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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Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
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27 views

Real version of Harish-Chandra-Itzykson-Zuber integral

I'm interested in an integral of the form $$ \int_{O(d)} \exp\left(-\frac{1}{2}\mathrm{trace}(CUAU^T)\right)dU $$ where the integration is with respect to the Haar measure on the orthogonal group, ...
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1answer
37 views

Why do roots span dual space of maximal toral subalgebra?

Suppose $\Phi$ is the root system of a semi simple Lie algebra with maximal toral subalgebra $H$. I read that $\Phi$ spans $H^\ast$. The Killing form on $H$ is nondegenerate, so $H\cong H^\ast$ by ...
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31 views

Computing the Cohomology of Lie groups

In Bredons "Topology and Geometry" [Chapter V, section 12] the following theorem is proven: If $G$ is a compact connected Lie group its $k$-th cohomology $H^k(G,\mathbb{R})$ is isomorphic to the ...
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1answer
36 views

Adding tori to semi-simple groups

Let $G$ be a complex, connected, semi-simple Lie group (throw in simply connected if you like) with Lie algebra $\mathfrak g$. Let $T \subseteq B$ be a maximal torus and choice of Borel, respectively. ...
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29 views

How to embed $U(1)$ (or other groups) into a bigger group, using Dynkin diagrams

I am trying to find the embedding and the branching rules for some group decompositions. For example, I consider $E_7$ and its maximally compact subgroup $SU(8)$ and I want to "see" how the Dynkin ...
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2answers
41 views

$\mathfrak g = [\mathfrak g,e]\oplus {\rm Ker}({\rm ad}f)$ for an $\mathfrak {sl}_2$-triple $\{e,h,f\}$.

Let $\mathfrak g$ be a finite dimensional semisimple lie algebra over $\mathbb C$. Let $\{e,h,f\}$ be an $\mathfrak{sl}_2$-triple in $\mathfrak g$ (i. e. with relations $[h,e] = 2e$, $[h,f]=-2f$ and ...
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Formula for $\theta:\mathfrak{P}(L)^{G}\to \mathfrak{P}(H)^{W}$ for $\mathfrak{sl}_2$; exercise in Humphrey

Let $L=\mathfrak{sl}(2,\mathbb{F})$ with standard basis $(x, y, h)$ and dual basis $(x^{*}, y^{*}, h^{*})$, $H$ a CSA, $W$ the Weyl group and $G=\operatorname{Int}L$. Let $\mathfrak{P}(L)^{G}$ be ...
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1answer
35 views

The fundamental vector fields of a principal bundle are vertical.

Let $p:P\to M$ be a principal $G$-bundle. To each $A$ in the Lie algebra of $G$ corresponds a fundamental vector field $A^*$ on $M$ defined by $$A^*_u=\frac{d}{dt}|_{t=0} u(exp(tA))$$ How can we see ...
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1answer
47 views

Left and right action?!

The adjoint of the adjoint representation $Ad^* : G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*, (g,x) \mapsto Ad^*_{g}(x)$ is a group action on the dual space of the Lie algebra. Now, we said ...
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27 views

Lie algebra operations from lie group

According to wikipedia, if $G$ is a closed subgroup of $GL(n, \mathbb{R})$ then the Lie algebra of $G$ can be thought of informally as the matrices $m$ of $M(n, \mathbb{R})$ such that $1 + εm$ is in ...
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1answer
8 views

Compute variation left action subgroup

I consider a Lie group $G$, with a group element $g$ parametrised in some manner with parameter $\theta_i$, $i=1,\cdots, \dim G$. Suppose that $K\subset G$. I want to compute the variation of an group ...
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1answer
23 views

Criterion for semisimplicity for $\mathfrak{so}_6(\Bbb C)$

I'm trying to prove that $\mathfrak{so}_6(\Bbb C)$ is semisimple. There exists a criterion which says that, given a Lie algebra $L\le\mathfrak{gl}(V)$, where $V$ is an irreducible $L$-module, then ...
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49 views

Standard cyclic module of sl2

Let $L=\mathfrak{sl}(2, \mathbb{F})$, $B$ a standard Borel subalgebra. I am trying to solve exercise 20.4 from J.E. Humphreys "Introduction to Lie Algebras and Representation Theory", but I am stuck. ...
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43 views

Ideals of Lie-algebras

I am wondering whether the following claim is true: Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra and $V$ some vector subspace of $\mathfrak{g}$. Claim: $V$ is an ideal of $\mathfrak{g}$ ...
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15 views

Are the elements of the adjoint represetnation normal operators

Given a Lie group $G$ with Lie algebra $\mathfrak{g}$ on has the adjoint action of each $g\in G$ given by $Ad_g(\mathfrak{g})$. Is $Ad_g: \mathfrak{g} \rightarrow \mathfrak{g}$ a normal operator ...
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14 views

The algebra of $W$-invariant polynomial funktions sl2

Let $L=\mathfrak{sl}(2,\mathbb{F})$, $H$ a borel subalgebra, $\Delta=\{\alpha\}$ a base of the corresponding root system and $W$ the Weyl group. Let $\lambda=\frac{1}{2}\alpha$ be the fundamental ...
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1answer
104 views

Equivalent definitions of positive root system

I begin with a definition of positive root systems of a root system over Euclidean space. A subset $\Delta$ of root system $\Phi$ is called a simple root system (or base) in $\Phi$ if (1) $\Delta$ ...
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2answers
119 views

Lie Groups/Lie Algebra - Applications?

I studied Lie Groups and Lie Algebras as part of my Masters back in the 1970s. Although very elegant and beautiful, it seemed to its own little world, I never saw the connection with other branches of ...
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1answer
54 views

Given adjoint action find original matrix.

Given the Adjoint action of a matrix; $\text{Ad}(g) X_1 = g \, X_1 \, g^{-1} = X_2 $. Where g is in a (matrix) Lie group, $X_1,\; X_2$ are from the Lie algebra, can a $g$ be written in terms of the ...
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33 views

Nonlinear Lie group from Fulton & Harris

On page 138 of my copy of the celebrated Representation Theory by Fulton & Harris, a proof is outlined to show that the real group of $3\times 3$ upper-triangular unipotent matrices modulo a ...
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17 views

Stable Cartan under two involutions

Let $\mathfrak{g}$ a complex semisimple Lie algebra, and let $\theta$ be an involutive automorphism of $\mathfrak{g}$. The following fact is known. Any parabolic subalgebra contains a ...
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152 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
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37 views

Universal enveloping algebra of sl2

I am currently trying to proof, that $x-1$ is not invertible in the universal enveloping algebra $\mathfrak{U}(\mathfrak{sl}(2,\mathbb{F}))$ of $\mathfrak{sl}(2, \mathbb{F})$, but I still struggle ...
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1answer
18 views

Matrix representation in exponential form

So having worked out beforehand that $Λ(v) = \begin{pmatrix} γ&0&\frac{-γv}{c}\\ 0&1&0\\\frac{-γv}{c}&0&γ\end{pmatrix}$ where $Λ(v) ∈ SO(2,1)$ is a matrix representation of a ...
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2answers
76 views

Lie group step in proof

Let $X_e,Y_e \in T_eG$ be vectors and $G = GL(n).$ Then the right translation is given by $Y_g = Y_e g$ and $X_g = X_e g.$ Now, I have a proof showing that $[X_e,Y_e] \in T_eG$ is the element ...
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1answer
29 views

Computing plethysms of the adjoint representation using the Littlewood Richardson rule

Let $N$ be an integer (let's imagine very large), and let $G$ be the group $\mathrm{GL}_N(\mathbb{C})$. I would like to compute various plethysms of irreducible representations which are not ...
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1answer
45 views

Lie Algebra to Lie Group Mapping.

When I map a lie algebra vector in se3 to SE3 using exponentiation and map it back to se3 using log, why do I get significantly different results? I followed this and coded an implementation in ...
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Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$? In the case of $G = SL_2$, we have $\mathbb{C}[SL_2] = \langle a,b,c,d\rangle / (ad-bc-1 )$ and ...
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1answer
22 views

A computation for Manin triple.

I am reading the book. I have some questions about the computations in (4.1) on page 40. The computation are in the following. I don't know why $$ ([[e_r^*, e_k], e_s^*]+[e_r^*, ...
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1answer
59 views

Two more questions on Kontsevich's “Noncommutative Identities” (Derivations on $\mathbb{C}\langle X,Y \rangle$) [Solved]

The following two questions regard once more the following article: arXiv:1109.2469. In the second chapter we are dealing with the Lie Algebra $\mathfrak{g}$ of derivations $\delta$ of ...
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1answer
17 views

representation theory of two-step nilpotent Lie algebras

Does anyone know of any good reference about the representation theory of two-step nilpotent Lie algebras, like whether their irreducible representations can be classified?
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2answers
42 views

Every semisimple Lie algebra of dimension at most 5 is simple.

How does one argue that every semisimple Lie algebra of dimension $\leq 5$ is simple. Since any simple algebra has dimension at least $3$, we have to show that any semisimple algebra of dimension ...
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2answers
41 views

Matrix representations of $\mathfrak{g}^*$.

Let $g = sl_2$. Then there is a matrix representation of $g$ as follows. The Lie algebra $g$ is a three dimensional vector space with a basis $E, F, H$ such that $[E,F]=H$,$[H,E]=2E$,$[H,F]=-2F$. The ...
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1answer
27 views

Sum of nilpotent ideals in a Lie algebra is nilpotent

When trying to show that $I+J$ is nilpotent, whenever $I,J$ are nilpotent ideals of a Lie algebra $L$, I did it brute force: By induction we can show the following: An element of $(I+J)^{2N}$ is a sum ...
3
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0answers
32 views

How to understand Weyl chambers? [duplicate]

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 ...
3
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0answers
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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1answer
24 views

Weightspace decomposition of a semisimple Lie algebra

$\DeclareMathOperator{\ad}{ad}$ Let $L$ be a (finite dimensional) semisimple Lie algebra. Let $H$ be a maximal toral subalgebra of $L$. Consider a representation $\pi: L \to \mathfrak{gl}(V)$. It is ...
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1answer
29 views

dimension of lie algebra

I am studying lie algebra myself and question is about finding dimension of lie algebra . While i read Wikipedia link about lie algebra and lie group i saw statement Lie algebra $\mathfrak{g}$ is ...
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1answer
70 views

Fundamental group of a Root System and determinant of the Cartan matrix

This is the first time I am posting, so I hope I didn´t get the formatting wrong. I am currently reading J. E. Humphreys "Introduction to Lie Algebras and Representation Theory" and got stuck at ...
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2answers
49 views

Solvable equivalent to nilpotency of first derived Lie algebra?

The Wikipedia "Solvable Lie Algebra" page lists the following property as a notion equivalent to solvability: $\mathfrak{g}$ is solvable iff the first derived algebra $[\mathfrak{g},\,\mathfrak{g}]$ ...
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Etingof problem 2.16.2: Irreps of Two-dimensional Lie algebra over a field of positive characteristic

This is problem 2.16.2 in Etingof's introduction to representation theory. Note that problem 2.16.1 is a proof of Lie's theorem. I'm having trouble with the second case, where the base field has ...
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Description of free Lie algebra in Weibel's book

In Exercise 7.3.2 in Weibel's book An Introduction to Homological algebra the following description of the free Lie algebra over some $k$-module $M$ is given (where $k$ is any commutative ring): ...
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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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1answer
32 views

Possible angles between roots in a root system

Given a Root System $\Phi$ let $\alpha,\beta \in \Phi$ with $\alpha \neq \pm \beta$ and $||\beta||\geq ||\alpha||$. Let $\theta$ be the angle between $\alpha$ and $\beta$. Since $<\alpha,\beta> ...
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1answer
26 views

$\operatorname{Rad}(k)=\operatorname{Rad}(L)$

Given a Lie Algebra $L$ on a field $F$, we define the radical of $L$ $\operatorname{Rad}(L)$ as the largest solvable ideal of $L$. We define the adjoint representation ...
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How can Clebsch-Gordan Decompositions be combined?

In section 4 of this paper the authors use a given list of Clebsch-Gordan coefficents for the $27 \otimes 27$ of $E_6$ from an old paper and combine it with their own list of Clebsch-Gordan ...
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What is corresponding Lie group for Lie algebra of vector fields in dynamical systems?

According to Ado's theorem, for every finite dimensional abstract Lie algebra there is a Lie group. Finite dimensional analytic (or meromorphic) Vector fields (in dynamical systems) over the filed of ...
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A Lie Algebra $L$ is reductive iff it is completely reducibile as an $\operatorname{ad}_L(L)$-module

Given a Lie Algebra $L$ we say it is reductive if $\operatorname{Rad}L=Z(L)$. How can we prove that $L$ is reductive iff it is an $\operatorname{ad}_L(L)$-module completely reducibile? Suppose $L$ ...
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22 views

Question about the theorem of highest weights

I have some confusion from reading Theorem 7.3 in Sepanski's Compact Lie groups and would appreciate it if someone could clarify. In part (e) the book says "for $w\in W$, $wV_\lambda=V_{w\lambda}$, ...