Questions tagged [lie-algebras]

Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.

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2 votes
1 answer
159 views

Lie algebras for which all one-dimensional extensions split

I was recently trying to prove the following "well-known" theorem for myself, given that I could not find a proof in the literature that I could understand. In what that follows, I will ...
3 votes
2 answers
110 views

When do there exist $ n $ commuting, derivations which locally generate $ T_{A/k} $ as an $ A $-module?

Let $ \operatorname{Spec}(A) $ be a non-singular, $ n $-dimensional, affine variety over a field $ k $ of arbitrary characteristic. For the definition of "variety" I use Hartshorne's ...
2 votes
1 answer
137 views

Difference between two definitions of affine Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, we have the notion of affinization of $\mathfrak{g}$, which is the central extension of the corresponding loop algebra. ...
3 votes
0 answers
101 views

Modules over the unipotent subalgebra as direct summands of modules over a semisimple Lie algebra

Let $\mathfrak g$ be a semisimple finite-dimensional Lie algebra over the field of complex numbers $\mathbb C$. Let $\mathfrak n\subset\mathfrak g$ be the maximal unipotent subalgebra of $\mathfrak g$...
1 vote
0 answers
68 views

Carleman approximation for functions from $\mathbb R$ to (closed convex subset of) a Lie algebra

I am looking for an approximation result dealing with continuous functions of a real parameter with values in (some subset of) the unitary algebra. However, I wouldn't be surprised if the following ...
5 votes
1 answer
196 views

Step in the Bruhat decomposition for reductive Lie groups

Err, not research but if anyone has read this part of Knapp's book recently, I'd be obliged if they could help me out. Also posted on MSE. I'm stuck on a line in the proof of Theorem 7.40 in Knapp's '...
7 votes
1 answer
261 views

Non(skew)commutative Lie algebras?

The Lie operad $\text{Lie}$ is generated by a binary operator $[\ ,\ ]$, modulo a degree two relation (skew commutativity $[x,y]=-[y,x]$) and a degree three relation (Jacobi $[x,[y,z]]+[y,[z,x]]+[z,[x,...
6 votes
0 answers
246 views

Tits construction of algebraic groups of type D₆ and E₇ via C₃

As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
2 votes
0 answers
79 views

Complex semisimple Lie algebra modules with non-semisimple Cartan action

Let $\frak{g}$ be a complex semisimple Lie algebra. I would like to know about infinite-dimensional representations $M$ of $\frak{g}$ for which the Cartan $\frak{h} \subseteq \frak{g}$ does not act ...
1 vote
0 answers
106 views

Fiberwise exponential map for vector bundle automorphisms

Let $p:E \to B$ be a smooth vector bundle of rank $n$ over a manifold $B$ and we identify $B$ with the image of the corresponding zero section. For $b\in B$ denote by $E_b = p^{-1}(b)$ the fiber over $...
4 votes
0 answers
109 views

Why are all "non-swinging" representations self-dual?

Let $\mathfrak{g}$ be a semisimple (say complex) Lie algebra, and $V$ an irreducible finite-dimensional representation of $\mathfrak{g}$. Denote by $w_0$ the longest element of the Weyl group, i.e. ...
9 votes
1 answer
426 views

$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$

While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...
1 vote
1 answer
97 views

Problem in understanding a fact about Belavin-Drinfeld triple

A Belavin-Drinfeld triple associated to a simple Lie algebra $L$ is a triple $(\Gamma_1, \Gamma_2, \tau)$ where $\Gamma_1, \Gamma_2 \subseteq \Gamma$ ($\Gamma$ is a set of simple roots or fundamental ...
7 votes
1 answer
520 views

Infinite dimensional representations of $\frak{sl}_2$

The finite-dimensional representations of a complex semisimple Lie algebra $\frak{g}$ are well known to be classifiable by their highest weight vectors, giving a convenient countable indexing set. I ...
1 vote
1 answer
126 views

About certain elements in the zero weight space of an irreducible representation of the complex simple Lie algebra of type G$_2$

$\newcommand{\fg}{\mathfrak g}\newcommand{\ee}{\varepsilon}$Let $\fg$ be the complex simple Lie algebra of type G$_2$. We consider its root system as follows (though it is probably not necessary to ...
0 votes
0 answers
76 views

Methods for calculating (one-parameter subgroup) actions

For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form \begin{equation} \mathrm{e}^{t L(z)} f(z) \end{equation} ...
3 votes
1 answer
207 views

Cartan subspace of graded Lie algebras

Suppose $\mathfrak{g}$ is a complex reductive Lie algebra and $\theta$ is an automorphism of order $2$. Let $\mathfrak{g} = \mathfrak{g_0} \oplus \mathfrak{g}_1$ be the corresponding $\mathbb Z_2$-...
6 votes
0 answers
337 views

Asymptotically nilpotent Lie sets of matrices

A matrix $A\in\textbf{Mat}_n(\mathbb{R})$ is called asymptotically nilpotent if for each vector $v$, ${\lim}_{k\to\infty}A^k(v) = 0$. Question 1. Assume that $\mathcal{A}$ is the subset of $\textbf{...
3 votes
0 answers
130 views

Is there a classical version of Yetter-Drinfeld modules?

One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$. If we think ...
7 votes
0 answers
181 views

Combinatorial bases of simple Lie algebras

The Lie algebra $\mathfrak{so}_3$ has a basis $x_1,x_2,x_3$ with the multiplication table $[x_1,x_2]=x_3$, $[x_2,x_3]=x_1$, $[x_3,x_1]=x_2$. Moreover there is an isomorphism $\mathfrak{so}_3(\mathbb C)...
4 votes
2 answers
1k views

Sum of all root lengths in simple Lie algebra

Part of one of my calculations involves (the innocent looking) expression $\sum_{\alpha\in\Sigma} (\alpha,\alpha)$ for simple Lie algebras. I have two methods of calculating it -- which don't agree. ...
0 votes
1 answer
117 views

Sub-coroot lattices

[This is a sequel to the previous question sub-coroot systems, that has been answered! :-) ] Let $T$ be a maximal torus of a compact Lie group $K$, and let $\Lambda \subset {\mathfrak t}$ be the ...
3 votes
1 answer
150 views

Sub-coroot systems

Let $T$ be a maximal torus of a compact Lie group $K$, and let $\Psi \subset {\mathfrak t}$ be the (finite) set of coroots for $(K,T)$, where $\mathfrak t$ is the Lie algebra of $T$. Assume now that $...
6 votes
4 answers
422 views

The dimension of the Grassmannian cohomology ring $H^*(\mathrm{Gr}_{n,d})$ and the fundamental $\frak{sl}_n$-representation $V_{\pi_d}$

The vector space dimension of the cohomology group of the $2$-plane Grassmannian $\mathrm{Gr}_{2,n}$ is given by the number of tuples $(\lambda_1,\lambda_2)$ satisfying $$ n - 2 \geq \lambda_1 \geq \...
19 votes
2 answers
2k views

Dual versions of "folding" symmetric ADE Dynkin diagrams?

Start with the Dynkin diagram of an irreducible root system, typically associated with a simple Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE diagrams ...
1 vote
0 answers
791 views

Differential and pre-differential of Jacobi identity

Let M be a manifold. To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied? That is a Lie algebra structure for which $[X,fY]=f[X,Y]$. (For every ...
3 votes
1 answer
250 views

Decomposition of tensor powers of the vector representation of $\frak{sl}_n$

Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as $$ V(\pi_1) \...
1 vote
1 answer
197 views

Lie algebras and pulled back group schemes

Suppose I have an extension of fields $L/K$, a group scheme $G_K$ over $\operatorname {Spec} K$. Let $G_L$ denote the pullback of $G_K$ to $\operatorname{Spec} L$. Then, under what conditions on the ...
1 vote
1 answer
187 views

Tensoring irreducible representations corresponding to root lattice elements

Let $\frak{g}$ be a complex semisimple Lie algebra with root lattice $Q$ and positive weight space $P^+$. Let $\lambda, \mu \in Q \cap P^+$, with corresponding respective fin-dim irreducible ...
0 votes
1 answer
109 views

Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules

Let $\lambda \in \mathcal{P}^+$ be a dominant weight for $\frak{sl}(n,\mathbb{C})$. When does it hold that the zero weight space, of the associated finite-dimensional $L(\lambda)$, is non-trivial? ...
4 votes
1 answer
232 views

Number of representations of a semisimple Lie algebra of any given dimension

For a semisimple complex Lie algebra $\frak{g}$ it is well known that irreducible finite-dimensional representation are not characterised by their dimension. More formally, let us define an ...
2 votes
0 answers
66 views

Identifying the adjoint representation of an affine Kac-Moody algebra

We work over $\mathbb{C}$. Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra (the question is still relevant for the non-affine case, but I'm specifically interested in the affine case). Suppose ...
9 votes
2 answers
7k views

How to find Casimir operators?

Given a general Lie algebra, is there a general procedure to find all its Casimir operator?
2 votes
1 answer
146 views

Casimir operators for Jacobi Lie algebra

I am looking for a complete description (if possible with proof) of the Casimir operators (i.e., generators and relations for the center of the universal enveloping algebra) for the Jacobi Lie algebra ...
5 votes
1 answer
1k views

The Casimir invariant of an irreducible representation of a compact Lie group

Let $G$ be a compact Lie group (not necessarily connected) and $\rho:G\to \mathrm{End}(V)$ an irreducible (hence finite-dimensional) unitary representation of $G$. Let $\mathfrak{g}$ be the Lie ...
2 votes
1 answer
213 views

What do the Pauli matrices say about the Threefold Way?

The Pauli matrices $$\sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_2=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_3=\begin{pmatrix} 1 & 0 \\ 0 & -1 \...
1 vote
0 answers
129 views

A question about Araki's 1962 paper on classification of irreducible symmetric spaces

I am looking at Sôhô Araki's 1962 paper for the classification of real semisimple lie algebras. Here's the link to the paper: On root systems and an infinitesimal classification of irreducible ...
3 votes
0 answers
138 views

Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation $$ G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) . $$ Basic ...
3 votes
0 answers
158 views

Explicit formula for the inverse of Harish-Chandra map in case $\mathfrak{g}=\mathfrak{gl}_n$

Let $\mathfrak{g}$ be a complex reductive Lie algebra (however, we are mainly interested in the case $\mathfrak{g}=\mathfrak{gl}_n$) and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. It ...
2 votes
1 answer
104 views

Proof of restrictableness of Lie algebra without basis

$\DeclareMathOperator\ad{ad}$According to the Strade and Farnsteiners' textbook, a Lie algebra $L$ is restrictable if and only if $\ad(L)$ is $p$-subalgebra of $\operatorname{Der}(L)$. Also, the ...
0 votes
1 answer
1k views

The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)

Original question (without additional information from Wendy): Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way: Taking the E8 as {128,...
2 votes
1 answer
69 views

How to define inverse of a non-degenerate triangular structure?

Let $\mathfrak {g}$ be a non-degenerate triangular Lie bialgebra with the non-degenerate triangular structure $r \in \bigwedge^2 \mathfrak {g}.$ Then how does it induce $r^{-1} \in \bigwedge^2 \...
17 votes
2 answers
833 views

Counting degrees of freedom in Lie algebra structure constants (aka why are there any nontrivial Lie algebras of dim >5?)

This is a question about the true number of constraints imposed by the Jacobi identity on the structure constants of a Lie algebra. For an $n$-dimensional Lie algebra, there are $\frac{n^2(n-1)}{2}$ ...
0 votes
0 answers
65 views

Why is $\Pi_r^L$ a non-degenerate Poisson structure on $G\ $?

Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a non-degenrate triangular structure on $\mathfrak {g}$ i.e. $r$...
9 votes
1 answer
214 views

Yang-Mills algebra and lower central series of surface groups

Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area. First, in "...
2 votes
1 answer
232 views

Characterization of reductive Klein geometries

In my struggle to understand Cartan/Klein geometries, I have the intuition that reductive Klein geometries are the link to connect the "classical" differential geometry approach with this &...
0 votes
0 answers
46 views

Left translations respect the Schouten bracket

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and $r \in \bigwedge^2 \mathfrak{g}$. For $x \in G$ let $\lambda_x$ denote the left multiplication by $x$. Let $[\cdot, \cdot]$ ...
18 votes
0 answers
977 views

"Special" meanders

One of the open problems in combinatorics is enumeration of meanders. Here on MO I only could find them under the heading not-especially-famous-long-open-problems-which-anyone-can-understand. Since my ...
4 votes
1 answer
254 views

Complexification of a Lie subalgebra of a compact real form

I'm currently reading the paper Lie algebra Cohomology and the Generalized Borel–Weil theorem written by B. Kostant, and I have a question about Remark 3.9 he made. In this paper, $\mathfrak{g}$ is a ...
3 votes
1 answer
128 views

Does every nilpotent orbit have an element supported on the simple root spaces?

Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and $\mathfrak{g}$ its Lie algebra. Let $O \subset \mathfrak{g}$ be an orbit of a nilpotent element. Let $\Pi = \{\alpha_1, \dots ,\...

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