All Questions
82 questions
5
votes
0
answers
147
views
Tensor product of a Verma module of the highest weight and a Verma module of the lowest weight, $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$
$\DeclareMathOperator\sl{\mathfrak{sl}}\newcommand\hw{\mathrm{hw}}\newcommand\lw{\mathrm{lw}}$Consider $\mathfrak{g}=\sl_2(\mathbb{C})$. Fix $\lambda,\mu\in\mathbb{C}\setminus \frac{1}{2}\mathbb{Z}_{\...
- 51
16
votes
0
answers
188
views
Representation theory of Pin groups
I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point.
Proctor describes this rule for $m = 2$...
- 1,574
1
vote
1
answer
114
views
Source for highest weight vectors for $\text{SL}_n(\mathbb{C})$ representations
I have a list of a lot of irreducible $\text{SL}_n(\mathbb{C})$ representations and need to know what all of their highest weight vectors are. For example, using the notation from Fulton and Harris, I ...
- 181
0
votes
0
answers
117
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+...
- 43.2k
5
votes
0
answers
146
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...
- 211
4
votes
0
answers
158
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 ...
- 141
2
votes
0
answers
102
views
Category O for (Yangian) toroidal Lie algebras?
Suppose throughout that $g$ is a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let us denote:
$$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$
$$g_{[2]}^+ := g \...
- 1,516
3
votes
1
answer
140
views
Asymptotics of Haar moments on general Lie groups
I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are ...
- 179
10
votes
1
answer
399
views
Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — Reference request
It is well known that the principal block $\mathcal{O}_0$ of the BGG category $\mathcal{O}$ of a semisimple Lie algebra is equivalent to the category of finitely generated modules over a certain ...
- 103
5
votes
1
answer
792
views
Constructing a Kac-Moody group as a quotient of the free product of its root subgroups
The paper "Regular Functions on Certain Infinite-dimensional Groups" by Kac and Peterson describes the construction of a group associated to the datum of a Kac-Moody algebra in a way I haven'...
- 53
2
votes
0
answers
108
views
Invariants of Lie superalgebras
Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
- 673
0
votes
1
answer
187
views
Matrix representations of Lie groups of type $B_n$
For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices:
\begin{align}
\left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\...
- 6,201
7
votes
0
answers
171
views
$\mathfrak{sl}_2(\mathbb{Z})$'s properties as a Lie algebra over a ring
I was wondering what was known about $\mathfrak{sl}_2(\mathbb{Z})$ as a Lie algebra; in particular, what is known about its representation theory? I know of some texts which treat Lie algebras ...
- 71
11
votes
1
answer
617
views
Is there a unique "natural" action of $\mathsf{SL}_{n+1}$ on $\mathbb{R}^n$?
Context
By acting naturally via $\mathsf{SL}_3$ on $\mathbb{RP}^2=\{[x:y:z]\}$ and by taking the induced action on the affine hyperplane $z=1$ (which we identify with $\mathbb{R}^2$), one can realize ...
- 2,527
7
votes
3
answers
599
views
Root system of fixed point Lie sub-algebra
It is known that a non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can ...
- 103
9
votes
1
answer
355
views
Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$
Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
- 239
6
votes
2
answers
331
views
Lie powers of a graded vector space and Klyachko's theorem
Let $V$ be a $\mathbb{Z}_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}_2$-graded Lie algebra (aka super Lie algebra). The free super Lie algebra is also graded by ...
- 21.5k
5
votes
1
answer
346
views
Restricting representations to a principal $\mathfrak{sl}(2)$
Let $\mathfrak{g}$ be a semi-simple Lie algebra over $\mathbb{C}$ with simply connected group $G$ and suppose that
$$\mathfrak{g} = \bigoplus_i\mathfrak{g}_i$$ is a $\mathbb{Z}$- or $\mathbb{Z}/n\...
- 2,516
2
votes
0
answers
91
views
Simple modules for universal enveloping algebras and Weyl algebras
Let $A$ be the universal enveloping algebra of a fintie dimensional Lie algebra (simple if needed) or the Weyl algebra.
Question: Are there recent survey articles about the (possibly infinite ...
- 26.5k
3
votes
1
answer
158
views
Twisted affine Lie algebras, Lie bracket and normalized standard invariant form
I am reading the book: Infinite-Dimensional Lie Algebras (Kac, third edition) and the article: Affine Lie algebras and the Virasoro algebras I (Wakimoto, link). The formulas they wrote for the Lie ...
- 320
5
votes
1
answer
283
views
Finite order automorpisms of affine Kac-Moody Lie algebras
It is known that for a finite order automorphism $\phi$ of a complex semisimple Lie algebra $L$, the fixed point subalgebra $L^{\phi}$ is a reductive Lie algebra and the centralizer of a Cartan ...
- 59
1
vote
0
answers
46
views
What is the maximal weight submodule of $\text{Hom}_{\mathfrak{g}}(M,N)$?
Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. For ...
- 507
4
votes
1
answer
436
views
Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie algebra: a reference request
Let $G$ be a (connected) semisimple algebraic group over an algebraically closed field $k$ of characteristic 0.
We consider the adjoint representation
$$ {\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$
...
- 14.2k
4
votes
0
answers
123
views
Bijections of Littlewood-Richardson coefficients
Let $c^{\lambda}_{\mu\nu}$ be the Littlewood-Richardson coefficients, where $\lambda,\mu,\nu$ are partitions. We know that $c^{\lambda}_{\mu\nu}= c^{\lambda}_{\nu\mu}$. Up to now, what are the ...
- 320
8
votes
1
answer
591
views
History of the study of Verma modules in terms of Kazhdan Lusztig Theory
Let $\mathfrak{g}$ be a complex finite dimensional semisimple Lie algbera, $W$ be the Weyl group, $\rho$ be the half sum of positive roots, $M(\eta)$ be the Verma module of weight $\eta$ and $L(\eta)$ ...
- 1,875
4
votes
1
answer
302
views
On maximal closed connected subgroups of a compact connected semisimple Lie group?
Let $G$ be a compact connected semisimple Lie group and let $\mathfrak g$ denote its Lie algebra.
Is the following result true? Does it follows directly from Dynkin's classification of maximal Lie ...
- 2,446
5
votes
2
answers
964
views
Weight spaces of representations of finite dimensional simple Lie algebras
This question has probably been asked before on this website, but I could not find any solution and neither can I solve this question. So again I am asking the following question:
Let $\mathfrak{g}$ ...
- 153
6
votes
3
answers
772
views
Existence of a weight of a representation in the fundamental Weyl chamber
Let $\mathfrak g$ be a complex simple Lie algebra.
Fix a Cartan subalgebra $\mathfrak h$ of $\mathfrak g$, let $\Delta$ denote the corresponding root system.
Pick a partial order on $\mathfrak h$, ...
- 2,446
5
votes
0
answers
218
views
Lusztig's completion for universal enveloping algebra
In Arkhipov, Bezrukavnikov and Ginzburg's paper "Quantum Groups, the loop Grassmannian and the Springer resolution", they mentioned that Lusztig introduced a certain completion for universal ...
- 677
6
votes
0
answers
179
views
Tensoring Harish-Chandra bimodules with Verma modules
The question is about the functor $T_\lambda$ defined by Bernstein and Gelfand in the paper Tensor Products of Finite and Infinite Dimensional
Representations of Semisimple Lie Algebras.
Setup: Let $\...
- 335
3
votes
1
answer
274
views
References for representations of Heisenberg Lie algebra
Please suggest some reference material for the representations of the infinite dimensional Heisenberg Lie Algebra or the oscillator algebra. I already looked at Kac and Rainas book, any other ...
- 427
9
votes
3
answers
576
views
Reference Request: Structure constants for G2
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...
- 6,180
14
votes
1
answer
544
views
Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$?
This might be forced to migrate to math.SE but let me still risk it.
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism ...
- 18.4k
13
votes
1
answer
753
views
Tilting Objects in BGG Categories $\mathcal{O}$
Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this book by ...
- 295
8
votes
0
answers
411
views
Which representations of the Lie algebra of a Lie group come from representations of the group itself?
I think this is very classic mathematics, but I can't find a complete answer in the literature.
Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
- 1,586
30
votes
1
answer
2k
views
Is there an accessible exposition of Gelfand-Tsetlin theory?
I'm hoping to start an undergraduate on a project that involves understanding a bit of Gelfand-Tsetlin theory, and have been tearing my hair out looking for a good reference for them to look at. ...
- 44.7k
6
votes
1
answer
169
views
Existence of a real eigenvalue is a necessary condition for the density of all the orbits of a Lie subgroup of $GL(\mathbb{R},d)$
Good morning,
I would like to pose the following (maybe naive) question. Let $\mathfrak{a}\subset \mathfrak{gl}(\mathbb{R},d)$ be any lie subalgebra, and $A$ be the connected, simply connected ...
user17697
5
votes
3
answers
849
views
Weyl's Branching Rule for $SU(N)$-Setting
On the Wikipedia page for restricted representations
https://en.wikipedia.org/wiki/Restricted_representation
there is presented a number of explicit "branching rules". In particular, there is the ...
- 835
3
votes
0
answers
97
views
Reference Request: Branching Rules of $\mathfrak{s}\mathfrak{l}_n$ in $\mathfrak{s}\mathfrak{l}_{n+1}$
I have heard that the branching rules are well-known for the simple Lie algebra $\mathfrak{s}\mathfrak{l}_n$ in $\mathfrak{s}\mathfrak{l}_{n+1}$ over fields of characteristic zero. Where can I find a ...
user117370
5
votes
2
answers
439
views
Difference of adjacent dominant weights is a root?
The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 ...
- 52.9k
9
votes
0
answers
470
views
Branching rules for compact Lie groups
Let $G$ be a compact connected Lie group, and let $H\subset G$ be a closed subgroup. For an irreducible representation $\pi:G\to\mathrm{End}_\mathbb{C}(V)$ of $G$ ($\dim\pi<\infty$) I want to know ...
- 1,959
2
votes
1
answer
208
views
Dimension of restricted root spaces of split Lie algebras
Let $\mathfrak g$ be a real simple split Lie algebra. Let $\mathfrak g = \mathfrak k \oplus \mathfrak p$ be the Cartan decomposition. Let $\mathfrak a\subseteq \mathfrak p$ be a maximal abelian ...
- 1,782
6
votes
1
answer
428
views
Reference on Highest Weight Module of Kac-Moody Algebra
I am trying to understand this paper. The construction requires the understanding of the following concepts in the representation theory of simple and affine Lie algebras:
The construction of Verma ...
- 989
9
votes
1
answer
497
views
Highest weight representations of Kac—Moody algebras: what is inside the weight spaces?
Let $V(\lambda)$ be the unique irreducible representation of a Kac—Moody algebra $\mathfrak{g}$ with the highest weight $\lambda$. If $\mathfrak{g}$ is not of finite type, then even for $\lambda$ one ...
- 3,186
5
votes
1
answer
1k
views
The Casimir invariant of an irreducible representation of a compact Lie group
Let $G$ be a compact Lie group (not necessarily connected) and $\rho:G\to \mathrm{End}(V)$ an irreducible (hence finite-dimensional) unitary representation of $G$. Let $\mathfrak{g}$ be the Lie ...
- 1,942
2
votes
0
answers
573
views
Clebsch–Gordan(CG) coefficients for SO(N) and Sp(N) group
I know how to calculate the CG coefficients for $SU(N)$, but there are other simple Lie group like $SO(N)$ and $Sp(N)$. But up to now I can't find any textbook tells me how to calculate these and I ...
- 249
3
votes
0
answers
116
views
Extension of representations of certain compact Lie groups
Let $G$ be a real, connected, non-compact, semisimple Lie group with finite center and real rank $1$. Let $G=KAN$ be an Iwasawa decomposition, then $K\subset G$ is a maximal compact subgroup, and $\...
- 1,942
15
votes
2
answers
762
views
Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$
I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}_n(\mathbb C)$ (Okounkov-Vershik style, if you wish).
Consider the chain $$\mathcal U(\...
- 3,513
11
votes
1
answer
480
views
Uncle of Witt algebra
A Witt algebra W is an infinite-dimensional Lie-algebra defined by the generator relations:
W: $[L_{j},L_{k}]:=(j-k)\cdot L_{j+k}$
And my first thought was: What about the analogous algebra defined by
...
- 4,829
4
votes
0
answers
318
views
Action of orthogonal group on the free Lie algebra
This question is somewhat related and inspired by this post of professor Montgomery.
The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of https://en....
- 277