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)

12
votes
0answers
166 views

Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
12
votes
0answers
345 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
votes
0answers
102 views

Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
11
votes
0answers
150 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 ...
8
votes
0answers
121 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
votes
0answers
169 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
7
votes
0answers
119 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 ...
7
votes
0answers
185 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 ...
7
votes
0answers
164 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 ...
7
votes
0answers
129 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. ...
7
votes
0answers
176 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
votes
0answers
80 views

A subspace is invariant by the Lie group if it is invariant by the Lie algebra

Let $G$ be a connected Lie group and $$\varphi:G\to \mathrm{GL}(V)$$ a representation on a finite dimensional real vector space $V$. Let $$\psi:\mathfrak{g}\to\mathrm{End}(V)$$ be the associated Lie ...
6
votes
0answers
83 views

Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and ...
6
votes
0answers
69 views

Solving a matrix differential equation

I am trying to solve: $\frac{d U_t}{dt} = Tr(G^{\dagger}U_t)G - Tr(U_t^{\dagger}G)U_t G^{\dagger} U_t$ Where $U_t \in SU(4)$ and $G \in SU(4)$ is given and constant. Is it possible to solve this ...
6
votes
0answers
728 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 ...
6
votes
0answers
215 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
votes
0answers
280 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
votes
0answers
60 views

How to calculate the Lie algebra of a neural network?

Define $F$ as the standard multi-layer feed-forward perceptron: \begin{equation} F(\mathbf{x}) = \Theta( W_1 \circ \Theta( W_2 \circ .... W_L(\mathbf{x}))) \end{equation} where $\Theta$ is the sigmoid ...
5
votes
0answers
86 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
votes
0answers
69 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
votes
0answers
56 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
votes
0answers
94 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
votes
0answers
47 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
votes
0answers
127 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
votes
0answers
84 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 ...
5
votes
0answers
43 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 $G_2$ ...
5
votes
0answers
165 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 ...
5
votes
0answers
106 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 ...
5
votes
0answers
79 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 ...
5
votes
0answers
119 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
votes
0answers
160 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
votes
0answers
41 views

Open problems in Lie theory

I been studying lie theory for some time. Beside classification related problems what are some examples of open problems in the lie world? Especifically in the topological/differentiable structure of ...
4
votes
0answers
41 views

Multiplicities in weight diagram of representations of $\mathfrak{sl}(3,\mathbb{C})$

In the weight diagram of an irreducible (finite dimensional, complex) representation of $\mathfrak{sl}(3,\mathbb{C})$, there are 'rings' of weights in the shapes of triangles or hexagons. Is there an ...
4
votes
0answers
43 views

No local optima in quantum control?

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation: $\frac{d x(t)}{dt} = F(x,u)$ one can consider the ...
4
votes
0answers
21 views

Representation of a Kac-Moody algebra

Let $n$ be an integer $\geq 3$ and let $\mathfrak{g}$ be the Kac-Moody algebra with cartan matrix $C$ given by $C_{ij} = 2 \delta_{i,j} - \delta_{i,j+1} - \delta_{i,j-1} - \delta_{|i-j|,n-2}$. For ...
4
votes
0answers
62 views

Baker Campbell Hausdorff formula for unbounded operators

Baker Campbell Hausdorff formula says that for elements $X,Y$ of a Lie algebra we have $$e^Xe^Y=\exp(X+Y+\frac12[X,Y]+...),$$ which for $[X,Y]$ being central reduces to ...
4
votes
0answers
53 views

Does a choice of measure on $\mathfrak{g}$ induce a measure on $G$?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. One can put a (left) Haar measure $\mu$ on $G$ and a Lebesgue measure $\lambda$ on $\mathfrak{g}$ which are both unique up to constants. My ...
4
votes
0answers
52 views

Basic application of Weyl-Character-Formula

(I did not find a solution of my problem in any forum so far. Sorry if it exists...) I am new to Lie-Algebras and representations and actually do not need the mathematical background... I need only ...
4
votes
0answers
51 views

Ideals of Lie-algebras

I am wondering whether the following claim is true: Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra and $V$ some vector subspace of $\mathfrak{g}$. Claim: $V$ is an ideal of $\mathfrak{g}$ ...
4
votes
0answers
34 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
votes
0answers
154 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) ...
4
votes
0answers
46 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; ...
4
votes
0answers
185 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
votes
0answers
110 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 ...
4
votes
0answers
64 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 ...
4
votes
0answers
96 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 ...
4
votes
0answers
111 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.
4
votes
0answers
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 ...
4
votes
0answers
256 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 ...
4
votes
0answers
46 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 ...