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2 votes
0 answers
159 views

The Cartan is a complex vector space but the root system is real?

Let $\frak{g}$ be a complex semisimple Lie algebra with some choice of Cartan subalgebra $\frak{h}$. The dual space $\frak{h}^* = \mathrm{Hom}_{\mathbb{C}}(\frak{h},\mathbb{C})$ is a complex vector ...
Jake Wetlock's user avatar
  • 1,144
3 votes
2 answers
492 views

Pairing a root with the half-sum of positive roots

Let $\frak{g}$ be a finite-dimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (-,-))$. Also we denote the half-sum of positive ...
Didier de Montblazon's user avatar
9 votes
1 answer
245 views

Must a continuous variation through compact simply connected Lie groups preserve topology

Let $V$ be a finite dimensional vector space over $\mathbb{R}$. Let $S$ be the vector space of multilinear maps from $V\times V$ to $V$. Let $L:\mathbb{I}\rightarrow S$ be a continuous map such that ...
Amr's user avatar
  • 1,117
2 votes
1 answer
337 views

Knapp's proof that the fundamental group of a compact semisimple Lie group is finite

In Knapp's book Representation Theory of Semisimple Groups: An Overview Based on Examples, he proves the following theorem of Weyl: If $G$ is a compact connected semisimple Lie group, then $\pi_1(G)$ ...
babu_babu's user avatar
  • 241
6 votes
0 answers
139 views

Why are all representations of split groups of real type?

(I am using a throwaway account because I plan on possibly pointing to this answer in a referee report I am writing, and using my main account would be a bit too obvious.) Let $\mathfrak{g}$ be a ...
temporarily_anonymous's user avatar
4 votes
2 answers
412 views

Minimal non-abelian groups -> Lie groups/algebras

A group is called minimal non-abelian if it is non-abelian and all proper subgroups are abelian. Does this notion also exist with Lie groups or algebras? As an example, consider the Lie algebra ...
Hauke Reddmann's user avatar
6 votes
0 answers
306 views

Tits construction of algebraic groups of type D₆ and E₇ via C₃

As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
nxir's user avatar
  • 1,479
5 votes
1 answer
228 views

Step in the Bruhat decomposition for reductive Lie groups

Err, not research but if anyone has read this part of Knapp's book recently, I'd be obliged if they could help me out. Also posted on MSE. I'm stuck on a line in the proof of Theorem 7.40 in Knapp's '...
Calamardo's user avatar
  • 675
2 votes
0 answers
81 views

Complex semisimple Lie algebra modules with non-semisimple Cartan action

Let $\frak{g}$ be a complex semisimple Lie algebra. I would like to know about infinite-dimensional representations $M$ of $\frak{g}$ for which the Cartan $\frak{h} \subseteq \frak{g}$ does not act ...
László Szabados's user avatar
4 votes
0 answers
120 views

Why are all "non-swinging" representations self-dual?

Let $\mathfrak{g}$ be a semisimple (say complex) Lie algebra, and $V$ an irreducible finite-dimensional representation of $\mathfrak{g}$. Denote by $w_0$ the longest element of the Weyl group, i.e. ...
Ilia Smilga's user avatar
  • 1,574
7 votes
1 answer
738 views

Infinite dimensional representations of $\frak{sl}_2$

The finite-dimensional representations of a complex semisimple Lie algebra $\frak{g}$ are well known to be classifiable by their highest weight vectors, giving a convenient countable indexing set. I ...
László Szabados's user avatar
0 votes
0 answers
97 views

Methods for calculating (one-parameter subgroup) actions

For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form \begin{equation} \mathrm{e}^{t L(z)} f(z) \end{equation} ...
eriugena's user avatar
  • 679
0 votes
1 answer
134 views

Sub-coroot lattices

[This is a sequel to the previous question sub-coroot systems, that has been answered! :-) ] Let $T$ be a maximal torus of a compact Lie group $K$, and let $\Lambda \subset {\mathfrak t}$ be the ...
bernardorim's user avatar
3 votes
1 answer
161 views

Sub-coroot systems

Let $T$ be a maximal torus of a compact Lie group $K$, and let $\Psi \subset {\mathfrak t}$ be the (finite) set of coroots for $(K,T)$, where $\mathfrak t$ is the Lie algebra of $T$. Assume now that $...
bernardorim's user avatar
3 votes
1 answer
280 views

Decomposition of tensor powers of the vector representation of $\frak{sl}_n$

Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as $$ V(\pi_1) \...
László Szabados's user avatar
1 vote
1 answer
189 views

Tensoring irreducible representations corresponding to root lattice elements

Let $\frak{g}$ be a complex semisimple Lie algebra with root lattice $Q$ and positive weight space $P^+$. Let $\lambda, \mu \in Q \cap P^+$, with corresponding respective fin-dim irreducible ...
Martim Pereir's user avatar
0 votes
1 answer
130 views

Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules

Let $\lambda \in \mathcal{P}^+$ be a dominant weight for $\frak{sl}(n,\mathbb{C})$. When does it hold that the zero weight space, of the associated finite-dimensional $L(\lambda)$, is non-trivial? ...
Martim Pereir's user avatar
4 votes
1 answer
239 views

Number of representations of a semisimple Lie algebra of any given dimension

For a semisimple complex Lie algebra $\frak{g}$ it is well known that irreducible finite-dimensional representation are not characterised by their dimension. More formally, let us define an ...
Martim Pereir's user avatar
2 votes
1 answer
371 views

What do the Pauli matrices say about the Threefold Way?

The Pauli matrices $$\sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_2=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_3=\begin{pmatrix} 1 & 0 \\ 0 & -1 \...
Andrius Kulikauskas's user avatar
1 vote
1 answer
340 views

Problem in understanding the coadjoint action of $\mathfrak {g}^{\ast}$ on $\mathfrak {g}$

$\DeclareMathOperator\ad{ad}$Let $\mathfrak {g}$ be a Lie bialgebra. Then $\mathfrak {g}^{\ast}$ is also a Lie bialgebra which is dual to $\mathfrak {g}$. Let the brackets on $\mathfrak {g}$ and $\...
Anil Bagchi.'s user avatar
3 votes
0 answers
142 views

Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation $$ G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) . $$ Basic ...
André Schlichting's user avatar
0 votes
0 answers
70 views

Why is $\Pi_r^L$ a non-degenerate Poisson structure on $G\ $?

Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a non-degenrate triangular structure on $\mathfrak {g}$ i.e. $r$...
Anil Bagchi.'s user avatar
0 votes
0 answers
51 views

Left translations respect the Schouten bracket

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$ and $r \in \bigwedge^2 \mathfrak{g}$. For $x \in G$ let $\lambda_x$ denote the left multiplication by $x$. Let $[\cdot, \cdot]$ ...
Anil Bagchi.'s user avatar
4 votes
1 answer
278 views

Complexification of a Lie subalgebra of a compact real form

I'm currently reading the paper Lie algebra Cohomology and the Generalized Borel–Weil theorem written by B. Kostant, and I have a question about Remark 3.9 he made. In this paper, $\mathfrak{g}$ is a ...
Ji Woong Park's user avatar
3 votes
1 answer
147 views

Does every nilpotent orbit have an element supported on the simple root spaces?

Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and $\mathfrak{g}$ its Lie algebra. Let $O \subset \mathfrak{g}$ be an orbit of a nilpotent element. Let $\Pi = \{\alpha_1, \dots ,\...
user492133's user avatar
3 votes
0 answers
143 views

Summing over roots of a simple Lie algebra and Deligne series

For a simple Lie algebra $\mathfrak{g}$ we can define a Killing form $K(X,Y) \equiv \frac{1}{2 h^\vee}\operatorname{tr}(\mathfrak{ad}_X \mathfrak{ad}_Y)$, where $\mathfrak{ad}_X Y \equiv [X, Y]$ as ...
Lelouch's user avatar
  • 857
0 votes
0 answers
73 views

Problem in understanding the proof of cocycle condition for cocommutator

Let $G$ be a Poisson–Lie group with Poisson bivector field $\pi$. Let $\pi^{R} \colon G \longrightarrow \bigwedge^2 \mathfrak{g}$ be defined by $$\pi^R (x) = (d_x R_{x^{-1}} \otimes d_x R_{x^{-1}}) \...
Anil Bagchi.'s user avatar
5 votes
2 answers
732 views

Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?

Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to ...
AmorFati's user avatar
  • 1,379
2 votes
1 answer
357 views

Tensor product of fundamental representations

Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $V_1,\cdots, V_n$ be the fundamental representations (the irreducible ones with fundamental weights $\omega_1,\cdots,\omega_n$). Take a $k$-...
Jun Yang's user avatar
  • 391
4 votes
0 answers
111 views

How many diagrams interlace a given Young diagram?

For a fixed partition $\lambda=(\lambda_1\geq\dots\geq \lambda_n)$ we say $\mu=(\mu_1\geq \dots \geq \mu_{n-1})$ $\textit{interlaces}$ $\lambda$ iff $$\lambda_1\geq \mu_1\geq \dots \geq \mu_{n-1}\geq \...
Nicolas Medina Sanchez's user avatar
3 votes
1 answer
203 views

Free $S^1$-action on compact homogeneous spaces

Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$. If $r(G) > r(K)$ (...
abracadabra12345's user avatar
1 vote
0 answers
64 views

Is the Lie bracket on $\mathfrak g^{\ast}$ induced from a cocommutator defined on $\mathfrak g\ $?

Let $G$ be a Poisson-Lie group. Let $\mathfrak g = \text {Lie} (G) = T_1 G$ be the corresponding Lie algebra. Then the Poisson structure on $G$ gives rise to a Lie bracket $[\cdot, \cdot]$ on $\...
Anil Bagchi.'s user avatar
6 votes
2 answers
194 views

Counting adjoints in the symmetric or antisymmetric square of a Lie group representation

EDIT (November 1, 2022): Over the weekend I think I found a technique to determine the exact multiplicities, according to how conjugation acts on the fundamental weights. While I haven't done the ...
Bruno Le Floch's user avatar
3 votes
1 answer
298 views

Characterization of reductive Klein geometries

In my struggle to understand Cartan/Klein geometries, I have the intuition that reductive Klein geometries are the link to connect the "classical" differential geometry approach with this &...
A. J. Pan-Collantes's user avatar
5 votes
1 answer
310 views

Non-standard partial orders on root systems

Let $\frak{g}$ be a semisimple complex Lie algebra and let $\Delta$ be its associated root system with $\{\alpha_1, \dotsc, \alpha_l\}$ a choice of positive roots. As we all know - $\Delta$ admits a ...
Didier de Montblazon's user avatar
4 votes
0 answers
161 views

Differential invariants and Lie symmetries

I have the following question: Does differential invariants have the same Lie symmetries? I want to know about the relation of differential invariants of an action and their symmetry properties. ...
Mostafa's user avatar
  • 49
10 votes
0 answers
291 views

Each simple real Lie algebra as a representation of its maximal compact subalgebra

I am interested in a detailed description of the Cartan decomposition of each type of simple, real, finite-dimensional Lie algebra. (This is essentially a question about the classification of simple, ...
Greg Kuperberg's user avatar
2 votes
2 answers
353 views

Particular reduced expression of the longest element of Weyl group

Let $I$ be the Dynkin diagram vertex set and $K$ be a proper nonempty subset of it. Let $w_0^K$ be the longest word of the Dynkin subdiagram $K$, which might be a disjoint union of connected Dynkin ...
F.H.A's user avatar
  • 201
2 votes
1 answer
219 views

Are finite-dimensional real representations of semisimple real Lie algebras completely reducible?

Suppose $\mathfrak{g}$ is a real form of a semisimple Lie algebra $\mathfrak{g}_\mathbb{C} = \mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$. Then we have the following: There is an equivalence of ...
Alistair Savage's user avatar
1 vote
1 answer
330 views

On Euler angles decomposition of $\mathrm{SU}(N)$

$\DeclareMathOperator\SU{SU}$I am looking for a (generalized) Euler angles decomposition for $\SU(N)\ (N>1)$ in the following fashion: $$ \SU(N)\ni m = a\, u \, b $$ where $a,b$ are independent ...
IgnoranteX's user avatar
4 votes
1 answer
348 views

Verma modules and Borel–Weil

Let $\mathfrak{g}$ be a semisimple Lie algebra and fix a root system. Let $\mathfrak{b}:=\mathfrak{h}\oplus\bigoplus_{\alpha\in R^+}\mathfrak{g}_\alpha$. The complex irreducible representation of $\...
lw h's user avatar
  • 181
3 votes
0 answers
142 views

Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces

Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
Callum's user avatar
  • 954
5 votes
0 answers
254 views

Reference on "infinite dimensional Lie algebras" from a mathematical physics point of view

It happens that I stumbled on a class of infinite dimensional Lie algebras that are not Kac-Moody algebras and for which I was not really prepared for. I know some general results on infinite ...
Dac0's user avatar
  • 295
2 votes
1 answer
248 views

What is the Molien series of the SO(2)-invariant ring on the plane (sometimes written C[X]^{SO(2)} )?

Let SO(2) be the group of rotations in the plane. What is the Molien series (sometimes called the Hilbert-Poincare series) of the SO(2)-invariant ring of polynomials? N.B. The main goal being to ...
Victoria's user avatar
0 votes
0 answers
132 views

Lie algebra action Whittaker model

Let $(\pi, H)$ be an irreducible unitary generic representation of $G=\operatorname{GL}(r,\mathbb{C})$ and let $H^{\infty}$ be its subspace of smooth vectors. Let $W :G\to\mathbb{C} $ be the Whittaker ...
Akash Yadav's user avatar
2 votes
2 answers
336 views

Orthosymplectic superalgebra

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 ...
jack's user avatar
  • 673
2 votes
1 answer
160 views

Is the restriction of the Cartan 3-form on conjugacy classes exact?

Let $G$ be a complex semisimple group and $\mathcal{O} \subset G$ a conjugacy class, i.e. $\mathcal{O} = \{gag^{-1} : g \in G\}$ for some $a \in G$. Let $\Omega$ be the Cartan 3-form on $G$ defined by ...
Lorenz Haber's user avatar
4 votes
0 answers
128 views

Real Representation ring of $U(n)$ and the adjoint representation

I have two questions: It is well known that the complex representation ring $R(U(n))=\mathbb{Z}[\lambda_1,\cdots,\lambda_n,\lambda_n^{-1}]$, where $\lambda_1$ is the natural representation of $U(n)$ ...
user avatar
3 votes
1 answer
195 views

A representation of $\frak{sl}_n$ as partial derivatives on polynomials

As is known to all, the Lie algebra $\frak{sl}_2$ admits a very nice representation on $$ \mathbb{K}[X,Y] $$ the polynomials in two variables, given by $$ E \mapsto X\frac{\partial }{\partial Y}, ~~ F ...
Jake Wetlock's user avatar
  • 1,144
4 votes
1 answer
230 views

Are isotypic components of $S(\mathfrak{g})$ finite-dimensional?

Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic ...
Lorenz Haber's user avatar

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