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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Lie Algebras: How to compute the Killing Form on $\mathfrak{sl}_n(\mathbb{C})$ and Jordan Decomposition Theorem question.

I'm reading the Fulton and Harris Representation Theory book, trying to learn about Lie Algebras. On pg. 213, they compute the killing form on $\mathfrak{h}^*$ for $\mathfrak{sl}_n(\mathbb{C})$. I ...
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90 views

Irreducible representation and reductive Lie algebra

If we have complex reductive Lie algebra L and her finite dimensional representation $\phi$. How can we show that $\phi$ is irreducible iff restriction $\phi|_{[L,L]}$ is irreducible?
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198 views

Weyl group and Dynkin diagram

Can somebody help me with following questions: 1)Prove that two simple roots in a Dynkin diagram that are connected by a single edge are in the same orbit under the Weyl group. and 2)For an ...
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1answer
94 views

Is algebraic closure required in Weyl's theorem on complete reducibility? (Lie algebras)

Weyl's theorem states that finite-dimensional representations of finite dimensional semisimple Lie algebras are completely reducible (expressible as a direct sum of irreducible submodules), with some ...
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130 views

Isomorphism with Lie algebra $\mathfrak{sl}(2)$

Let $L$ be a Lie algebra on $\mathbb{R}$. We consider $L_{\mathbb{C}}:= L \otimes_{\mathbb{R}} \mathbb{C}$ with bracket operation $$ [x \otimes z, y \otimes w] = [x,y] \otimes zw $$ far all $x,y \in ...
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140 views

establishing an isomorphism

I request help with this is a question from Introduction to Lie algebra by Erdmann and Wildon. The question asks to show that show that $so(4,\mathbf{C})\cong sl(2,\mathbf{C}) \oplus ...
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110 views

Weyl group of $\mathfrak{sl}(2,\mathbb{C})$

$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{gl}(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the ...
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100 views

How can I find a Chevalley basis of $B_2$?

How can I find a Chevalley basis of a type $B_2$ when the related lie algebra is defined as a linear Lie algebra of elements of the form $x= \begin{pmatrix} 0 & b_1 & b_2 \\ c_1 & m & ...
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21 views

Finite dimensional representations of semi-simple Lie algebras

I've been trying to understand the proof of the following statement: An injective map of $\mathfrak{g}$-representations of a semisimple Lie algebra splits. I'm supposed to show this considering the ...
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45 views

How can I maintain linear independence through a commutator?

Consider a Lie algebra $\mathcal{L}$, a linearly independent generating set $\mathcal{G}$, and an element $X \in \mathcal{L}$. Edit: Note that $\mathcal{G}$ is not necessarily a basis; the generation ...
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21 views

basis for symmetric square

If we have the symmetric square Sym$^{2}V$ and $V=\mathbb{C}^{2}$, why is it that $\{x^{2}, xy, y^{2}\}$ form a basis for it? So symmetric square matrices are when the main diagonal acts as a ...
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56 views

Intuition for the exponential of a matrix

I'm trying to understand an algorithm that tries to map points from a lie group to its lie algebra using the exponential map. The background is the representation of 3d coordinate transformations as a ...
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30 views

Showing elements of $\mathfrak{gl}_n$ can be represented as a nilpotent and a semisimple matrix under addition

I want to use the Jordan form to show that every element $A\in \mathfrak{gl}_n$ can be written as $N+S$ where $N\in\mathfrak{gl}_n$ is nilpotent and $S\in\mathfrak{gl}_n$ is semisimple and $NS=SN$. ...
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24 views

Solving a large system of linear equations to satisfy the Lie bracket: am I doing it right?

I'm still working on a Lie algebra isomorphism from the Lie algebra of $SL_2(\mathbb C)$ into the Lie algebra of $O(3, \mathbb C)$. It has been suggested to me to use linear combinations of ...
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1answer
38 views

Affine transformation invariants and lie groups

Is it possible to generate geometric properties which are invariant under affine transformations? I'm trying to learn about lie groups and lie algebras with the example of the lie group of affine ...
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1answer
30 views

Symmetrizability of generalised Cartan matrix

How to prove that a generalized Cartan matrix whose diagram contains no cycles is symmetrizable? Any hint would be sufficient. Thanks in Advance.
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53 views

How does one define weights for a semisimple Lie group?

For compact Lie groups one considers a maximal torus to define the weight space decomposition of a representation. For a complex semisimple Lie algebra one considers a Cartan subalgebra. How does ...
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29 views

Lie group as a subset of its Lie algebra

Consider a (possibly infinite-dimensional) Lie group $\mathcal{G}$ and let $\mathcal{A}$ be an algebra with a product $\cdot$ and the bracket $[u,v]=u\cdot v - v\cdot u$. The following statement is ...
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56 views

General differentials operators (Grothendieck definition) and polynomial rings

Let $A$ be an algebra over some field $\mathbb{k}$. A linear map $f:A\to A$ is said to be a differential operator of an order $\le n$ if for all $a_0,a_1,\ldots a_n\in A$ we have ...
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21 views

Multiple Cartan sub-algebras

How is it that for a Semi-simple Lie Algebra there is not one Cartan Sub-Algebra? I assume since there are multiple CSA's of a SS Lie algebra that must mean some of the ss elements of the Lie ...
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26 views

Standard set of Generators

A standard set of generators for a semisimple Lie algebra $ L $ is defined as: {${x_\alpha}, {y_\alpha}, {h_\alpha} $} Where: $ x_\alpha \in L_\alpha, $ $ y_\alpha \in L_{-\alpha}, $ $ ...
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56 views

How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$?

Let $U$ be the positive unipotent radical of $SL_n$ and $\mathfrak{n}$ the Lie algebra of $U$. How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$? Here $\mathcal{O}_q[U]$ is the ...
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116 views

Lie group and Lie algebra automorphisms

Assume that $G$ is a connected Lie group and that $\alpha:G\rightarrow G$ is an automorphism of $G$. Furthermore let $\alpha_*:\mathfrak{g}\rightarrow\mathfrak{g}$ be the corresponding tangent map at ...
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72 views

Showing that an $\mathfrak{sl}(2,\mathbb{C})$-module is determined by eigenvalues of $h$

This question is essentially exercise 8.4 from the book "Introduction to Lie Algebras" by Erdmann and Wildon: "Suppose that $V$ is a finite-dimensional $\mathfrak{sl}(2,\mathbb{C})$-module. Show that ...
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21 views

Uniqueness of the Lie brackets in the quotient space of a Lie algebra

Suppose I have a Lie algebra $\mathfrak g$ which is an ideal of $\mathfrak a$. Then I consider the quotient set $\mathfrak g / \mathfrak a$ which is the set of all equivalence relations of $\mathfrak ...
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361 views

Heisenberg XXX spin model

Let $\pi$ be the standard representation of $sl_2(\mathbb{C})$ on $\mathbb{C}^2$. Let $p_1,p_2,p_3$ the three Pauli matrices. Define $S^a:=\frac{1}{2}\pi(p_a)$. What does such matrices looks like?
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97 views

How to find Casimir Operators and their degree.

Consider the quite general problem of computing all Casimir Operators of a given Lie Algebra $\mathfrak{g}$. How does one proceed, in general? And how is possible to compute the degree of a given ...
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27 views

Weyl function defintion

In Lie algebra book by Humphreys, in the section 24.1 (page number 136) he defines Weyl function $q$ as $ q = \Pi_{\alpha \gt 0}(\epsilon_{\frac{\alpha}{2}}-\epsilon_{-\frac{\alpha}{2}})$. I can't ...
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66 views

Is it true for solving differential equations by getting constant coefficient matrix with magnus expansion

The magnus expansion is given in detail http://en.wikipedia.org/wiki/Magnus_expansion. While implementing magnus expansion to differential equations we have an iteration formula as follows $$Y'(t) = ...
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144 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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101 views

Why is the dual space of Cartan subalgebra an irreducible representation of Weyl group

it is proposition 14.31 in Fulton-Harris book. The proof goes like this. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and assume $\mathfrak{z}\subseteq\mathfrak{h}^*$ were preserved ...
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75 views

Specific help in showing that Poisson Bracket is part of this Lie Algebra

Given in this exercise is the following set: $U = \{f(z) = z^TCz\ \vert\ C \in \textrm{Mat}_{2n}(\mathbb R),\ C^T=C\}$ is a Lie Algebra with $\left\{\cdot , \cdot \right\}$ where $$ \left\{f,g ...
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132 views

Basic Lie Algebra Question

Essentially, I'm trying to prove that when computing the tangent space for a group that there's nothing special about considering it at only the identity. Namely, there is an isomorphism of vector ...
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21 views

L-module definition

I have the following definition of an L-module We say that V is an L-module if there is a k-bilinear mapping L × V → V sending a pair (x, v) ∈ L × V to x.v ∈ V such that [x, y].v = x.(y.v) − y.(x.v) ...
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22 views

Can we represent the curl as a multiplication by skew-symmetric matrix?

Considering that two vectors $A \times B$ = $\hat A* B$, where $\hat A$ is a skew symmetric matrix containing elements of $A$ Can we then write the curl $\nabla \times A$ as $\partial \vec r *A$ ...
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22 views

Map to the submodule of invariants of a Lie algebra representation

If $G$ is a compact group and $V$ is a representation, the inclusion $V^G \to V$ has an easy-to-write-down retract: \begin{equation*} V \to V^G,\:\: v \mapsto \frac{1}{|G|} \int_G g\cdot v\;dg ...
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31 views

Dual of a matrix lie algebra

In fact I already calculate the dual space with a formula, but I did'd understand some steps of the formula. So, I want to calculate the dual space of The lie algebra of $SL(2,R)$. Knowing that ...
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22 views

A doubt from the Isomorphism theorems of Lie algebras.

Given an isomorphism of two irreducible root systems $\Phi$ and $\Phi$' we need to show that the corresponding simple Lie algebras $L$ and $L'$ are isomorphic. For that we take the subalgebra $D$ of $ ...
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22 views

Metrics on affine connections

In one of the paper's I read this statement: "the affine geodesics of the Cartan connections (group geodesics) are metric-free". What does this really mean? paper: ...
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45 views

Generalization of Schur's Lemma: finite dim. representations of real Lie algebras

Let $V$ be an irreducible finite dimensional real representation of a real finite dimensional Lie algebra $\mathfrak{g}$. From Schur's Lemma, what is $Hom_\mathfrak{g}(V,V)$ or ...
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20 views

Cohomology group of an algebra

I tried to guess what is a cohomology group of an algebra. I would like to find the correct definition of this. I know what is a cohomology group of a group, but I don't know how connect the second ...
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66 views

Is there a more intelligent way to compute the determinant of the Killing form of $\mathfrak{sl}(3,F)$?

Is there a more intelligent way to tackle exercise 7 of paragraph 5 of Humphreys (Introduction to Lie Algebras and Representation Theory)? Exercise 7: Relative to the standard basis of $L = ...
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33 views

Lie algebras with different bases

I am interesting to know that if a finite dimensional Lie algebra $L$ has two bases $\beta_1$ and $\beta_2$, how can we compare the cardinal of two sets $\{(x,y)\in \beta_1\times ...
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40 views

Embedding so(n) in su(n)

Is there any way of embedding $\mathfrak{so}(n)$ into $\mathfrak{su}(n)$ for any $n$ other than picking the antisymmetric matrices of $\mathfrak{su}(n)$? I know that for small $n$ one can use ...
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53 views

One parameter subgroup

I am new to Lie group and I am reading the "Lie Groups, Lie Algebras, and Representations" by Brian Hall. So what's the intuitive idea about one parameter subgroup? I understand all the definition but ...
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39 views

How can we compute a Lie bracket for powers of elements of given lie algebra?

Let $L$ be a lie algebra over finite field, for $ x,y$ in $L$ I want to solve the following bracket: $[yx^k,x]=?$ How can we describe that in the format of $[...[y,x],x],...,x]=[y,x]_i$ ($i-times$)
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58 views

Complex conjugation of positive roots

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...
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22 views

Restricted Universal Enveloping Algebras

Is there example of restricted universal enveloping algebra $uL$ of the $p$-Lie algebra $L$ over field $k$ of characteristic $p > 0$ such that $L$ hasn't nonzero $p$-algebraic elements and global ...
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34 views

Is adjoint map invertible?

I've already studied the group of automorphisms of a simple lie algebra on a finite field, but according to the definition of an adjoint representation of a Lie algebra, can we claim an adjoint map is ...
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29 views

Kac-Moody root datum introductory text?

I have been given a project to describe the construction of the Lie algebra associated to a Kac-Moody root datum $D=(I,A,\Lambda, (c_i)_{i\in I}, (h_i)_{i\in I})$. I only know basic definitions: that ...