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 ...

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4
votes
1answer
84 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
0answers
39 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 ...
0
votes
1answer
86 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
0answers
37 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
5answers
536 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
0answers
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
1answer
77 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
0answers
98 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
0answers
82 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
0answers
90 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
0answers
52 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
1answer
113 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
0answers
131 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
1answer
136 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
1answer
122 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
1answer
124 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
0answers
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
1answer
130 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
1answer
232 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
2answers
173 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
0answers
107 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
0answers
111 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
1answer
125 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
1answer
107 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
3answers
331 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
2answers
221 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
1answer
115 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
1answer
274 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
1answer
212 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
0answers
143 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
0answers
128 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
0answers
90 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
3answers
392 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
0answers
95 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
0answers
67 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
1answer
82 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
0answers
123 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 ...
2
votes
0answers
105 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
1answer
192 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
1answer
194 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 ...
3
votes
1answer
542 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
0answers
52 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
0answers
80 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
1answer
168 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
0answers
143 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?
1
vote
1answer
145 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
0answers
139 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 ...
3
votes
1answer
363 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 ...
3
votes
0answers
68 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
1answer
319 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 ...