For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

learn more… | top users | synonyms (1)

8
votes
1answer
402 views

Osp, USp, SU(,) and PSU

I would be glad if someone can give me some (hopefully easy to understand!) references for learning about these groups Osp, USp and PSU and their representations. I run into these mostly while ...
4
votes
1answer
127 views

How to compute the Gel'fand Models for a (quantum) Lie Algebra

Given a lie algebra $g$, how does one approach finding the Gel'fand models? For clarity, by this I mean $\bigoplus_{\lambda\in P^+}V(\lambda)$ where $P^+$ are the dominant weights, and ...
1
vote
1answer
191 views

Finding roots for a Lie algebra g, wrt toral subalgebra h

I'm trying to find the root space decomposition of a lie algebra wrt a toral subalgebra h. Both a matrix lie algebras. I'm confused about how do I find the linear forms $\lambda \in \mathfrak{h}^*$ ...
8
votes
3answers
2k views

How do you find the Lie algebra of a Lie group (in practice)?

Given a Lie group, how are you meant to find its Lie algebra? The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them? My group is the set ...
1
vote
0answers
131 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 $X \in gl(n,\mathbb{C})$ ...
4
votes
1answer
1k views

Representation theory of $SO(n)$

This is probably not a very ethical question to ask but I need to have a fast introduction to a range of concepts about the representation theory of the $SO(n)$ and I would be happy to see some online ...
1
vote
2answers
99 views

$A \otimes_k A \to \bigwedge^2 A$ homs

Let $k$ be a unital commutative ring, and A be a $k$-module. Is there a homomorphism $f: A \otimes_k A \to \bigwedge^2 A$ such that $f(a \otimes a) \neq 0$ for some $a \in A$? I can take a hint :) ...
1
vote
2answers
114 views

An equality about characters of representations

This is an equality that I am gleaning out of some papers that I have been reading. I am not sure I am reading it right. Hopefully people will correct it. Let $U$ be a group element and $R$ be a ...
3
votes
3answers
386 views

Reasoning about Lie theory and the Exponential Map

I'm having a little difficulty wrapping my head around Lie theory (I'm a computer scientist, so perhaps that's to be expected). Specifically, considering the following definition from Wikipedia for ...
4
votes
1answer
283 views

does every adjoint orbit of a Lie group go through the Cartan subalgebra?

A naive question from a physicist, so forgive the lack of rigor. Consider a Lie group, acting on its Lie algebra by the adjoint action. Does every orbit go through the Cartan subalgebra? ...
6
votes
1answer
408 views

Relationship between Riemannian Exponential Map and Lie Exponential Map

It is well known that for a matrix Lie group the Lie exponential map is $e ^z$. This maps a tangent vector $z$ at the identity to a group element. On the other hand the general Riemannian ...
2
votes
0answers
74 views

orbit of a Dynkin diagram automorphism

Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$. What is a set of representatives of the orbits of $\Delta$ under $f$ ? Thanks,
2
votes
3answers
355 views

How to differentiate a homomorphism between two Lie groups

Let $G$ and $H$ be two Lie groups and $\rho: G \to H$ be a homomorphism. How to differentiate $\rho$ to obtain a Lie algebra homomorphism $d\rho_e: T_eG \to T_eH$?
4
votes
1answer
202 views

what is the usual topology on a vector space?

I do not understand the topology of a Lie group clearly. Let $G$ be a Lie group and $T_eG$ be its tangent space at the identity $e \in G$. Why $Aut(T_eG)$ is an open subset of the vector space of ...
1
vote
0answers
335 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 ...
12
votes
2answers
378 views

Why are knot invariants best organized as polynomials?

Does anyone have a good explanation for why Knot invariants tend to be well organized as polynomials? What exactly is going on and why don't we often see polynomial invariants for classifying other ...
1
vote
1answer
390 views

Complexifying representations

Let me try to split the question in a few parts, I would like to understand the claim that all non-degenerate bilinear symmetric forms are equivalent over the complex while for the reals they can be ...
6
votes
2answers
449 views

sl(2,C) and the harmonic oscillator

I've been studying the finite-dimensional representations of the lie algebra sl(2,C). I've read that these representations are related to the harmonic oscillator and the associated raising and ...
3
votes
1answer
242 views

Synthetic differential geometry

I'm reading Serre's "Lie algebra and Lie groups" now, and I found a description of Lie algebra of an algebraic matrix group, given for a free algebra (free as a module) $k'$ with a basis $\lbrace ...
6
votes
1answer
412 views

On the relationship between the commutators of a Lie group and its Lie algebra

I was trying to teach myself some basic Lie theory, and I came across this statement on Mathworld, relating the commutator of a group, $\alpha\beta\alpha^{-1}\beta^{-1}$, to the commutator of its Lie ...
0
votes
1answer
246 views

Lie algebra of the bounded continuous functions

I can think of the set of bounded, continuous functions from $\mathbb R \to \mathbb R$ as a group, with composition as addition of functions. In other words, this group has the rule that the ...
28
votes
4answers
1k views

“Cayley's theorem” for Lie algebras?

Groups can be defined abstractly as sets with a binary operation satisfying certain identities, or concretely as a collection of permutations of a set. Cayley's theorem ensures that these two ...
6
votes
3answers
452 views

Lie algebras and infinitesimals

I have seen at many places the notions that Lie Algebras are infinitesimal objects and they look really close at a point. But I never understood this. They are abstract algebraic objects different ...
2
votes
2answers
262 views

Books on Lie Groups via nonstandard analysis?

Are there any books or online notes that cover the basics of lie groups using nonstandard analysis? Another thing I would like is a to see these things set in category theory (along the lines of ...
3
votes
1answer
583 views

Why do we use the commutator bracket for Lie algebra's

We define Lie algebras abstractly as algebras whose multiplication satisfies anti-commutativity and Jacobi's Identity. A particular instance of this is an associative algebra equipped with the ...