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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Discrete nonabelian free subgroups of semisimple Lie groups

I understand that the following is a theorem: If $G$ is a noncompact connected semisimple Lie group, then $G$ contains a discrete nonabelian free subgroup. I can find proofs that such a $G$ contains a ...
Iian Smythe's user avatar
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Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$

Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group). By the Tannakian formalism, $G(k)$ can be ...
Antoine Labelle's user avatar
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Equivalence of first two Lie theorems [closed]

Can you prove the subgroups subalgebras correspondence knowing the homomorphism theorem?
vertvolt's user avatar
1 vote
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An interesting identity involving skew-Schur functions

Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9) \begin{align*} \prod_{k\geq1}(1+...
T. Amdeberhan's user avatar
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39 views

Action of Hopf algebra of identity supported distributions on a Lie group

The Hopf algebra of identity supported distributions on a lie group is cocommutative. It is well known that it is a group object in the category of cocommutative coalgebras. Is there a canonical ...
Lefevres's user avatar
1 vote
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47 views

How can we construct a non-trivial central extension of a Lie group

Let $G$ be a connected and simply connected Lie group with its Lie algebra $\mathfrak{g}$. Assume that $[c]\in H^2 (\mathfrak{g};\mathbb{R})$ is a non-trivial 2-cocycle. Then we can construct a non-...
Mahtab's user avatar
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Friedrich Schur on the BCHD theorem (notes in English)

According to Sternberg in his book Lie Algebras, The formula [the BCHD formula] is named after three mathematicians, Campbell, Baker, and Hausdorff. But this is a misnomer. Substantially earlier than ...
Tom Copeland's user avatar
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1 vote
1 answer
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Derivatives and ODEs on Lie groups

I'll keep the question specific to the scenario I'm working with, which is the Lie group SO(3) and its Lie algebra so(3). Consider two rotation matrices $R_0$ and $R_1$, and define $R_t = \exp_{R_1} ((...
CComp's user avatar
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2 votes
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Whitehead lemma for simplicial Lie algebras

Let $R \to R'$ be a morphism of connected free simplicial Lie algebras. Then the analog of the Whitehead lemma states that if $\pi_* f_{\mathrm{ab}}$ is an isomorphism then $\pi_* f$ is an isomorphism....
thrw's user avatar
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3 answers
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Where do root systems arise in mathematics?

One often hears that root systems are ubiquitous in mathematics and physics. The most obvious occurrence of root systems is in the classification of complex simple Lie algebras. Where else do they ...
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Simple highest weight modules of quantum affine algebras

Let $U_q(\hat{\mathfrak g})$ be a quantum affine algebra, and let $L$ be an integrable simple highest weight module of $U_q(\hat{\mathfrak g})$. In [Lu], Lusztig proved that the limit of $L$ when $q\...
qweqwe's user avatar
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Second cohomology group of the contact Lie algebra $K_3$

Let $F$ be a field of characteristic zero and, for all $n>0$, consider the contact Lie algebra $K_{2n+1}$. It follows from Corollary 3 of the paper [V. Guillemin - S. Shnider: Some stable results ...
Rocky Smith's user avatar
4 votes
3 answers
260 views

Does every nilpotent lie in the span of simple root vectors?

Let $G$ be a reductive group (for simplicity, we can work over $\mathbb{C}$). Let $N \in \operatorname{Lie} G$ be nilpotent. Does there exist a Borel pair $(B,T)$ of $G$ such that $N$ lies in the span ...
Alexander's user avatar
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4 votes
1 answer
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Multiplication factors in folding root systems and Lie algebras by automorphisms

When Stembridge, in the paper Folding by automorphisms, considers folding by automorphism $\sigma$ he considers the root system generated by for each orbit $J$. $$\sum_{i \in J} \alpha_i .$$ Whereas ...
Smith's user avatar
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Extension of a type A Springer fibre

Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{...
Filip's user avatar
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5 votes
1 answer
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Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations

Let $\mathfrak{sp_n}$ be the symplectic Lie algebra, that is, the $C_n$ complex simple Lie algebra. Is it true that the first fundamental, which is to say the vector space, representation $V_1$ of $\...
Zoltan Fleishman's user avatar
2 votes
1 answer
137 views

Trivial representation of a maximal torus

Let G be a connected compact Lie group and $T\subset G$ a maximal torus. For an irreducible representation $V_\lambda$ of $G$, the multiplicity of the trivial representation of $T$ in $V_\lambda$ is ...
Local's user avatar
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3 votes
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108 views

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
2 votes
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39 views

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
7 votes
0 answers
193 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
170 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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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
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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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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
109 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
104 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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2 votes
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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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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
120 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
0 answers
170 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
0 answers
71 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
1 vote
1 answer
267 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
193 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
300 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
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9 votes
0 answers
347 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
330 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
80 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
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5 votes
0 answers
154 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
148 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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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
106 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
106 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
245 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
130 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
135 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
93 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
123 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
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