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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322 views

Abelian Cartan subalgebras

If a Lie algebra is semisimple or reductive, its Cartan subalgebras are Abelian, and their elements semisimple. Are there non-reductive algebras with Abelian Cartan subalgebras all of whose ...
11
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135 views

Why are parabolic subgroups called “parabolic” subgroups?

I used to think that things called "parabolic" must have something to do with parabolas or their defining quadratic equations. In fact, terms like parabolic coordinate, parabolic partial differential ...
9
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356 views

Campbell-Baker-Hausdorff formula for $\log(\exp(X+Y)\exp(X-Y))$

Given $X,Y\in \mathfrak g\mathfrak l_{\mathbb R}(n)$, and the CBH formula for $\log(\exp X\exp Y)$ (wiki), is it possible to derive the general term in the series of $\log(\exp(X+Y)\exp(X-Y))$ that ...
8
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115 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
7
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170 views

Why the Steinberg idempotent is idempotent?

Consider the group $GL_n(\mathbb{F}_p)$. We have the following subgroups : -$\Sigma_n$ the symmetric group (permutation matrices) -$B_n$ the Borel subgroup (upper triangular matrices) -$U_n$ the ...
6
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102 views

Are there common inequivalent definitions of Cartan subalgebra of a real Lie algebra?

I'm confused about seemingly different notions of a Cartan subalgebra of a real semisimple Lie algebra, and I'm wondering if there are common inequivalent definitions. In the book Lie Groups: Beyond ...
6
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140 views

Non-degenerate bilinear forms of Lie algebra with a degenerate Killing form

Definition: A Lie algebra is defined by: $$ [e_a,e_b]={f_{ab}}^ce_c $$ The Killing form is $$ g_{ab}=-{f_{ac}}^d {f_{bd}}^c $$ Set-Up: The type of Lie algebra of our interests (found out during a ...
6
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155 views

Generating function for characters of representations

One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ ...
6
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198 views

An exercise in Serre's Lie algebra book

Let $k$ be a commutative ring. Prove that a Lie $k$-algebra $\mathfrak{g} = 0$ iff $U\mathfrak{g} = k$. Use the adjoint representaion. Here is my attempt at it: The only non-trivial statement is ...
6
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262 views

Root Systems of Lie Groups.

Let $G$ be a compact Lie group assumed to be a subgroup of $U(n)$. Also, let $T$ be a maximal torus of G. Then there exists a basis $\{v_1, \ldots ,v_d\}$ of the Lie algebra of $G$, $\mathfrak{g}$, ...
5
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51 views

Geodesics on Homogeneous Spaces

Consider a homogeneous space $G/\text{Stab}_p \cong M$ where $G$ is a compact Lie group active transitively on $M$ (a compact manifold). If $F$ is a Finsler Metric on $G$ which pushes forward ...
5
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61 views

Is every complex Lie algebra a complexification?

I'm wondering if every finite-dimensional complex Lie algebra can be written as a complexification of a real Lie algebra. At the vector space level, clearly every $\mathbb{C}^n$ is a complexification ...
5
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53 views

Borel subalgebras inside the grassmannian

This is probably something standard and I just don't know where to look (so a reference would be just as appreciated as an answer), but... Let $\mathfrak{g}$ be a finite dimensional semisimple Lie ...
5
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81 views

duality for (co)homology of Lie algebras

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. What is the relationship between $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, ...
5
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42 views

Existence of product $\ast$ in a Lie algebra so that $[X,Y]=X*Y-Y*X$

I've been studying particle physics, and studying Lie algebra using physics text book doesn't give me enough information, so I'm asking my question here. Given a Lie algeba $\mathcal{A}$ where ...
5
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71 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
5
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104 views

— Cartan matrix for an exotic type of Lie algebra --

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra? For example, consider this Lie algebra: $$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = - f_{ik}{}^j Y^k \qquad\qquad ...
5
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81 views

Lie algebra associated to a linear form

Let $F$ be a finite dimensional vector space over a field $k$, if $f : F \to k$ is any linear form, I can define on $F$ a Lie algebra bracket by the following rule $$ [x,y]=f(x)y-f(y)x, $$ or in terms ...
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111 views

Motivation?: Lie algebra and algebraic group Cohomology

This is just an a-priori question to get a motivational heuristic idea: If an algebraic group G (more generally, G an affine group scheme), is connected over an algebraically closed base-field k. ...
5
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42 views

Does every element of $G_2 = \mathrm{Aut}(\mathbb{O})$ stabilize a quaternion subalgebra?

Has anyone heard of this result before? Let $\mathbb{O}$ denote the octonions and let $G_2$ denote its automorphism group (i.e. the 14 dimensional subgroup of $SO(7)$). Then any element of $G_2$ ...
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545 views

Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
5
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98 views

Quantum Adjoint Action of the Coordinate Algebra on the Enveloping Algebra

As is well known, any Lie group $G$ has a canonical action on its Lie algebra $\frak{g}$, namely the adjoint action $Ad$. Firstly, let me ask, does this extend to an action of $G$ on its enveloping ...
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77 views

Are there any infinite dimensional subalgebras of the Witt algebra?

The Lie bracket of elements of the Witt algebra is given by: $[L_m,L_n]=(m-n)L_{m+n}$ Are there any infinite dimensional subalgebras of the Witt algebra that are not isomorphic to the Witt algebra ...
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114 views

Pullback of a 3-form to SU2

I have a left invariant 3-form, $\sigma$ on an simply connected Lie group, $G$ whose value at the identity is $\sigma=\langle[x,y],z\rangle$, where $\langle\cdot,\cdot\rangle$ denotes an invariant ...
5
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148 views

On the root systems

Let $\Phi$ be a root system of $E$. $\alpha,\beta\in \Phi$. Let $\lbrace \beta+i\alpha | i\in \mathbb{Z}\rbrace\cap \Phi$, $\alpha$-string through $\beta$, be $\beta-r\alpha,\ldots,\beta+q\alpha$, ...
4
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25 views

Recipe to compute dimension and decompose product of $SO(N)$ group representations

As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts: There is a correspondence between irreps ...
4
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36 views

Are the two definitions of the BGG category equivalent?

Let $\mathfrak{g}$ be a finite dimensional complex semisimple lie algebra, the BGG category $\mathcal{O}$ is defined as the set of $\mathfrak{g}-$ module $M$ such that $M$ is finitely generated; ...
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110 views

Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
4
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94 views

Weyl orbits of integral dominant weights and convex polytopes

Let $\xi$ be an integral dominant weight of a root system $\Delta$, and let $\mathcal{O}_{\xi}$ be its orbit under the action of the Weyl group. The elements of the orbit are the vertices of an ...
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55 views

Weyl's Dimension Formula

I'm trying to grasp how Weyl's Dimension Formula works, and I'm having a bit of trouble. As an example, I was trying to calculate the dimension of V($\varepsilon_1$) for gl(3). First, I set the ...
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167 views

How to write $SO(2n)$ characters in terms of rotation angles?

Say one is working in a representation of $SO(2n)$ such that it has the highest weights $(h_1,...,h_n)$. And let $\{H_i\}_{i=1}^{n}$ be a basis in the Cartan of $so(2n) = Lie(SO(2n))$. Now one says ...
4
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83 views

Weyl group of a non-symmetrizable Generalized Cartan Matrix

Let $A$ be a generalized Cartan matrix on the index set $I$. Define the Weyl group of $A$ as the Coxeter group on the basis $I$ with $m(i,j)=2,3,4,6,\infty$ according to whether $A_{ij} A_{ji}$ is ...
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92 views

Cartan Subalgebra and regular elements

Let $L$ be semisimple Lie algebra, $x\in L$ semisimple. Prove that if $x$ lies in exactly one Cartan subalgebra, then $x$ is regular.
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41 views

Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
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223 views

Centralizer of an element in a Cartan subalgebra is reductive.

Let $\mathfrak{g}$ be a Lie algebra with Cartan subalgebra $\mathfrak{h}$ and root system $\Phi$. Show that $C_\mathfrak{g}(h)$ is reductive, that is $Z(C_\mathfrak{g}(h))=Rad(C_\mathfrak{g}(h))$, ...
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152 views

Irreducible representations of $\mathfrak{sl}_3\mathbb{C}$

I am working through the exercises in Fulton and Harris's Representation Theory, and am stuck on two on page 189. Let $\text{Sym}^2V$ denote the second symmetric power of the standard 3-dimensional ...
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201 views

Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with ...
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42 views

Is it possible to further simplify the product of three exponentials $e^A e^B e^C$ when $[A,C]=kB$ (k is a scalar)

The background is calculation of the little group elements of Poincare group for massless particles. I start with a bunch of exponentials of operators, and the end goal is to crunch them into the ...
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181 views

When is the adjoint representation self-dual?

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...
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432 views

Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.

I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question. To start with we work with the $\mathbb{Q}$ version of ...
4
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104 views

The Nambu bracket

Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $$\{F,G\}_c = \langle\nabla ...
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172 views

internal direct product of lie groups

If $G$ is a (edit: simply connected)Lie group, when does a direct sum decomposition of its Lie algebra (into a direct sum of subalgebras) correspond to a (semi)direct product decomposition of $G$? ...
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59 views

Proving $\mathscr L(C_G(H)) \subseteq \mathfrak c_{\mathfrak g}(\mathfrak h) = \{ \mathrm x \in \mathfrak g \mid [\mathrm x, \mathfrak h] = 0\}$

Let $H$ be a closed subgroup of the algebraic group $G$, $C = C_G(H)$. Prove that $\mathfrak{c} = \mathscr{L}(C_G(H)) \subseteq \mathfrak{c}_{\mathfrak{g}}(\mathfrak{h}) = \{ \mathrm{x} \in ...
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104 views

Why are root systems presented in this confusing way?

I quote Bjorner and Brenti, "Combinatorics of Coxeter Groups." We begin with a simple geometric lemma. Let $m \geq 3$ be an integer, let $\gamma = \pi/m$, and let $k, k'$ be real numbers ...
4
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339 views

Elementary proof of the third Lie theorem

I would like to understand a (not well known) proof due to G.M; Tuynman, of the third Lie theorem, which asserts that for any given finite dimensional Lie algebra $\mathcal{G}$ there exists a (simply ...
3
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0answers
18 views

Weight spaces of a irreducible representation of $\mathfrak{gl}(n, \mathbb{C})$.

Let $\mathfrak{gl}(n,\mathbb{C})$ be the general linear Lie algebra. Let $\{E_{s,t}\}_{1\leq s,t,\leq n}$ be the standard basis for it. And set its Cartan subalgebra $\mathfrak{h}$ to be ...
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33 views

Which elements of $su(n)$ commute with those of a subalgebra $su(2)$

Given a subalgebra $su(2) \subset su(n)$ , how many generators of $su(n)$ commute with any element in the subalgebra $su(2)$? I know that there are at least $n-2$ elements in $su(n)$ satisfying this ...
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27 views

Center of the dual of a Lie algebra

Let $\mathfrak{g}$ be a Lie algebra. Let $C \subset \mathfrak{g}^*$ be the subspace of linear forms that vanish on Lie brackets: $$C = \{\alpha \in \mathfrak{g}^*, ~ [\mathfrak{g}, \mathfrak{g}] ...
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63 views

When did the notion of the tangent space emerge?

When did the modern conception of/notation for the tangent space to a manifold come into use?
3
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56 views

If a Lie algebra L decomposes as a direct sum of its derived subalgebra and its center, is L reductive?

A Lie algebra $L$ is said to be reductive if for any ideal $\mathfrak{a}$ of $L$, there is an ideal $\mathfrak{b}$ of $L$ such that $L=\mathfrak{a}\oplus\mathfrak{b}$. It is known that a reductive ...