# 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.

2,182
questions

1
vote

0
answers

34
views

### 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 ...

3
votes

1
answer

67
views

### 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 ...

0
votes

0
answers

37
views

### Equivalence of first two Lie theorems [closed]

Can you prove the subgroups subalgebras correspondence knowing the homomorphism theorem?

1
vote

0
answers

77
views

### 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+...

0
votes

0
answers

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 ...

1
vote

0
answers

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-...

-3
votes

0
answers

118
views

### 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 ...

1
vote

1
answer

158
views

### 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} ((...

2
votes

0
answers

96
views

### 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....

18
votes

3
answers

2k
views

### 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 ...

1
vote

0
answers

37
views

### 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\...

4
votes

1
answer

108
views

### 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 ...

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 ...

4
votes

1
answer

208
views

### 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 ...

1
vote

0
answers

59
views

### 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_{...

5
votes

1
answer

99
views

### 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 $\...

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 ...

3
votes

0
answers

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 ...

2
votes

0
answers

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 $\...

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 $(\operatorname{Sym}^\bullet V)^W$...

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 $\{...

4
votes

1
answer

139
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 $\...

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*}
[...

1
vote

0
answers

61
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\{ \...

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 ...

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 ...

2
votes

0
answers

89
views

### 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 ...

0
votes

0
answers

244
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 "...

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 ...

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, ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 $L$ ...

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-...

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 ...

1
vote

0
answers

213
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 $k\in\mathbb{N}$ ...

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...

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, ...

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 $G=\SL(n,\mathbb{C})$. ...

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 $\...

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 $\leqslant 2$) ...

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 ...

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 ...

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 ...

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 ...

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 $\...