**1**

vote

**1**answer

117 views

### Is the endpoint map smooth

Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE:
$\frac{d U_t}{dt} = (a + w(t)b)U_t$
consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...

**2**

votes

**0**answers

95 views

### Symmetric and antisymmetric powers of SU(2) representations [closed]

Recently, I took a course in representation theory at Imperial College, and on the first homework the questions were about certain sneaky relationships when it came to representations of SU(2).
...

**0**

votes

**1**answer

55 views

### Centralizer of the derived algebra in a non-perfect Lie algebra

Is there a non-perfect Lie algebra for which the centralizer of the derived algebra is trivial?

**0**

votes

**0**answers

137 views

### Reference request: the formula $\langle x, [f, g] \rangle = \langle \delta(x), f \otimes g \rangle$

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}^*$ the dual vector space of $\mathfrak{g}$ which is also a Lie algebra with natural brackets. Let $\delta: U(\mathfrak{g}) \to U(\mathfrak{g}) \...

**4**

votes

**1**answer

107 views

### Characterizations of Jacobson-Morozov parabolics associated to a nilpotent

Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a Jacobson-Morozov parabolic ...

**0**

votes

**1**answer

102 views

### Dimension of Span of Adjoint orbit in $\mathfrak{su}(n)$

Given two elements $A,B \in \mathfrak{su}(n)$ what is the dimension of the span of the following adjoint orbit: $\{Ad_{e^{sA}}(B) \ | \ s \in [0,t]\}$ for different values of $t$. Does it ever change ...

**11**

votes

**6**answers

950 views

### Representation Theory of Lie Groups: Reference Request

I am looking for a reference that describes the correspondence between the (finite-dimensional) representations of (real) Lie groups and the representations of their Lie algebras. More precisely, ...

**1**

vote

**0**answers

60 views

### Largest dimensional Lie subgroup of $SU(N)$ [duplicate]

What is the largest (Lie) subgroup of $SU(n)$ in the sense of its dimension.
I am aware of this potential duplicate subgroup of SU(N) with maximal manifold dimension , however, the title of this ...

**0**

votes

**1**answer

91 views

### About Kahler curvature operator

I have problems on how to consider the Kahler curvature operator. I know that one can consider the Riemannian curvature operator $R$ as a linear transformation from $\mathfrak{so}(n,\mathbb{R})$ to $\...

**3**

votes

**0**answers

136 views

### scalar multiple of Young symmetrizer

The following is a lemma from Fulton and Harris' book -Representation theory,a first course (page 53):
Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar \;...

**2**

votes

**0**answers

105 views

### Contraction of the maximal submodule in a Verma module

Suppose $\mathfrak{g}$ is a real semisimple Lie algebra, $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ is a Cartan decomposition, and $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{k}$. ...

**6**

votes

**0**answers

160 views

### What is the importance of the number $k+h^{∨}$ (level+dual Coxeter number)?

The number $k+h^{∨}$ appears at many places in the representation theory of affine Lie algebras (and probably elsewhere). Here $h^{∨}$ is the dual Coxeter number of the root system, and $k$ is the ...

**2**

votes

**1**answer

125 views

### Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$

Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which:
$K \left(A, Ad_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the ...

**7**

votes

**0**answers

167 views

### How to show the compatibility between Duflo isomorphism and Harish-Chandra isomorphism for semi-simple Lie algebras?

I was told that the Duflo isomorphism is compatible with the Harish-Chandra isomorphism when the Lie algebra $\mathfrak{g}$ is semi-simple. However I cannot see why this is true. All I can show is ...

**0**

votes

**1**answer

155 views

### Existence of a Lie algebra element orthogonal to the adjoint orbit of another element

Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Denote the Killing form by $K$.
Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) ...

**1**

vote

**1**answer

154 views

### Generating function for $t$-residues of partitions using Heisenberg + $\hat{sl_t}$ representation theory

Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let $(n_0,\...

**2**

votes

**1**answer

179 views

### Understanding representations of affine Lie algebras

Please reference this paper for notation in this question.
I'm trying to understand two claims made in the above paper (they may be related). First, in the construction of $\mathcal{H}_\lambda$ on ...

**1**

vote

**0**answers

45 views

### Reference for using an algebra of meromorphic functions to extend a Lie algebra

For example, let $\mathfrak{g}=\mathfrak{sl}_{2}\left(\mathbb{C}\right)$, let $s_{0}=1$, $s_{1}=-1$, $s_{2}$=0, $s_{3}=\infty$ in $\mathbb{P}_{1}\left(\mathbb{C}\right)$ and $\mathcal{R}$ is the ...

**3**

votes

**1**answer

139 views

### Interpreting Frobenius pullback as an invariant differential in the case of an elliptic curve

Let $S$ be an affine scheme of characteristic $p > 0$, let $E \rightarrow S$ be an elliptic curve over $S$, and let $F$ denote the absolute Frobenius. Since $E$ is its own $\mathrm{Pic}^0$ there is ...

**1**

vote

**1**answer

268 views

### A subalgebra of the Virasoro algebra

Let $L_n$ ($n\in\mathbb{Z}$) and $c$ be the standard generators of the Virasoro algebra ${\rm Vit}$. In the literature one usually considers the involutive authomorphism given by $\tau(L_n)=-L_{-n}$, $...

**3**

votes

**2**answers

264 views

### Solvable Lie algebras: embedded in upper triangular matrices?

Let $K$ be an arbitrary field and $\mathfrak{g}$ a finite-dimensional Lie $K$-algebra.
Let $\mathfrak{nil}_n\leq\mathfrak{sol}_n\leq\mathfrak{gl}_n$ be the Lie algebras of all ((strictly) upper-...

**0**

votes

**0**answers

117 views

### A solvable Lie algebra

Consider the Lie algebra $\mathfrak{g}$ span by $x_1,\ldots,x_n,y_1,\ldots,y_n$ with the commutator
$\left[y_i,x_i\right]=y_i, \left[x_i,x_{i+1}\right]=y_i, \left[y_i,x_{i+1}\right]=-y_i,\quad 1\leq ...

**2**

votes

**0**answers

124 views

### $l$-weights and $l$-character of finite-dimensional highest $l$-weight representation of $L\mathfrak{g}$

I am trying to solve the following problem, which is related to relatively recent results, but I am not sure how to do it.
Problem
In this problem, $\mathfrak{g}=\mathfrak{sl}_{2}$. We study finite-...

**0**

votes

**1**answer

132 views

### Equivalence of Lie subalgebras, within a (irreducible) representation

Lie subalgebras inside simple Lie algebras (of type ABCDEFG) have been classified up to equivalence, and linear equivalence (by Dynkin et al). How does one classify embeddings of a Lie algebra h ...

**0**

votes

**1**answer

113 views

### Minimal dimension of a Lie algebra of matrices, with a restrictive property

Let $\mathfrak{g}$ be a sub-Lie-algebra of $\mathfrak{gl}_n(\mathbb{C})$, the Lie algebra of complex $n\times n$ square matrices.
Let us call $(H)$ the hypothesis: for all $x, y\in\mathbb{C}^n$, ...

**2**

votes

**3**answers

564 views

### Root in positive Weyl chamber

Let $\mathfrak{g}$ be a complex simple Lie algebra. We fix a Cartan subalgebra $\mathfrak{t}\subset \mathfrak{g} $. Let $R\subset \mathfrak{t}^*$ the set of roots. We fix $\Pi\subset R$ the set of ...

**6**

votes

**2**answers

244 views

### Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation?

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and $\phi_{2}:\mathfrak{sl}_2(\mathbb{C})\rightarrow\...

**3**

votes

**1**answer

144 views

### Prescribed spherical representations, symplectic group $Sp(n)$

An irreducible representation $(\pi,V_\pi)$ of a compact group $G$ is called spherical with respect to the pair $(G,K)$, $K$ is closed subgroup of $G$, if $V_\pi$ has a non-zero vector invariant by $K$...

**18**

votes

**1**answer

298 views

### On a drawing in Dixmier's Enveloping Algebras

This image
comes from Dixmier's book, 'Enveloping Algebras' ('Algèbres enveloppantes').
Dixmier writes that
The curves shown on p. XIV have their origin in the study of U(sl(3)).
They are ...

**4**

votes

**1**answer

224 views

### When do two non-degenerate quadratic forms give rise to isomorphic Lie algebras?

Let $V$ be a vector space over some number field $k$. (I'm fine with $\mathbb{Q}$.)
Let $\phi \colon V \to k$ be a non-degenerate quadratic form. Associated with $\phi$ is the orthogonal group $\...

**1**

vote

**0**answers

152 views

### How the exceptional simple Lie groups/ algebras were first discovered and by whom?

I am wondering whether exceptional simple Lie groups/ algebras were first discovered in order to obtain a complete list of such objects, or they appeared as answers to completely different questions.
...

**2**

votes

**0**answers

136 views

### Kernel of the Weil homomorphism for compact symmetric spaces

Let $X = G/K$ be a Riemannian symmetric space of compact type and consider the "Weil homomorphism" $$w^\bullet: H^\bullet(BK; \mathbb R) \to H^\bullet(X; \mathbb R),$$ i.e. the map in cohomology ...

**2**

votes

**3**answers

410 views

### First Explicit Irreducible Representations

Although the classification of simple Lie Algebras and their representations is fully understood, I wonder whether there is some book with exhaustive tables describing explicit irreducible ...

**0**

votes

**0**answers

118 views

### Reference about a formula of coroot in an affine root system

Let $\delta$ be the null of an affine root system and let $\alpha + p\delta$ be a real affine root, $p$ is an integer. It is said that
$$
(\alpha + p\delta)^{\vee} = \alpha^{\vee} + \frac{2p}{(\alpha,...

**0**

votes

**0**answers

70 views

### Cayley graphs for finitely presented Lie algebras

I have seen that an important tool of finitely presented groups consists in writing down its Cayley graph with respect to a given set of generators, and then try to extract data like the coarse ...

**2**

votes

**1**answer

121 views

### Bounds on Hilbert-Schmidt norm of difference of products of matrices

I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in).
I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and $R_{1},\ldots,R_{k}...

**6**

votes

**0**answers

136 views

### vanishing of Lie algebra cohomology with coefficients in an infinite-dimensional module

Let $G$ be a real semisimple Lie group, $K$ its maximal compact subgroup, $\mathfrak g, \mathfrak k$ the corresponding Lie algebras. Let $V$ be a locally convex, Hausdorff vector space, which is a $G$-...

**2**

votes

**0**answers

116 views

### Exploiting a finite presentation of a Lie algebra

In "my youth" I computed a finite presenation of the Poisson algebra on $S^2$ (finite presentation as a Lie algebra).
In what ways might this be useful? Does this allow you to extract information ...

**0**

votes

**1**answer

233 views

### generalization of highest weight theorem for semisimple lie algebras

Let $\mathfrak g$ be a real semisimple Lie algebra (without compact factors) with Iwasawa decomposition
$\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak u$.
Let $\mathfrak p$ be a
...

**3**

votes

**1**answer

225 views

### The Jordan Plane and Enveloping Algebras

Let $k$ denote a field of characteristic $0$ (assume algebraically closed for convenience). Define $J=k\langle x,y|[x,y]=y^{2}\rangle$. This noncommutative algebra (which can be viewed as a derivation ...

**4**

votes

**1**answer

659 views

### Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...

**1**

vote

**0**answers

64 views

### How to define the determinant of a morphism between graded Lie algebras?

I have the following question. Suppose $g_1$ and $g_2$ are two finite dimensional, nilpotent, stratified Lie algebras and $A:g_1\to g_2$ is a morphism of the graded Lie algebra. I wonder whether there ...

**5**

votes

**0**answers

105 views

### How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...

**-1**

votes

**1**answer

178 views

### Reductive space & Reductive Lie algebra

If $M=G/H$ is a reductive space and $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ be the canonical decomposition, then are $\mathfrak{g}$ or $\mathfrak{h}$ or both reductive lie algebras? (in this case, ...

**2**

votes

**0**answers

163 views

### How to write BRST-BV for dg-Lie?

The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc.
Where is there written a corresponding formula incorporating the differential of
a dg Lie algebra and module?

**2**

votes

**1**answer

165 views

### Explicit generators of $Z(U(\mathfrak{g}))$

Let $\mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field. By Harish-Chandra, the center of its universal enveloping algebra $Z(U(\mathfrak{g}))$ is a polynomial ring and the ...

**2**

votes

**0**answers

159 views

### Free-field representations: how to study highest-weight submodules of the Fock module?

Suppose we have a representation of some affine Lie algebra $\mathfrak{g}=\mathfrak{n}_- \oplus \mathfrak{h} \oplus \mathfrak{n}_+$ on a Fock space $V$. The module $V$ will contain a lot of highest-...

**3**

votes

**1**answer

372 views

### Is $M=E_{7(7)}/SU(7)\times\mathbb{R}^{+}$ a (pseudo)Kähler-Hodge manifold? Open problem

I have been told that the following is an open problem in mathematics, but I am pretty sure that experts in the topic surely know the answer.
Is the manifold
$$M=\frac{E_{7(7)}}{SU(7)}\times \mathbb{...

**3**

votes

**0**answers

93 views

### Homology of derivations of Differential Graded Lie algebra

Let $(L,d)$ be a Differential Graded Lie Algebra ($L=\bigoplus L_i$ and $d:L_i \to L_{i-1}$ satisfying the graded Leibniz rule).
On the algebra $\mathrm{Der}L$ of derivations of $L$ define a grading ...

**6**

votes

**1**answer

344 views

### Torsion in profinite groups

Is there a finitely generated profinite group $G$ with a closed subgroup of infinite index $K \leq G$ such that for every $g \in G$ there exists some $n \in \mathbb{N}$ for which $g^n \in K$ ?
Can $G$...