All Questions
19 questions
20
votes
6
answers
4k
views
Polynomial invariants of the exceptional Weyl groups
Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}...
- 2,160
8
votes
1
answer
374
views
Chevalley restriction theorem for exterior algebras
Suppose $G$ is semisimple Lie group, $\mathfrak{g}$ is its Lie algebra, $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, and $W$ is the correspondent Weyl group.
Chevalley restriction theorem ...
- 1,276
8
votes
0
answers
145
views
Semisimple Lie groups admitting a free algebra of invariants
Assume we work over an algebraically closed field of characteristic zero.
I know that for a connected semisimple algebraic group there is an upper bound for the number of isomorphism classes of ...
- 189
7
votes
2
answers
418
views
About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$
Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) $...
- 9,019
6
votes
2
answers
794
views
Tensor algebra and universal enveloping algebra
Let $\mathfrak g$ be a Lie algebra which is not reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
- 673
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 ...
- 755
4
votes
1
answer
333
views
Orbits in the adjoint representation of $SU(2,1)$
How can one describe the orbits of the Lie group $G=\mathrm{SU}(2,1)$ in its Lie algebra $\mathfrak{g}=\mathfrak{su}(2,1)$ with respect to the adjoint representation?
- 14.2k
4
votes
1
answer
381
views
The existence of a finite dimensional Lie algebra with a given symmetric invariant metric
The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...
- 755
4
votes
1
answer
1k
views
Invariant polynomials with respect to group actions on matrices
Let $\mathfrak{gl}_n(\mathbb{R})$ be the Lie algebra of matrices with real entries and $GL_n(\mathbb{R})$ its associated Lie group. Recall that a linear subgroup $G \subseteq GL_n(\mathbb{R})$ acts by ...
- 463
3
votes
2
answers
285
views
Invariant theory for parabolics
Let $G$ be a connected reductive group over $\mathbb{C}$ of (reductive) rank $\ell$. Let $P$ be a parabolic of $G$ and let $P=LN$ denote the Levi decomposition. Let $\mathfrak{g}, \mathfrak{p}, \...
- 2,751
3
votes
1
answer
698
views
slice theorem for proper actions
I'm trying to understand the slice-theorem for proper Lie-group actions.
Having a smooth manifold $M$ and a Liegroup $G$ acting on $M$ in a proper way, we have the slice theorem, saying that at each ...
- 95
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
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 ...
- 21
2
votes
1
answer
359
views
Characterization of the weight orbit in the projective space via second order Casimir.
This is the spin-off of the question I previously asked.
First, let me remind you some notation from that question:
$G_0$ - compact, simply connected Lie group giving rise (by complexification) ...
- 965
2
votes
0
answers
180
views
Howe duality vs first fundamental theorem in invariant theory
I'm working on Howe duality, and R. Howe proved that the Howe duality of $\mathrm{GL}_n$ is equivalent to the first fundamental theorem (FFT) in invariant theory. So, Howe duality gives a ...
2
votes
0
answers
406
views
Invariants of the group $SO(2)$
Let $V_d$ be the complex vector space of binary forms of degree $d$ endowed with the natural
action of the special orthogonal group $SO(2).$ Consider the corresponding action of the
group $SO(2)$ on ...
- 656
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 ...
- 255
2
votes
0
answers
143
views
Diagonal invariants of $SO(n)$
Consider a Lie algebra $\mathfrak g$ (I am mostly interested in the case $\mathfrak g=so(n)$), its universal enveloping algebra $U$ and its center $C$. There is an adjoint action of $\mathfrak g$ on $...
- 1,488
0
votes
0
answers
128
views
How to find the polynomials that define a compact Matrix Lie group from its Lie algebra?
Consider a compact (connected) Lie group, or more generally, a linear algebraic Lie group. Suppose we are given the Lie algebra corresponding to the Lie group. How can we find a set of polynomial ...
- 1