**0**

votes

**0**answers

31 views

### A question on an set of 8 matrices related to the SU(3) generators

SU(2) and SU(3) differ quite a bit.
The Lie algebra of SU(2) formed by the three generators g_n is the same as the algebra formed by the SU(2) matrices/elements F_n=exp (pi * i * g_n / 2). In fact, ...

**7**

votes

**0**answers

77 views

### One identity in Lie algebras

Let $L$ be a (non-restricted) Lie algebra over a field of prime characteristic $p,$ $UL$ be its universal enveloping algebra and $a_1,\dots, a_p \in L$ (the number of elements is equal to the ...

**4**

votes

**2**answers

197 views

### $A \wedge A \wedge A$ in Chern-Simons

I am confused with the wedging operations of Lie algebra valued differential forms. Especially, for instance, I have some problems with the Chern-Simons 3-form
$$A \wedge dA + \frac{2}{3}A \wedge A ...

**3**

votes

**1**answer

189 views

### Earliest source for a Lie algebra construction

I am looking for the earliest reference to the fact that any associative algebra becomes a Lie algebra with bracket $AXB-BXA$, where $X$ is a fixed element of the algebra. This is observed in the ...

**2**

votes

**0**answers

61 views

### A different Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$

In this question $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$ is the space of all smooth vector fields on the plane or sphere. A limit cycle for a vector field $X$ is an isolated closed ...

**3**

votes

**0**answers

93 views

### n-homology of a Harish-Chandra module

Let $G$ be a connected real reductive Lie group and let $K$ be its maximal compact subgroup.
Let $P=MAN$ a parabolic subgroup. Let $K_M^0=M^0\cap K$ be connected component of the maximal compact ...

**0**

votes

**1**answer

105 views

### Generalization of the Lie group exponential map and its derivative

Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$, and $exp:\mathfrak{g}\to G$ be its exponential map. The group $G$ could be finite or infinite dimensional. Let $G$ have the property that
...

**3**

votes

**2**answers

112 views

### About supersolvable Lie algebras

A colleague of mine asked me the question below, and since I could not answer it, I thought I might have more luck on MO.
In Encyclopedia of Mathematics, a finite dimensional Lie algebra $L$ over a ...

**1**

vote

**1**answer

84 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 ...

**1**

vote

**0**answers

54 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

44 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

117 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}) ...

**3**

votes

**0**answers

35 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

**0**answers

31 views

### Motivation for a representation of the Witt algebra

I'm reading through these notes and I'm confused by the definition of a
representation of the Witt algebra on page 6.
Precisely, the sentence
This is the representation that one would discover by ...

**1**

vote

**1**answer

70 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 ...

**0**

votes

**0**answers

30 views

### Maximal nilpotent substructures

In finite group theory of solvable groups maximal nilpotent subgroups are important. These are Carter subgroups, nilpotent injectors, nilpotent projectors and Fischer-subgroups.
In Lie-theory there ...

**8**

votes

**5**answers

501 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**

votes

**0**answers

29 views

### Generating pairs of $\mathfrak{su}(N)$ and $\mathfrak{so}(N)$ Lie algebras - characterization

It is known that any simple Lie algebra can be generated by two elements (Kuranishi paper). Is it know how to characterize ALL generating pairs for $\mathfrak{su}(N)$ and $\mathfrak{so}(N)$ algebras?

**1**

vote

**0**answers

57 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

70 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

91 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

73 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 $\mathfrak{k}$. ...

**5**

votes

**0**answers

71 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 ...

**0**

votes

**0**answers

43 views

### Centralizer of a non-regular Lie algebra element

It is well understood that the centralizer of a regular element $A$ of a Lie algebra of complex (square, diagonalizable) matrices consists of polynomials $p(A)$ in that element of degree less than $n$ ...

**2**

votes

**1**answer

107 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

113 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

134 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

113 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 ...

**2**

votes

**1**answer

110 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

39 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

123 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

213 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

148 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) ...

**0**

votes

**0**answers

95 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

109 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 ...

**0**

votes

**1**answer

119 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

105 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 $x, y\in\mathbb{C}^n$, ...

**1**

vote

**3**answers

285 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

211 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 ...

**3**

votes

**1**answer

107 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 ...

**17**

votes

**1**answer

261 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

203 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

141 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

126 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 ...

**1**

vote

**0**answers

83 views

### Irreducible decomposition of $\Lambda^i(\mathfrak{p})$

Let $G$ be a connected semisimple real Lie group with finite center with Lie algebra $\mathfrak{g}$. Take $K$ its maximal compact subgroup of $G$, and $\mathfrak{k}$ its Lie algebra. We denote by
...

**2**

votes

**3**answers

386 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

89 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} + ...

**0**

votes

**0**answers

65 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

74 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 ...

**6**

votes

**0**answers

116 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 ...