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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A complex lie algebra is the direct sum of simple ideals iff it is semisimple

So I am wanting to show that a complex lie algebra is the direct sum of simple ideals iff it is semisimple. In fact I have already proved <= It remains for me to prove => $\textbf{Currently I ...
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Given $L$ a complex finite dimensional Lie algebra. Then suppose $L$ is solvable. Show $L^{(1)}$ is nilpotent.

Given $L$ a complex finite dimensional Lie algebra. Then suppose $L$ is solvable. Show $L^{(1)}$ is nilpotent. Okay, so I have the existence of a flag of ideas in $L$. Can I deduce from this that ...
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207 views

Lie algebra of Euclidean group

From the book "Spinning Tops" by Audin, she claims that $$\mathfrak{so}(3)[\epsilon]/\epsilon^2$$ with coefficientwise Lie bracket is a Lie algebra of a Lie group that is $TSO(3)$ (group action not ...
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67 views

Proof of Lie theorem on solvable Lie algebra

I am reading a book of Helgason. As you know, solvable Lie algebra $g \subset V= {\bf C}^n$ have a nonzero $v$ such that $v$ is an eigenvector of any element of $g$. I can follow the proof in ...
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99 views

The Lie algebra of the commutator subgroup

If $G$ is a connected Lie group with Lie algebra $g$, then is its commutator subgroup $[G,G]$ a closed subgroup with Lie algebra $[g,g]$?
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Exterior and symmetric powers of $\mathfrak{sl}(4,\mathbb{C})$ representation

I am taking a course on representation theory, and going through Lecture 15 of Fulton and Harris's Representation Theory. One of the topics we're currently covering is the example of ...
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61 views

Projection map $\text{Sym}^2(\text{Sym}^3V)\to \text{Sym}^2V$ viewed as a Hessian

Exercises 11.21 and 11.22 in Fulton's Representation Theory are the following: Let $V$ be the standard representation of $\mathfrak{sl}_2\mathbb{C}$. The projection map from ...
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88 views

If $\mathfrak{g}$ admits a decomposition then it is semisimple

I want to show: If $\mathfrak{g}$ is a Lie algebra that has an abelian subalgebra $\mathfrak{h}$ such that $\mathfrak{g}$ has a Cartan decomposition $\mathfrak{g}=\mathfrak{h}\oplus(\bigoplus_{\alpha ...
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114 views

Conjugate Representations for $\mathfrak{sl}(2,\mathbb{C})$

Let $\mathfrak{sl}(2,\mathbb{C})$ be the complex Lie algebra of $SL(2,\mathbb{C})$ and $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ be its realification; that is $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
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43 views

Not null Killing form.

I have to find an example of solvable Lie algebra $L$ such that the Killing form of $L$ isn't null. If we take the Borel subalgebra of $\mathfrak{sl}(2)$, we have that the Killing form of $L$ is the ...
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jacobian involving SO(3) exponential map: $\log(R * \exp(m))$

I would like to compute the 3x3 Jacobian of $$ \log(R * \exp(m)) $$ with respect to the 3-vector $m$, evaluated at $m=0$. In the above, $\exp$ is the exponential map from so(3) to SO(3), $\log$ is ...
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Root system of a Lie Algebra

Could anybody help me to solve this problem with roots system? Be $\Phi$ an irreducible root system. $\Phi^{+}$ a choice of positives roots in $\Phi$. Prove that if $(\alpha,\beta)\ge0$ $\forall ...
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36 views

Symmetric algebra and complex polynomial on Lie algebra

Let $G$ a Lie group and $\mathfrak{G}$ its Lie algebra. How can I identify the symmetric algebra on $\mathfrak{G}$ ($S(\mathfrak{G})$) with the algebra of complex polynomials on $\mathfrak{G}$, that I ...
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47 views

Invariants of representation theory of Lie groups

How to compute the determinant of a representation of an element of the special linear group? How do I argue that it doesn't change? (@Marek: @rschwieb: Yes well, given one represenation (with ...
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Lie Derivative in Projective Hilbert Space

In considering a projective Hilbert space, $P(H)$, for linear maps (tensors) of vectors in the space, $A^{a}_{b}v_{a}=u_b$, is there a natural definition for the Lie Derivative for such linear maps? ...
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96 views

Submanifold of a Lie group - tangent space

Let $G$ be a compact Lie group and $H, H' \leq G$ Lie subgroups. Consider the set $M = H' \cdot H = \{h\cdot h' \ \vert \ h \in H, h' \in H'\}$. Is it possible to describe explicitly the tangent space ...
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Is there any Lie algebra that is not constructed from an associative algebra

I see in Wikipeida that every Lie algebra is either constructed from an associative algebra by defining: $[x,y]=xy-yx$, or a subalgebra of a Lie algebra thus constructed. Where can I find a proof? ...
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Followup question in Brian Hall's Lie Groups and Algebras.

In ex 9, page 60, he writes down that in order to prove that each invertible matrix $A$ can be written as $A=e^X$, where $X\in M_{n\times n}$, one need to use the fact that if $A$ is unipotent then ...
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53 views

Basics of Lie 2-algebras?

Could somebody (simply) explain the basics foundations of Lie 2-algebras, and some basic practical applications ? For instance, does it exist a 3-map (equivalent to the 2-map commutator for Lie ...
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136 views

The universal enveloping algebra of a loop algebra as a quotient of the free associative algebra.

Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra and set by $\tilde{\mathfrak{g}}:=\mathfrak{g}\otimes_{\mathbb C} \mathbb{C}[t,t^{-1}]$ its loop algebra. How to express the ...
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$1$-parameter subgroups in $GL_n(\mathbb{C})$

I came across this link on planetmath and a few facts on that link are confusing me. According to planetmath, any $1$-parameter subgroup in $GL_n(\mathbb{C})$ arises from the exponential map. That ...
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The nonexistence of nontrivial solvable series in $M_n(k)$

I am a bit confused about semisimple Lie algebras. For the sake of simplicity, let's take $\mathfrak{g}=M_n(k)$ where $k=\bar{k}$. According to Wiki, $M_n(k)$ is solvable if the radical of $M_n(k)$ ...
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Inverse boson operator realization of $\mathfrak{so}(3)$

This is actually a homework problem. The inverse boson operators $a^{-1}$ and $\left(a^\dagger\right)^{-1}$ are defined as $$a^{-1} |n\rangle = \frac{1}{\sqrt{n+1}} |n+1\rangle$$ ...
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set of roots satisfying a minimal condition related to the induced Killing form

Let $\mathfrak{g}$ a finite-dimensional complex simple Lie algebra with Cartan subalgebra $\frak h$. Let denote $(\cdot,\cdot)$ the non-degenerate bilinear form on $\frak h^*$ induced by the Killing ...
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63 views

Question about root space

Let $\mathfrak{g}$ be a Lie algebra and consider $\operatorname{Rad}(\mathfrak{g})$, the radical of $\mathfrak{g}$, that is, the sum of all solvable ideals in $\mathfrak{g}$. Suppose that we have the ...
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180 views

How to prove the lie algebra of $n\times n$ traceless matrices is semi-simple?

The Lie algebra of all the $n \times n$ matrices is not semi-simple. However, if we restrict ourselves to traceless $n\times n$ matrices, we do obtain a semi-simple (in fact, simple) Lie algebra which ...
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186 views

$SU(2)$ is a covering space of $SO(3)$.

The method of topology is very clear.Then there's a question asking to use adjoint representation of lie group $SU(2)$ $(\operatorname{adj}:SU(2)\to GL(su(2)))$to prove this. I can't solve this .
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$G_1$-Scalar factors for Clebsch-Gordan coefficients for $ U(n)$

when evaluating the $G_1$ scalar factors for CGC's of $U(n)$ it seems that some of the factors are undefined. The explicit formula for the evaluation of the scalar factors is Eq. (6) in 18.2.8 of N.J. ...
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123 views

First-order derivatives in differential forms calculus

Let $d$ denote the Cartan differential, and let $\delta$ denote the codifferential. The underlying domain is not important for what follows. The canonical generalization of the Laplace-operator ...
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Subspace of a homogeneous space.

Given two homegenous spaces $\frac{G}{H}$, $\frac{A}{B}$ with $A\subset B$ is there a way to prove that $\frac{A}{B}\subset \frac{G}{H}$ ie that $B\subset A\cap H$? In particular I would like to ...
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109 views

PBW Theorem applied to graded Lie algebras

Fix a $\mathbb Z_+^n$-graded Lie algebra ${\frak a}=\oplus_{r \in\mathbb Z_+^n}^{} {\frak a}[r]$ such that ${\frak g}:={\frak a}[0]$ is a finite-dimensional semisimple Lie algebra over the complex ...
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Cartan subalgebras of a loop algebra.

For an algebraically closed field $\mathbb F$ of characteristic zero, a finite-dimensional Lie algebra $\frak G$ has a Cartan subalgebra and these subalgebras are conjugated in a certain sense. Let ...
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Complete reducibility of finite-dimensional representations of $\mathfrak{sl}_2(\mathbb{C})$

By Weyl's theorem every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ is completely reducible, because $\mathfrak{sl}_2(\mathbb{C})$ is a (semi) simple Lie algebra. It seems there ...
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The form of a subgroup of $GL(n,K)$ when the derived group is of certain form

The famous Lie-Kolchin theorem in the theory of algebraic groups states: Let $G$ be a connected solvable subgroup of $GL(V)$, $0 \neq V$ finite dimensional. Then $G$ has a common eigenvector in ...
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When does $C_G(s) \times Cl_G(s)s^{-1}$ equal $G$

I have read on James E. Humphreys' Linear Algberaic Groups If $G$ is an algebraic subgroup contained in $GL(n,K)$, and $s$ is a semisimple element of $G$, then $\mathfrak{g}$ has the ...
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Does triality survive in product Lie groups?

Look at the following diagrams of Lie groups ...
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Finite order automorphisms of Lie algebras

Let $\Gamma$ be a Dynkin diagram automorphism of diagram type $A_{2n}$ and let $\sigma$ be a non-trivial finite order automorphism of $\Gamma$. Let $g$ the Lie algebra associated to $\Gamma$ and ...
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239 views

How to show that the structure constant of SU(3) is invariant?

So suppose $f_{ijk}$ is the antisymmetric structure constant of SU(3), and $D^8_{ij}(g)$ is the matrices of 8-dimensional adjoint representation of SU(3), then how to show that ...
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126 views

Do the classical Lie algebras all satisfy $XM + MX^T = 0$?

I'm working on a homework assignment in which part of the question statement says that each of the classical Lie algebras can be described as the set of all matrices $X \in gl(n,\mathbb{C})$ ...
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328 views

Length of root strings

Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra $g$ of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the length of an $a$-string through $b$ is at most ...
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33 views

To check d^2= 0 in the standard complex of Lie superalgebras.

For a Lie superalgebra $\mathfrak{g}$ and a $\mathfrak{g}$-module $V$ we can define the cohomology $H^i(\mathfrak{g}, V)$ with coeffiecient in $V$ to be the cohomology space of the following complex: ...
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23 views

reflection(reflection) = rotation

Lel $\alpha$ and $\beta$ be two distinct simple roots in a root system $\Phi$. How to prove that i) $S_{\alpha} S_{\beta}$ is a rotation in $\mathbb{R}\Phi$ ii) Composition of two reflection is a ...
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A refernce about Cartan matrix

There exist an approach to "Cartan Matrix" in Carter's book "Finite groups of Lie type, conjugacy classes an complex characters" p.23, which seems be different to other definitions of Cartan matrix I ...
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Differential operators on the polynomial ring

Let $A$ be a commutative algebra over complex numbers. If $a\in A$ we define $m_a$ to be a linear map which sends each $x$ to $ax$. The zero map $A\to A$ is said to be a differential operator of an ...
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Dimension of maximal tori

Let $G$ be a compact Lie group. $T$ $-$ its maximal torus. Is there a simple reasoning to show that dimensions of $T$ and $G$ have the same parity? I am sorry if this quesion is for children, but ...
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Confusion in infinite-dimensional Lie algebra notes

I've been going through these notes and everything was pretty much fine until I saw line labeled (5.1) on page 6. What kind of object is $z^\alpha (dz)^\beta$? How does exactly differential operator ...
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9 views

Cayley-Hamilton type decomposition of SL(3,R) matrices

Given an element $\lambda = \theta_a T_a$ of SL(3,R) Lie algebra, where $T_a$s are the generators and $\theta_a$s are parameters, is there a general formula to determine the coefficients A,B and C ...
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Elements of $gl(2l+1,\mathbb{C}): x^tS =- Sx$, How they are found and Erdman exercise 4.2

On page 130 of Erdman's book "Introduction to Lie Algebras" we have: Let $L = gl_S(2l+1,\mathbb{C})$ for $l \geq 1$ where $S = \left(\begin{array}{cc} 1 & 0 & 0 \\ 0 & 0 & I_3 \\ 0 ...
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Convolution and Characters

I am confused about the purpose of the Formal Character, character functions, and the convolution in representation theory of Lie algebras. Is the Character function different than just the Character? ...
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Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...