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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In how many ways does a Lie algebra decompose as an orthogonal direct sum of Cartans?

For a prime $p$, the Lie algebra $\mathfrak{su}(p)$ can be decomposed into an orthogonal direct sum of $p+1$ Cartan subalgebras as follows. Consider the clock and shift matrices — these are a pair of ...
Theo Johnson-Freyd's user avatar
1 vote
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An infinitely generated Lie algebra, its finitely generated envelope

If a Lie algebra $\mathfrak g$ is finitely generated, its enveloing algebra $U\mathfrak g$ is finitely generated as an associative algebra. In fact, taking the enveloping algebra of the surjection $\...
Qwert Otto's user avatar
6 votes
0 answers
171 views

Chevalley-Solomon formula and Weyl character formula

Let $\Phi\subset V$ be a root system of rank $r$ with Weyl group $W$, a choice of positive roots $\Phi_+$ and exponents $d_1, \ldots, d_r$ (i.e. the invariant algebra $(\operatorname{Sym}^\bullet V)^W$...
Antoine Labelle's user avatar
6 votes
1 answer
158 views

Does every Lie algebra appear as centralizer of an element in a semisimple Lie algebra?

Given a finite dimensional, complex, semisimple (fcss) Lie algebra $\mathfrak{g}$ and an element $x\in\mathfrak{g}$, denote by $\mathfrak{g}^x$ the centralizer of $x$ in $\mathfrak{g}$ i.e. the set $\{...
Hugo MTV's user avatar
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4 votes
1 answer
126 views

Which Lie groups admit finite generation by a set of Lie algebra elements? And what are some known choices of generators which realize this?

Consider a (finite-dimensional) real connected Lie group $G$ with Lie algebra $\frak{g}$. Take a generating set $\mathcal{G} = \{ X_1, \cdots X_n \} $ of $\frak{g}$, i.e. such that any element of $\...
Another User's user avatar
17 votes
0 answers
1k views

Relations in a certain Lie algebra

Let ${\mathfrak g}$ be the (real) Lie algebra generated by infinitely many generators $D_i, E_i$ with $i=1,2,3,\dots$ subject to the following relations for any natural numbers $i,j$: \begin{gather*} [...
Terry Tao's user avatar
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1 vote
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59 views

Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?

$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$. Let us consider a Schwartz space $\mathcal{S}$ defined as \begin{equation} \mathcal{S}:= \Bigl\{ \...
Isaac's user avatar
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3 votes
0 answers
99 views

Lie algebra cohomology of formal non-commutative vector fields

Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
Qwert Otto's user avatar
3 votes
0 answers
97 views

Nilpotent orbits in characteristic $0$ vs. positive characteristics

Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group ...
Dr. Evil's user avatar
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Techniques for computing integrals on $G/K$

Edit: I originally tried writing a simpler question, but to make more clear what I want I wrote the general question below Let $G$ be a compact group with compact subgroup $K\subseteq G$ (everything ...
freeRmodule's user avatar
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240 views

Any "inherent" definition of $\mathrm{SU}(2)$ independent of any matrix representation?

$\DeclareMathOperator\SU{SU}\SU(2)$ is explained in detail here. However, if I know right, this definition itself is known the "fundamental representation". I wonder if there is any "...
Isaac's user avatar
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1 vote
0 answers
110 views

Representation theory of reductive lie algebras

I have studied Lie algebras at the level of Humphrey's Introduction to Lie Algebras and Representation Theory. This only really includes representation theory of semisimple lie algebras. In the ...
Smith's user avatar
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2 votes
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166 views

Nonlinear Poisson brackets associated with nilpotent (matrix) Lie algebras?

With every finite-dimensional Lie algebra $\mathfrak{g}$ one can associate a linear Poisson bracket on $\mathfrak{g}^\ast$. With some more restrictions on $\mathfrak{g}$ and some extra ingredients, ...
Ricardo Buring's user avatar
2 votes
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66 views

Quantum Schubert cell algebra and quantum odd-dimensional euclidean space

De Concini, Kac, Procesi introduced quantum Schubert cell algebra associated to a complex Lie algebra $\mathfrak{g}$ which is denoted by $\mathcal{U}^{w}_{\epsilon}$ where $w$ is an element of Weyl ...
snehashis mukherjee's user avatar
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1 answer
259 views

Chevalley restriction theorem

$\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\Sym{Sym}$I'm having a hard time understanding the proof of Chevalley's restriction theorem given by Humphreys in "Introduction to Lie Algebras and ...
Trinity-Slifer 's user avatar
5 votes
2 answers
183 views

Enveloping algebra of affine Lie algebra is (not) noetherian

I work over an algebraically closed field of characteristic $0$. Let $\mathfrak{g}$ be a semisimple Lie algebra, $\hat{\mathfrak{g}}=\mathfrak{g}[t,t^{-1}]\oplus\mathbb{C}K\oplus\mathbb{C}D$ the ...
Estwald's user avatar
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7 votes
1 answer
291 views

Do you know a survey of modular Lie algebras and its representations?

When I was an university student, I liked reading some books about the representation theory of finite groups or Lie algebras and I was interested in explicit constructions of irreducible ...
Ozzie's user avatar
  • 71
9 votes
0 answers
340 views

How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?

$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
user509184's user avatar
5 votes
0 answers
327 views

Have you seen this Lie algebra?

Computing something I have come across a Lie algebra $\def\L{\mathfrak L}\def\CC{\mathbb C}\L_N$ that I would like to identify. Fix an integer $N$ such that $N\geq2$, let $\L_N$ be the free complex ...
Mariano Suárez-Álvarez's user avatar
1 vote
0 answers
79 views

Representation theory for symmetries of probability distribution functions

I would like to parameterize all the possible modifications to a probability density function. Is there a representation theory for this? Something along the lines of, these are all the operators $L$ ...
Alex's user avatar
  • 119
5 votes
0 answers
151 views

Are there known minimal models for the cohomology of semisimple Lie algebras?

My student and I recently found a cute construction of a minimal model for the cohomology of a Lie algebra $\mathfrak{g}$. This is a "minimal model" in the sense that it is a minimal chain-...
user509184's user avatar
3 votes
1 answer
142 views

Symmetric tensor of highest weight modules for $\mathrm{SU}(2)$

Let $V_i$ be the $(i+1)$-dimensional representation of the special unitary group $\mathrm{SU}(2)$ with the highest weight $i$. Is there any uniform way to compute the irreducible decomposition for the ...
Hebe's user avatar
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1 vote
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212 views

Is the span of all nilpotent ideals also a nilpotent ideal?

Given a non-zero Lie algebra $\mathcal{L}$ over $\mathbb{C}$, we define $\mathcal{L}^2 = \big[\mathcal{L}, \mathcal{L} \big] = \big\{ [x, y]: x, y\in \mathcal{L} \big\}$, and for any $k\in\mathbb{N}$ ...
Sanae Kochiya's user avatar
5 votes
0 answers
105 views

On Soergel's results concerning projectives modules in category $\mathcal{O}$

I am looking for a translated proof of two results of Soergel usually referred to as endomorphismensatz and struktursatz. Both of those results were shown in the paper Soergel, W. (1990). Kategorie 𝒪...
alerouxlapierre's user avatar
6 votes
1 answer
1k views

If two Lie algebras are isomorphic, under which conditions will their Lie groups also be isomorphic?

Let $G$ and $G'$ be compact connected Lie groups (which are not necessarily simply connected) with Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$. Suppose that the two Lie algebras are isomorphic, ...
user32415's user avatar
0 votes
1 answer
104 views

Conjugacy of Cartan subgroups in $\mathrm{GL}(n)$

$\DeclareMathOperator\SL{SL}$I have probably a very basic question on the structure of semisimple Lie groups. Sorry if it is too elementary. Let either $G=\SL(n,\mathbb{R})$ or $G=\SL(n,\mathbb{C})$. ...
asv's user avatar
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2 votes
1 answer
241 views

Reasons about the difference between twisted affine algebras of $A_{2l}$ and other types

I am reading about Kac–Moody algebras. I have a question about the construction of twisted affine algebras. Let $\sigma$ be a graph automorphism of the simple Lie algebra $\mathfrak{g}$. When $\...
fusheng's user avatar
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2 votes
1 answer
235 views

Non-example to PBW theorem

I am interested in a (simple) example of an associative algebra $A$ with 1 generated by $x_1, \ldots, x_n$ which is quadratic (i.e. all relations between $x_1, \ldots, x_n$ have degree $\leqslant 2$) ...
Alex-omsk's user avatar
0 votes
0 answers
29 views

Hilbert representation with compactness and *-property

Given a continuous representation $\pi$ of a Lie group $G$ on a Hilbert space $H$, not unitary. Suppose that it is a $\star$-representation for the Lie algebra, i.e., $\pi(X)^\star=\pi(-X)$ on the ...
user avatar
2 votes
1 answer
127 views

Solvability of derivation Lie algebras of local finite-dimensional commutative algebras

Let $A$ be a finite-dimensional local commutative algebra (with one) over a characteristic zero field $k$. Is it true that the Lie algebra $\operatorname{Der}_k(A)$ of $k$-derivations of $A$ is ...
მამუკა ჯიბლაძე's user avatar
4 votes
0 answers
134 views

Relation between two Harish-Chandra homomorphisms

Let $\mathfrak{g}$ be a Lie algebra admitting a triangular decomposition $\frak g = N^-\oplus h\oplus N^+$ and $\gamma$ the classic Harish-Chandra isomorphism defined on the center $\frak Z$ of the ...
S. D. Z's user avatar
  • 141
1 vote
0 answers
92 views

Explicit central elements of $\mathcal{U}(\mathfrak{so}(4,1))$

I am interested in finding the central elements of the universal enveloping algebra of the Lie algebra $\mathfrak{so}(4,1)$. Notation: the 10 generators are $D, J_i, P_i, K_i$ ($i=1,2,3$), satisfying ...
Edward Lilley's user avatar
2 votes
0 answers
116 views

What is a Gelfand-Tsetlin subalgebra?

For context on general Gelfand-Tsetlin theory, see for instance this MO post. Let's work over $\mathbb{C}$. Fix $n>0$. There is a natural chain of embeddings of the general linear Lie algebras $\...
jg1896's user avatar
  • 2,683
0 votes
1 answer
157 views

What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?

Let $\mathfrak g$ be a Lie algebra. $\mathfrak g^{(1)}=\mathfrak g$ and $\mathfrak g^{(n+1)}=[\mathfrak g,\mathfrak g^{(n)}]=\mathbb R$-span$\{[X,Y]:X\in\mathfrak g,Y\in\mathfrak g^{(n)}\}$. The ...
enihcamemit's user avatar
2 votes
0 answers
119 views

Classification of restricted Lie algebras of reductive groups

$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
Martin Ortiz's user avatar
7 votes
1 answer
297 views

Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?

In this popular 2019 MO question, user მამუკა ჯიბლაძე asked: The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...
Alvaro Martinez's user avatar
0 votes
0 answers
100 views

Basis of Lambda algebra for a programmer

First of all, I'm not a specialist in alg. top., but I try to apply computational math to it, so if I'm wrong in something you'd be doing a better thing explaining it to me instead of blaming me :) ...
Dmitry Vilensky's user avatar
7 votes
0 answers
674 views

What's the point of geometric representation theory?

Please forgive the provocative title, what I mean is the following: One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
Béla Fürdőház 's user avatar
3 votes
1 answer
192 views

Relation between enveloping algebras and algebras of differential operators

I asked this question on math stack exchange about 3 years ago, but received no answer. Our base field $\mathsf{k}$ will be algebraically closed of zero characteristic. Let $X$ be an smooth affine ...
jg1896's user avatar
  • 2,683
2 votes
1 answer
196 views

Wedderburn–Artin like theorem for infinite dimensional Lie algebras?

The Wedderburn–Artin Theorem is one of the cornerstones of the structure theory of (associative) rings. Wedderburn–Artin Theorem : Let $R$ be a left Artinian ring with zero Jacobson radical. Then $R$ ...
jg1896's user avatar
  • 2,683
2 votes
1 answer
97 views

Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces

For the special linear algebra $\frak{sl}_{m}$ which finite dimensional irreducible representations $V_{\mu}$ have non-trivial zero weight spaces? For $\frak{sl}_2$ this is clear: $V_{2k\pi}$ for $\pi$...
Béla Fürdőház 's user avatar
1 vote
0 answers
78 views

Generators of simple Lie groups and finite word length

Let $G$ be a connected simple Lie group with finite center. Let $a=\mathrm{exp}(X)$ be a semisimple element. Then we can decompose the lie algebra of $G$ into the direct sum of the eigenspaces of $\...
Van's user avatar
  • 31
3 votes
0 answers
75 views

Why do most eigenspaces of a Lie algebra automorphism have finitely many orbits?

I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper on the Local Langlands Conjectures (omitting the "well-known" proof). Suppose $G$ is a ...
David Schwein's user avatar
0 votes
0 answers
39 views

Relation between real forms of Lie algebras and root systems on pseudoeuclidean vector spaces

This might be trivial but I cannot see it clearly. Simple complex Lie algebras are fully classified by the root systems arising from the Cartan subalgebra for which the Euclidean norm is the Cartan-...
Dac0's user avatar
  • 285
2 votes
0 answers
57 views

graded reps of Lie algebras literature

I am currently studying 'advanced' representation theory from a physicist's perspective, including topics like super-Lie algebras. I've come across various gradings (excluding the ℤ2 grading), such as ...
iron's user avatar
  • 21
0 votes
0 answers
167 views

What does the set of all fundamental coweights look like?

Let $\Phi$ be an irreducible root system in a Euclidean vector space $V$. Let $W$ denote its Weyl group. Choose a base $\Delta=\{\alpha_1,...,\alpha_r\}$ for $\Phi$. Then $\Delta$ is a basis for $V$. ...
Dr. Evil's user avatar
  • 2,641
2 votes
0 answers
106 views

Is there an explicit description of a gauge transformation $F$ such that $U_{\hbar}(\mathfrak{g})$ and $(U(\mathfrak{g})[[\hbar]])_F$ are isomorphic?

Let $\mathfrak{g}$ be a semisimple Lie algebra, let $t$ be its canonical 2-tensor, and let $\Phi_{KZ}$ be a Drinfeld associator.When $R_{KZ}=e^{\hbar t/2}$, $(U(\mathfrak{g})[[\hbar]],\Phi_{KZ},R_{KZ})...
yohei ohta's user avatar
3 votes
2 answers
715 views

The adjoint representation of a Lie group

Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
rr314's user avatar
  • 131
2 votes
0 answers
85 views

Category O for (Yangian) toroidal Lie algebras?

Suppose throughout that $g$ is a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let us denote: $$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$ $$g_{[2]}^+ := g \...
Dat Minh Ha's user avatar
  • 1,472
1 vote
0 answers
57 views

A certain subalgebra of $\mathfrak{e}_8$ over a p-adic field

Can $\mathfrak{e}_8(\mathrm{k})$ have a maximal subalgebra isomorphic to $\mathfrak{sl}_1(\mathrm{D})\oplus\mathfrak{g}_2(\mathrm{K})$, where $\mathrm{k}$ is a finite extension of some $\mathbb{Q}_p$, ...
Daniel Sebald's user avatar

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