All Questions
298 questions
3
votes
2
answers
686
views
Real Adjoint representations of complex type
Let $G$ be a semi-simple compact Lie group. Let $V$ be a real vector space and let:
$\rho : G \to Aut_{\mathbb{R}}(V)$
be an irreducible real representation of $G$ on $V$. We say that $\rho$ is a ...
1
vote
1
answer
791
views
On the centre of the universal enveloping algebra
I posted the following question on MSE but I was advised to ask the question here.
Let $\mathfrak{g}$ be a lie algebra. Consider the anti-automomorphism
$$\phi: \mathfrak{g} \rightarrow \mathfrak{g}...
- 8,524
5
votes
3
answers
1k
views
Question on irreducible representation of tensor products
Question:
Suppose that $V_1$ and $V_2$ are two finite dimensional representations of lie algebra $\mathfrak{g}$ generated by highest weight vectors $v_1^*$ and $v_2^*$ of weights $\mu_1$ and $\mu_2$ ...
- 27.9k
11
votes
1
answer
571
views
Diagram folding of simple Lie algebras
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 be constructed by ...
- 15.5k
6
votes
2
answers
517
views
The Analog of Borel Subgroup in a Compact Real Form
I recently learned that there is a one-to-one correspondence between isomorphism classes of complex reductive groups and isomorphism classes of compact connected real Lie groups given by taking a ...
- 27.9k
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
3
votes
0
answers
130
views
About the purpose of introducing '"groups of Heisenberg type"
I would like to know, can we say that the "groups of Heisenberg type" where introduced by A. Kaplan in "Kaplan, A. (1980). Fundamental solutions for a class of hypoelliptic PDE generated by ...
- 650
2
votes
0
answers
808
views
Casimir operators of a given Lie Algebra
I am a Physicist, so let me apologize in advance for some possible imprecisions.
I'm working on a 10-dimensional Lie Algebra. Each element of the algebra represents a quantum mechanical operator, and ...
CommunityBot
- 1
5
votes
2
answers
491
views
Kostant's $G$-invariant part in the sym power ring of adjoint representation?
Let $g$ be a Lie algebra, say $sl_n(\mathbb C)$. It is considered as the adjoint representation of $G=SL_n(\mathbb C)$.
A famous theorem of Kostant from "Lie Group Representations on Polynomial ...
1
vote
1
answer
232
views
Large spin expansion of affine $\mathfrak{su}(2)_k$ characters
There is a problem I am trying to solve for some time now which in a few words boils down to computing the coset characters for
$$
\frac{\mathfrak{su}(2)_k\oplus\mathfrak{su}(2)_\ell}{\mathfrak{su}(2)...
- 71
3
votes
1
answer
572
views
Dimension of the zero weight space in $V_{2\rho}$
Let $\rho$ be the half sum of the positive roots (also the sum of fundmental weights) for $SL_n(\mathbb C)$. Then $2ρ$ is in the root lattice. Then what is the dimension of the zero weight space for ...
- 27.9k
5
votes
1
answer
284
views
If $N_G(S)/Z_G(S)$ is a reflection group, is it a Weyl group?
Let $G$ be a compact, connected Lie group and $S$ a torus in $G$ not assumed maximal. Then conjugation in $G$ induces a faithful representation of $N = N_G(S)/Z_G(S)$ in the Lie algebra $\mathfrak s$ ...
CommunityBot
- 1
5
votes
0
answers
167
views
Plancherel formula for $L^2(G/N)$
Let $G$ be a connected real semisimple or reductive Lie group. Let $TA$ be a Cartan subgroup, where $T$ is compact and $A$ is split. Let $MA$ be the centralizer of $A$ in $G$, and let $N$ be the ...
- 51
5
votes
2
answers
961
views
Relationship between parabolic subgroups and parabolic subalgebras over non-algebraically-closed fields
Let $\mathbb{K}$ be an arbitrary field, and $G$ an algebraic group (or group variety?) over this field. A Borel subgroup of $G$ is a connected solvable subgroup variety $B$ of $G$ such that $G/B$ is ...
- 8,597
5
votes
1
answer
340
views
Irreducible representations containing simple actions of $\mathrm{SL}(2,\mathbb{C})$
Let $G$ be a complex semisimple Lie group and let $\rho: G \longrightarrow \mathrm{SL}(n,\mathbb{C})$ be a faithful irreducible representation of $G$ with $n \geq 3$. Suppose that $G$ contains a copy ...
- 11.2k
6
votes
1
answer
194
views
Involutions and Little Adjoint Representations of Simple Algebras
In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$.
Is easy to see that ...
- 4,729
3
votes
0
answers
274
views
Invariant functions on the dual Lie algebra
Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$.
...
- 95
6
votes
2
answers
334
views
Multiplication in universal enveloping algebra in terms of PBW isomorphism
Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt ...
- 8,524
7
votes
1
answer
606
views
Motivating the existence of Canonical Bases for Representations
In Representation Theory, the theme of the existence of a canonical basis has been explored quite a lot. I will limit myself in this question to the kind of canonical bases that arise from the ...
- 27.9k
11
votes
2
answers
881
views
From Weyl groups to Weyl groupoids?
I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed.
Let $\mathfrak{g}$ be a semisimple lie algebra.
...
CommunityBot
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1
vote
0
answers
84
views
Integrable modules and comodules
Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? ...
- 6,201
6
votes
1
answer
1k
views
Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$
Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...
CommunityBot
- 1
6
votes
1
answer
596
views
Vector fields, diffeomorphism subgroups and lie group actions
Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:
Let $\{X_j\} \in Vect(M)$ be a ...
- 7,789
2
votes
1
answer
251
views
Characterization of restricted weights of representations of real semisimple Lie groups
I need to use the following theorem:
Let $\mathfrak{g}$ be a semisimple real Lie algebra, $\Sigma$ a set of restricted roots for $\mathfrak{g}$. Let $\rho$ be any finite-dimensional representation of ...
- 1,574
3
votes
0
answers
126
views
Irreducible representations in BGG category $\mathcal{O}$ over (finitely) direct sum of general linear Lie superalgebra
Let $\mathfrak{g} = \oplus_i^k\mathfrak{gl}(m_i|n_i)$ be a direct sum of general linear Lie superalgebras $\mathfrak{gl}(m_i|n_i)$'s with the Cartan subalgebra $\mathfrak{h} = \oplus_i^k \mathfrak{h}...
- 1,472
27
votes
1
answer
891
views
Why do the adjoint representations of three exceptional groups have the same first eight moments?
For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...
- 148k
3
votes
0
answers
236
views
Deligne-Simpson problem for classical groups
Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n =...
- 181
2
votes
0
answers
135
views
Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations
I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations:
$[U_i, U_j] = 0$ for $|i-j|>1$
$U_iU_{i+1}U_i=...
- 71
19
votes
5
answers
2k
views
Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?
Let $G$ be a simple algebraic group group over $\mathbb C$.
Let $V$ be a self-dual representation of $G$.
Let $\lambda$ be the highest weight of $V$.
Write $\lambda$ as a sum of fundamental weights: $...
- 14.2k
6
votes
2
answers
729
views
Relationship between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?
Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the Lie algebra of vector fields on $M$; that is $\text{Lie}(\...
CommunityBot
- 1
5
votes
1
answer
368
views
Regular functions on nilpotent orbits and their covers
Let $G$ be a complex semisimple algebraic group with Lie algebra $\mathfrak{g}$.
In 1989 McGovern described the structure (as $G$-module) of the ring of regular functions on a finite cover of the ...
- 52.9k
1
vote
1
answer
147
views
Non-graded representations over Lie superalgebra $\mathfrak{gl}(m,n)$
I have the following questions:
Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra $U(...
- 19.6k
7
votes
1
answer
824
views
Infinite-dimensional admissible representations of SL(2,C)
I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper "Infinite Irreducible ...
- 52.9k
0
votes
0
answers
53
views
Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action
I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
- 2,099
6
votes
1
answer
1k
views
Understanding the Weyl Character Formula
Let $G$ be a compact (connected) Lie group with a maximal torus $T$. For each (analytically) integral weight $\lambda$ the Weyl character formula
$$\Theta_{\lambda}(H)=\frac{\sum_{w\in W(G)}\epsilon(w)...
- 44.7k
0
votes
2
answers
1k
views
Representation Theory of $U(N)$
(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...
- 273
3
votes
0
answers
170
views
The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$
It is commonly known that we have a chain of embeddings
$$SU(4)\subset Spin(7)\subset SO(8)$$
(there is more than one possible $Spin(7)$, just take one).
Which is the explicit analog for the Lie ...
- 2,091
3
votes
1
answer
281
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 $K$...
- 52.9k
7
votes
0
answers
167
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 ...
- 9,019
5
votes
3
answers
1k
views
classifying space and cohomology of integer general linear group
I have obtained that the classifying space
$$
BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty)
$$
is the Grassmannian.
I have also obtained that the mod 2 cohomology is the polynomial ...
- 17.4k
12
votes
5
answers
4k
views
Weight lattice and the fundamental group
Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively. Then, $\frak{t}$ can be considered as a Cartan subalgebra of $...
CommunityBot
- 1
6
votes
1
answer
255
views
Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky
The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, \...
- 4,612
3
votes
2
answers
704
views
Closure relations between Bruhat cells on the flag variety
Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...
CommunityBot
- 1
4
votes
1
answer
1k
views
Reductive Lie Groups and Complexification
Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, $G$...
- 4,612
1
vote
1
answer
178
views
Reducible reductive Lie subalgebras of so(p,q)
Is it true that $S(O(p) \times O(q))$ is the only proper subgroup of $SO(p,q)$ of full rank acting on the natural representation $\mathbb{R}^{p+q}$ of $SO(p,q)$ that stabilizes a $p$-dimensional ...
- 14.2k
5
votes
1
answer
459
views
Some questions on analytic vectors and the integrability of Lie-algebra representations
I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this ...
- 1,396
7
votes
1
answer
1k
views
When is the Ad (Adjoint Representation) Morphism a Closed Map
Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...
- 63.9k
1
vote
1
answer
264
views
labeling state vectors in representation space of a simple lie algebra
Given a simple lie algebra (over ${\mathbb C}$ or ${\mathbb R}$). What is the number
of operators such that their eigenvalues sufficiently label all state vectors in the algebra's representation space....
CommunityBot
- 1
5
votes
0
answers
428
views
Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book
This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
- 33.8k
7
votes
3
answers
2k
views
Characterising the adjoint representation of SU(N)
One can show that the adjoint representation of $\mathrm{SU}(n)$, the image of the map $\mathrm{Ad}:\mathrm{SU}(n) \rightarrow \mathrm{Aut}(\mathrm{su(n)})\subset \mathrm{GL}(\mathrm{su}(n))$, is an $...
- 71