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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11
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1answer
550 views

Inscrutable proof in Humphrey's book on Lie algebras and representations

This is a question pertaining to Humphrey's Introduction to Lie Algebras and Representation Theory Is there an explanation of the lemma in §4.3-Cartan's Criterion? I understand the proof given there ...
9
votes
1answer
157 views

Partial derivatives on Manifolds

Let $F : A \times B \to C$ be a map of smooth manifolds. Define the following maps ("partial derivatives"): $E_1 F: TA \times B \to TC$ $E_1 F(a,v,b) = D_a F(-,b) v $ where $v \in T_a A$ $E_2 F: A ...
6
votes
1answer
146 views

Lie algebras of reductive groups

Let $k$ be an algebraically closed field of positive characteristic and let $G$ be a connected split reductive group. We know $G$ is the product of its center $Z(G)$ and derived group $[G, G]$ and ...
5
votes
1answer
217 views

Lie Group Multiplication in Coordinates

I'm having a bit of trouble with the last bit of Problem 3.2 in Kirillov Jr.'s Introduction to Lie Groups and Lie Algebras. (3.2) Let $f: \mathfrak{g} \rightarrow G$ be any smooth map such that ...
4
votes
1answer
63 views

Weights in the Dynkin Basis and Eigenvalues of the Cartan Generators for SU(3)?

The Cartan Generators of $SU(3)$ in the three dimensional rep have eigenvalues $(1,-1,0)$ and $\frac{1}{\sqrt{3}} (1,1,-2)$. Therefore we have the weights: $$ (1,\frac{1}{\sqrt{3}}) \quad ...
4
votes
1answer
82 views

Linear structure on the category of formal groups

Let $R$ be a commutative ring. If $R$ is a $\mathbb{Q}$-algebra, then the category of formal groups over $R$ (or the category of formal group laws) carries the structure of an $R$-linear category; ...
4
votes
1answer
89 views

Perturbation in characteristic p, or Why, really, does Lie's theorem fail?

While recalling some basics of Lie theory, I found a funny proof of the main lemma in Lie's theorem on triangularity of representations of solvable Lie algebras. It turns out that this proof has a ...
4
votes
1answer
62 views

solvable subalgebra

I want to show that a set $B\subset L$ is a maximal solvable subalgebra. With $L = \mathscr{o}(8,F)$, $F$ and algebraically closed field, and $\operatorname{char}(F)=0$ and $$B= ...
4
votes
1answer
147 views

Lie Algebras : Showing L is nilpotent if every maximal Lie subalgebra of L is an ideal.

Given a finite dimensional Lie algebra $L$, suppose that each maximal lie subalgebra of $L$ is an ideal. Suppose the adjoint map, $ad_y$ is not nilpotent. Then pick a maximal subalgebra $M$ ...
3
votes
1answer
47 views

Why can we write the weights of a representation in terms of the simple roots?

I'm currently trying to get my head around the fact that we can write the weights of any representation in terms of the simple roots of the algebra. Is there any, not too-technical, explanation? I ...
3
votes
1answer
63 views

Set of non fixed points of an automorphism

I am trying to prove the following "For an orbifold chart $ (\tilde{U},G,\phi)$ the set of non fixed point of $ g : \tilde{U} \rightarrow \tilde{U} $ where $ 1 \neq g \ \in G$ is dense in $\tilde ...
3
votes
1answer
47 views

How to obtain a Lie algebra homomorphism from a Lie group homomorphism

In class we learn a theorem tells us one can cook up a Lie algebra from a Lie group: If $f: G\to H$ is a homomorphism of a Lie group then $T_I f: T_I G\to T_I H$ is a homomorphism of Lie algebra. ...
3
votes
1answer
65 views

Cartan subalgebra of semisimple Lie algebra

My question is: How can I construct the Cartan subalgebra of a semisimple Lie algebra L which is the direct sum of simple Lie algebras, such as for example su(2)⊕su(2)⊕su(2)?
2
votes
1answer
32 views

The deconposition of $\mathfrak{so}(V \oplus V^*)$

Let $V$ be an n dimensional real vector space and $V^*$ be the dual vector space. We have a non degenerate inner product $(\centerdot,\centerdot)$ in $V\oplus V^*$ such that $(v+\xi , ...
2
votes
1answer
41 views

High Dimensional Rotation Matrices As Product of In-Plane Rotations

Lately I've been thinking a lot about how to find high-dimensional rotation matrices. In particular, can any rotation in $n$-dimensional space be represented as the product of $2$D plane rotations? ...
2
votes
1answer
57 views

The maximality of Cartan subalgebras of lie algebras

In some lecture notes of mine we define a Cartan subalgebra $\mathfrak h$ for semisimple $\mathfrak g$ as an abelian subalgebra of $\mathfrak g$ containing ad-diagonizable elements which are maximal. ...
2
votes
1answer
91 views

Different definitions of Casimir element

I read about the Casimir element just recently and I found it a bit difficult to wrap my mind around the definition(s). In fact, I have seen two different definitions. For concreteness, let ...
2
votes
1answer
58 views

Complete reducibility of a field extension of an lie algebra representation

Let $\mathfrak{g}$ be a lie algebra over a field $k$ with characterstic $0$ and $k\subset k'$ a finite field extension. Suppose $\mathfrak{g}\otimes k'$ has the property, that all finite dimensional ...
2
votes
1answer
93 views

Reducing size of ODE system by using symmetries: examples, references help request.

We know: A high order differential equation can be expressed as an ODE system. Knowledge of a symmetry allow one to reduce the order of a differential equation. So if we do $n$-order ODE ...
2
votes
1answer
112 views

With the branching rules of subalgebra, how can I write down explicit matrix elements for a representation?

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
2
votes
1answer
104 views

Equation On Root Systems (Humphreys Exercise 9.10)

I am stuck in the following problem from Humphreys. Let $\alpha, \beta$ be roots in a root system $\Phi$. Let the $\alpha$-string through $\beta$ be $\beta - r\alpha, \ldots, \beta + q\alpha$ and let ...
2
votes
1answer
253 views

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 ...
2
votes
1answer
97 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?
2
votes
1answer
209 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 ...
2
votes
1answer
131 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 ...
2
votes
1answer
143 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 ...
2
votes
1answer
115 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 ...
2
votes
1answer
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 & ...
1
vote
1answer
20 views

Every base of a root system arises as indecomposable positive roots of a regular element?

I'm confused about a line in the Theorem p48 in Humphrey's's book on Lie Algebras. He's proving that every base $\Delta$ of a root system $\Phi$ arises as the set of $\Delta(\gamma)$ of ...
1
vote
1answer
19 views

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 ...
1
vote
1answer
30 views

Structure constants for and the adjoint representation and meaning in $sl(2,F)$

First, what I know is that given the basis: $$e = \left(\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right),f = \left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}\right),h = ...
1
vote
1answer
27 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 ...
1
vote
1answer
63 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 ...
1
vote
1answer
28 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 ...
1
vote
1answer
45 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 ...
1
vote
1answer
31 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.
1
vote
1answer
61 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 ...
1
vote
1answer
30 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 ...
1
vote
1answer
62 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 ...
1
vote
1answer
23 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 ...
1
vote
1answer
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}, $ $ ...
1
vote
1answer
57 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 ...
1
vote
1answer
123 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 ...
1
vote
1answer
75 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 ...
1
vote
1answer
23 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 ...
0
votes
0answers
41 views

The simplicity of $\bigwedge^i \mathbb{C}^{n+1}$ as a representation of $\mathfrak{sl_{n+1}}$ and its weight vectors

I want to show that $\bigwedge^i \mathbb{C}^{n+1}$ is a simple representation for $\mathfrak{sl}(n+1,\mathbb{C})$ for each $1\le i \le n+1$ but I'm already stuck at determining the weight vectors. So ...
0
votes
0answers
50 views

Map algebras between scheme and Lie algebra

Kindly asking for any hints about the following questions: Suppose, $X$ be an arbitrary scheme over an algebraically closed field $k$. 1- In general, what is the structure of $A= ...
0
votes
0answers
58 views

Lie algebra homomorphism as an scheme morphism

Kindly asking for any hints about the following questions: Assume $g$ is a finite-dimensional Lie algebra. We denote the group of Lie algebra automorphisms of $g$ by $\rm Aut_k g$. Any Lie algebra ...
0
votes
0answers
79 views

a question on weyl group and its action on $\mathfrak{t}$

$\mathfrak{g}$ is a complex semisimple lie algebra which is a subalgebra of some $\mathfrak{g}l(n,\mathbb{C})$, we have chosen a compact real form $\mathfrak{l}$ of $\mathfrak{g}$ and let $K$ be the ...
0
votes
0answers
59 views

Understanding quotients $\mu^{-1}(0)/G$

This question is related to this link: Geometric difference between two actions of $GL_n(\mathbb{C})$ on $G\times \mathfrak{g}^*$ Further analyzing Scenerio 1: Let $G=GL_n(\mathbb{C})$ act on $G$ by ...