Lie Groups that I'd like to take, but it seems that for various reasons it's a good idea to take Lie algebras along with it. But after having a brief look at the contents of the ...
1
vote
1answer
33 views

Using isometric group to describe E7.

I read John C. Baez's paper, The Octonions, and I am wondering the following statement: $$E_7\simeq Isom(\mathbb{(H\otimes O)P}^2).$$ In his contents, I can only figure out $$E_7\hookrightarrow Isom(\...
-3
votes
1answer
64 views

Help finding an article [closed]

Hello Recently I have been studying algebra and am in search of the following paper : Kac, V. G. Classification of simple $Z$-graded Lie superalgebras and simple Jordan superalgebras. Comm. Algebra 5 ...
3
votes
0answers
32 views

What are the units of $U(\mathfrak{sl}_2)$?

Let $U(\mathfrak{sl}_2)$ be the Universal Enveloping Algebra of $\mathfrak{sl_2}$ over a field $K$, i.e. the (non-commutative) algebra generated by three generators $E,F,H$ subject to the commutator ...
0
votes
1answer
24 views

Why the induced metric from the lie algebra of lie group $G$ is left invariant.

we say that a Riemannian metric on $G$ is left invariant if $<u,v>_y = <d(L_x)_y u,d(L_x)_y v>_{L_x(y)}$ to introduce a metric on $G$, take any arbitrary inner product $< , >_e$ on ...
0
votes
0answers
18 views

Is a reductive subalgebra of a semisimplie Lie algebra semisimple?

I have the following doubt. Say that $\mathfrak g$ is a semisimple Lie algebra, $\mathfrak k$ a reductive subalgebra, and suppose further that any Cartan subalgebra in $\mathfrak k$ is a Cartan ...
0
votes
1answer
31 views

Derived algebras and solvable Lie algebras

The idea of a Solvable Lie algebra hinges on the definition of the sequence: $$ g \ge [g,g] \ge [[g,g],[g,g]] \ge [ [[g,g],[g,g]] , [[g,g],[g,g]] ] \ge \ldots $$ and its limiting group. If an ...
0
votes
1answer
21 views

Where I can find classifications of Lie algebra $A_{4.7}^{-1}$?

My Lie algebra with commutation relation $[e_2, e_3] = e_3,\;[e_2, e_4] = -e_4,\;[e_3, e_4] = -e_1$ is isomorphic to Lie algebra $A_{4.7}^{-1}$ through transformations $e_1\mapsto e_1,\;e_2\...
0
votes
0answers
28 views

Limit of the commutator of two elements?

Given a Lie group $G$ such the $\mathfrak{g}$ denoted its Lie algebra. Let $[g,g']_{G}$ the commutator of two elements $g,g' \in G$ and denoted by $[X,X']_{\mathfrak{g}}$ the Lie bracket of two ...
1
vote
0answers
25 views

compact lie group -> real analytic orbits in $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Now, $G$ also acts on $\mathfrak{g}^*$, the dual of the Lie algebra, by the coadjoint-action. My question now is: are ...
1
vote
1answer
57 views

Irreducible representations of Heisenberg group

Lately, I've been struggling with the following problem. Let $H$ be the 3 dimensional Heisenberg group and let $\rho:H\to\text{GL}(n,\mathbb{C})$ be a irreducible representation. Show that $n=1$. I ...
2
votes
0answers
59 views

Involutions and Representation of Lie Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$. Is easy to see that ...
1
vote
1answer
23 views

what if the infinitesimal generator of a vectorfield vanishes?

Let $(M,g)$ be a riemannian manifold and $H$ a Lie group acting on $M$. Denote by $l \colon H \times M \to M$ and $l_h \colon M \to M$ the action of $H$ on $M$. Now $H$ acts on $TM$ by derivations, ...
0
votes
1answer
66 views

Is the alternating group a lie group

Is the alternating group a lie group. If so what is the lie algebra corresponding to it? This is not a homework questions. I need the dimension of the lie algebra (if one exists) to prove some ...
0
votes
0answers
21 views

Extension of Lie algebras

Let $L$ be a Lie algebra and $A$ be its subalgebra. Let consider derivation $d: A \to L$. How can we interpret this extension $$H=\langle L,t | [t,a]=d(a), \forall a \in A \rangle? $$ How will the ...
0
votes
1answer
35 views

Radical of a direct sum of Lie algebra

If we take $L$ a finite dimensional Lie algebra on $\mathbb{R}$, $A$ a sub-abgebra and $I$ an ideal of $L$ such that $L=A \oplus I$ as vector spaces. We have that $rad(L)=rad(A) \oplus rad(I)$ as ...
1
vote
1answer
36 views

If G is a compact semisimple Lie group, then its Killing form is negative definite

Theorem: If $G$ is a compact semisimple Lie group, then its Killing form is negative definite. In its proof: Since $G$ is compact there is an Ad-invariant inner product on $\mathfrak g$. Since each ...
0
votes
1answer
12 views

Ideal spanned by monomial of degree $n$ in the Universal Enveloping Algebra

I'm trying to understand the Ado theorem proof which uses the universal enveloping algebra of a Lie algebra. In this proof we use the ideal spanned by all monomial of degree $n$ in the universal ...
0
votes
0answers
27 views

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$?

On a Lie group $G$, is $\{v-Ad_gv\mid v\in\mathfrak{g},g\in G\}=[\mathfrak{g},\mathfrak{g}]$? This question is inspired by noting that if we have a Hamiltonian Lie group action $G\curvearrowright (M,\...
3
votes
1answer
61 views

Cartan decomposition of SO(2n)

I am trying to understand the Cartan decomposition theory, on the following example : $G=SO(2n)$, and $K=U(n)$, and I'm interested in the manifold $G/K$ (an hermitian symmetric space). 1) How do we ...
2
votes
1answer
48 views

Showing a rep of $sl(2,\mathbb{K})$ is irreducible

Let $V$ be a $m+1$-dim $K$-vector space with char$K=0$. Let $(v_0,v_1,\dots,v_m)$ be a basis of $V(m)$. Now suppose I construct a representation of $sl(2,K)$ on this representation. How do I show ...
1
vote
0answers
25 views

By which the Heisenberg group is introduced? [closed]

I want to know who was the first who introduced the Heisenberg group and in what year. In the Wikipedia there is just an indication that this group was named in honor of the famous German physicist ...
1
vote
1answer
49 views

Representation of of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$

Certain part in my textbook implies that a representation of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$, where $S \in \mathbb Z$, is possible. I am trying to find a path that leads to this ...
1
vote
2answers
41 views

The Lie Algebra of $O(n)$ is the set of $n \times n$ skew-symmetric matrices

I'm trying to show that the Lie Algebra for $O(n)$ is the set of $n \times n$ skew-symmetric matrices. Here is what I have so far. Since $O(n)$ is the union of two disjoint subsets, the matrices with ...
0
votes
0answers
10 views

Graph properties of Bruhat order for the general linear Lie algebra $\mathfrak{gl}$ on $\mathbb{Z}^n$

Let $P = \oplus_{i\in \mathbb{Z}}\mathbb{Z}\epsilon_i$ the free abelian group of infinite rank. Then we have a natural partial order $\leq'$ on $P$, that is, $a \leq' b $ if and only if $b \in a+\sum_{...
0
votes
1answer
51 views

Regular elements of a Lie algebra

I'm currently trying to learn about regular elements of a Lie algebra but i'm finding the definition quite abstract and can't seem to find many examples anywhere. One thing i'm really unsure about ...
1
vote
0answers
29 views

The linear Lie algebra of a closed linear group is closed

I was reading Knapp's Lie groups beyond an introduction and, in the first pages, he shows that the set of all tangent vectors to a given closed linear group $G$ at the identity matrix, that is $$\...
2
votes
1answer
75 views

Sandwich rule for Lie algebras

On an infinite dimensional vector space an operator can be onto but not one-to-one (and vice versa). This arises the following question. Let $L_1$ and $L_2$ be Lie algebras (infinite dimensional, over ...
0
votes
0answers
16 views

Relation between integrable representations and highest weight representations.

Let $g$ be a simple Lie algebra and $U_q(g)$ the corresponding quantum group. What are the relation between integrable representations and highest weight representations of $U_q(g)$? Are all highest ...
4
votes
1answer
75 views

$E_8$ and theta functions

The root lattice $\Gamma_8$ of the exceptional Lie algebra $E_8$ is an eight-dimensional lattice which consists of lattice points in $\mathbb{R}^8$ which with respect to an orthonormal basis $e_1, \...
0
votes
1answer
68 views

How many three and four dimensional Lie algebras are there?

Patera and Winternitz have carried out extensive classification of three and four dimensional Lie algebras. When I tried to look for classification for three dimensional Lie algebra with non-zero ...
2
votes
0answers
26 views

Finite dimensional, irreducible representations of the Lie superalgebra gl(1|1)

I am wondering how the finite dimensional, irreducible representations of the Lie superalgebra gl(1|1) are parametrized. I understand that they are all highest weight, and that the only non-trivial ...
0
votes
1answer
37 views

How to write quotient algebra for normalizer?

For Lie algebra $\mathfrak{g}=\{e1,e2,e3,e4\}$ with commutations $[e1, e3]=\,e1, [e2, e3]=\,\alpha\,e2$, I have calculated normalizer for sub-algebra $\mathfrak{q}=\,\{e1+e2\}$ as $\text{Nor}_{\...
0
votes
1answer
27 views

Splitting and non-splitting extensions in Lie algebras

For Lie algebra $S=\{e_{1}, e_{2}, e_{3}, e_{4}\}$ with non-zero commutations: $[e_{1}, e_{3}]=e_{1}, [e_{2}, e_{3}]=\alpha\, e_{2}$ we have $S=e_{4}\oplus L_{3}$, such that $L_{3}=\{e_{1}, e_{2}, ...
1
vote
1answer
30 views

Orthogonal basis of a Cartan of a Lie algebra with respect to Killing form.

I am trying to understand orthogonal basis of a Cartan of a Lie algebra with respect to Killing form. For example, let $g=sl_2 = \text{Span}\{h, E, F\}$. Then a Cartan of $g$ is $\mathfrak{h} = \...
1
vote
0answers
8 views

Finding a polytope in the Cartan Subalgebra

The finite Coxeter groups can be realized as symmetry groups of (semi)-regular polytopes. Not all semi-regular polytopes can be realized this way, but all regular polytopes can. Some examples of ...
0
votes
1answer
16 views

How subsets are defined in Lie algebra?

Consider four dimensional Lie algebra with non-zero commutations: $[e_{2},e_{3}]=e_{1}, [e_{2}, e_{4}]=e_{2}, [e_{3}, e_{4}]=-e_{3}$ having sub-algebras $S_{1}=\{e_{1}, e_{2}\}, S_{2}=\{e_{1}+e_{2}\}...
0
votes
3answers
52 views

If a Lie Algebra is solvable, is the corresponding Lie group solvable in the group theoretic sense?

I just started working with Lie Algebras with a professor. The way we defined them is probably the standard way; treat Lie Algebras as tangent spaces at the identity of the Lie Group. Now, consider ...