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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104 views

Why are root systems presented in this confusing way?

I quote Bjorner and Brenti, "Combinatorics of Coxeter Groups." We begin with a simple geometric lemma. Let $m \geq 3$ be an integer, let $\gamma = \pi/m$, and let $k, k'$ be real numbers ...
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374 views

Elementary proof of the third Lie theorem

I would like to understand a (not well known) proof due to G.M; Tuynman, of the third Lie theorem, which asserts that for any given finite dimensional Lie algebra $\mathcal{G}$ there exists a (simply ...
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15 views

Constructing a coset representative of $SO(n,4)/(SO(n) \times SO(4))$.

In $\mathcal N = 2$ Supergravity the scalar components of Hypermultiplets form a quaternionic Kaehler manifold. Only isometries of this so-called target manifold can be gauged. I am interested in ...
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35 views

How do roots act on weights?

In Lie theory it's possible to compute things very explicit using tensor methods. For example, we can use an explicit matrix for each generator $T^a$ and compute the "action" of this generator on an ...
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125 views

Is it possible to chose the value of $(\alpha, \alpha)\in \mathbb{F}$ for the root system $A_{1}$?

The question is based on what I tried to solve two exercises in James E. Humphreys "Introduction to Lie Algebras and Representation Theory": chapter 26 exerise 1 and chapter 9 exercise 2. I am looking ...
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110 views

Humphreys Introduction to Lie Algebras - Conjugate Borel subalgebras sl(2,F)

Let $L$ be a Lie Algebra and let $E(L)$ denote the subgroup of the inner automorphisms, generated by all $\exp(\operatorname{ad}(z))$ for $z\in L$ being strongly ad-nilpotent. Let ...
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87 views

How to understand Weyl chambers?

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane ...
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24 views

Weight spaces of a irreducible representation of $\mathfrak{gl}(n, \mathbb{C})$.

Let $\mathfrak{gl}(n,\mathbb{C})$ be the general linear Lie algebra. Let $\{E_{s,t}\}_{1\leq s,t,\leq n}$ be the standard basis for it. And set its Cartan subalgebra $\mathfrak{h}$ to be ...
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34 views

Which elements of $su(n)$ commute with those of a subalgebra $su(2)$

Given a subalgebra $su(2) \subset su(n)$ , how many generators of $su(n)$ commute with any element in the subalgebra $su(2)$? I know that there are at least $n-2$ elements in $su(n)$ satisfying this ...
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29 views

Center of the dual of a Lie algebra

Let $\mathfrak{g}$ be a Lie algebra. Let $C \subset \mathfrak{g}^*$ be the subspace of linear forms that vanish on Lie brackets: $$C = \{\alpha \in \mathfrak{g}^*, ~ [\mathfrak{g}, \mathfrak{g}] ...
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64 views

When did the notion of the tangent space emerge?

When did the modern conception of/notation for the tangent space to a manifold come into use?
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30 views

Root system of a simple lie algebra is irreducible

The proposition is from Humphreys. I don't understand how to prove the highlighted statements. How can I express a general element of K? I tried using Cartan decomposition of L but it doesn't work. ...
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60 views

If a Lie algebra L decomposes as a direct sum of its derived subalgebra and its center, is L reductive?

A Lie algebra $L$ is said to be reductive if for any ideal $\mathfrak{a}$ of $L$, there is an ideal $\mathfrak{b}$ of $L$ such that $L=\mathfrak{a}\oplus\mathfrak{b}$. It is known that a reductive ...
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44 views

The bijection between central characters and linkage classes over a semisimple Lie algebra

I have a question about the modules over a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra. For a given $\lambda \in ...
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28 views

Homologie of Lie algebra with coefficients in tensor product of modules

I'd like to prove that $H_i(\mathfrak g, M\otimes N)=\operatorname{Tor}_i^{U \mathfrak g}(M,N).$ My idea was that $H_i(\mathfrak g, M\otimes N)=\operatorname{Tor}_i^{U \mathfrak g}(k,M\otimes N)$. ...
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67 views

Derivative of exponential map

Somehow I've gotten myself confused trying to take the derivative of the exponential map on $\mathfrak{so(3)}$. For vector $\theta$, $\delta \theta$, and $p \in \mathbb{R}^3$, define $$R(\theta, p) ...
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42 views

Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
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54 views

How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...
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103 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) ...
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45 views

Finding a basis and weight space for $L = \mathfrak{so}_6(\mathbb{C})= \{x \in \operatorname{End}(\mathbb{C}^6)\ |\ ^{\mathrm t}xS + Sx = 0 \}$

The question: Let $S = \left(\begin{array}{cc} 0 & I_3 \\ I_3 & 0 \end{array}\right)$ and let $$L = \mathfrak{so}_6(\mathbb{C})= \{x \in \operatorname{End}(\mathbb{C}^6)\ |\ ^{\mathrm t}xS + ...
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89 views

Weyl Group of Parabolic subgroups

Let $G=SL(n,\mathbb R)$ with Lie algebra $\mathfrak{g}=\mathfrak{sl}(n,\mathbb R)$. The classical minimal parabolic subgroup $B$ consists of the upper triangular matrices. The parabolic subgroups $P$ ...
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49 views

Compact Lie algebras and Lie groups

A simple or semisimple Lie algebra is said to be compact if the $\mathrm{Tr}\left \{ T^\mathrm{adj}_{a}, T^\mathrm{adj}_{b}\right \}$ is positive definite where $T^\mathrm{Adj}_{a}$ are the generators ...
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65 views

Actions of Weyl group

I get a feeling what I am going to ask is very standard and classic, but I am not able to find any reference. Any answer or reference would be appreciated. Let us assume that $G$ is a simply ...
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59 views

Recovering a restricted Lie algebra from its restricted enveloping algebra

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $\mathfrak{g}$ be a restricted Lie algebra, with restricted enveloping algebra $u(\mathfrak{g})$. We can place a Hopf ...
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46 views

Would the transformation of a differential equation obey the same algebra?

I've found that the algebra of this differential equation $$\frac{d^2y}{dz^2}-(3z^2+\gamma)\frac{dy}{dz}+(cz+\alpha)y=0$$ is in $sl(2)$ because it is possible to use the generators of the $sl(2)$ ...
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92 views

How was this Lie algebra found?

In a paper the author lists, without justification, generators for a Lie algebra. I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself. ...
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85 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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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 ...
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70 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 ...
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74 views

A Isomorphism between the extension group and cohomology group of Lie algebras

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove ...
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55 views

PBW theorem for restricted Lie algebras

I'm looking at the proof of the PBW theorem for restricted Lie algebras to be found in Ponto and May's "More Concise Algebraic Topology", page 361 (367 in linked file). I either see an error in their ...
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216 views

Representation of Complexification of Lie Algebra

Is the following obvious? I think it is, but wanted to make sure before an exam tomorrow! "There is a bijection between the complex representations of a real Lie algebra and the complex ...
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74 views

Weights set spans

Definition Let $T$ be a torus and $R: G \to GL(V)$ a representation. $R(T)$ is a collection of commuting matrices and therefore can be simultaniously diagonalized. For a character $\lambda \in ...
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100 views

adjoint representation completely reducible

Let $\mathcal{A}$ be a Lie algebra. Suppose that adjoint representation of $\mathcal{A}$ is completely reducible (or semisimple). Show that $\mathcal{A}$ can be written as a direct sum of semisimple ...
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66 views

Usage and determination of “rank” and “dimension” of groups & representations

Physicist here. I seem to see conflicting statements about the rank of some groups I've come across lately. A paper I'm reading states that $SO(6)$ is rank 3 and therefore its Cartan subalgebra ...
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110 views

Regarding the definition of vector field flow

To make the connection to the Lie derivative, let $t \mapsto \Phi^X_t$ be the 1-parameter group of diffeomorphisms (or flow) generated by the vector field $ X $. The differential $ d\Phi^X_t ...
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147 views

The enveloping algebra of a finite dimensional Lie algebra has no zero divisor

Let $L$ be a complex, finite dimensional Lie algebra. It is well-known that the graded associative algebra of the enveloping algebra $U(L)$ is isomorphic to the symmetric algebra $S(L)$. Therefore ...
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105 views

finding highest weight of dual of a representation of a semisimple lie algebra

If $V$ is an irreducible representation of a semi simple lie algebra having highest weight $\lambda$ then what will be the highest weight of the corresponding irreducible representation $V^*$ (Dual of ...
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286 views

Fundamental vector fields

I have a question related to fundamental vector fields. For that I first setup the notations and properties etc. Let $G$ be a lie group acting smoothly on the manifold $M$. Let $\mathfrak{g}$ be its ...
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129 views

Integral forms of loop algebras.

The question following is about integral forms for semisimple Lie algebras and loop algebras constructed from them. Let $\frak g$ a finite-dimensional Lie algebra over $\mathbb C$ and $L(\frak ...
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107 views

Question concerning semisimple Lie algebras

I'm currently solving a problem in Fulton's Representation Theory A first course and I'm not sure why a particular result is true. One part of the problem (exercise 14.15 if anyone is interested) ...
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117 views

The Weyl Group of $F_4$

The Weyl Group of $F_4$ is of order $1152=2^{7} \cdot 3^{2}$. By Burnside's theorem the group is solvable. Is there a way to see solvability from the root system? Is it possible to see the order of ...
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334 views

Invariant inner product $\langle\,,\rangle$ on a Lie algebra

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We can use the Killing form to identify $\mathfrak{h}$ and $\mathfrak{h}^*$ ...
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19 views

Local Lie derivative on $G$-space at the zero of a vector field

Let a Lie group $G$ act on a manifold $M$ and let $X\in Lie(G)$. For now suppose $G=T$ is a torus (but the answer to this question should hold for $G$ abelian). $L_X$ is a vector field on $M$, at a ...
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55 views

Derivatives of solution to Schrödinger equation

Consider the differential equation (Schrödinger, but rewritten to be pleasing to Lie algebraic eyes): $\frac{d U(t)}{dt} = c(t)U(t)$ where $c(t)=a+w(t)b(t)$, $a,b \in \mathfrak{su}(n)$ and $w$ is a ...
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37 views

Relationship between solutions of two matrix differential equations

Given a ($4\times4$ in the important case) matrix differential equation: $\frac{d U_t}{dt}= A_t U_t$ where $U_t \in SU(n)$ and $A_t \in \mathfrak{su}(n)$. What is the relationship between the ...
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59 views

Why doesn't the “naive” scalar product for $SO(n)$ yield something invariant?

By definition, for $SO(n)$ we have $g^T g=1$ for $g \in SO(n)$. Given some vector $v \in V$ and some representation $R: SO(N) \rightarrow \mathrm{Lin}(V)$, the defining condition above tells us ...
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36 views

Tangent space of coadjoint orbit

Let $\xi \in T_xOx$ be a tangent vector at $x \in O_x :=\{\mathrm{ Ad} _{u}^*(x); u \in G\}$ for $x \in g^*.$ ($g$ is the Lie-Algebra) Then I read that this $\xi$ can be represented as the velocity ...
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26 views

List of simple roots in the H-basis for various Lie algebras?

There are four usual bases one can use to express the roots and weights of a given algebra. The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$. The ...
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31 views

Map for roots of a Lie group to roots of a special subalgebra?

For regular subalgebras $h$ of some group's Lie algebra $g$, $$ h \subset h $$ the root system of the subalgebra is a subset of the root system of the original's group algebra. Subalgebras whose ...