New answers tagged lie-algebras
2
votes
Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
For a reductive group the category of representations is semisimple so the algebra of endomorphisms of the fiber functor is just a product of matrix algebras, one for each irreducible representation.
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- 137k
1
vote
Accepted
Counting adjoints in the symmetric or antisymmetric square of a Lie group representation
The exact formula for $p_\Lambda$ and $p_S$ is given in Theorem 3.3 in D. Panyushev and O. Yakimova "The PRV-formula for tensor product decompositions and its applications", Funct. Anal. ...
1
vote
Lie's third theorem via graded geometry
A sketch of how to use graded geometry to integrate dg manifolds and, in particular, Lie algebroids appeared first in the legendary paper https://arxiv.org/abs/math/0105080
More details appeared later ...
- 11
0
votes
Is there a reasonable way to define "reductive Lie algebra" in prime characteristic?
Serre (1994) (« Sur la semi-simplicité des produits tensoriels de représentations de groupes », Invent. math. 116, p. 513–530) — see also his 1997 Bourbaki seminar (Complète réductibilité, https://...
- 12.8k
4
votes
Accepted
Derivatives and ODEs on Lie groups
$\newcommand\SOg{\mathrm{SO}}\newcommand\sog{\mathfrak{so}}\newcommand\TT{\mathsf{T}}$
Key points to remember:
$\sog(3)$ is the tangent space to $\SOg(3)$ at the identity element.
$\SOg(3)$ is a Lie ...
- 37.6k
11
votes
Where do root systems arise in mathematics?
They arise in the representation theory of quivers: Gabriel's theorem says that a connected quiver has finite representation theory type if and only if it is of type ADE, and then the indecomposable ...
16
votes
Where do root systems arise in mathematics?
I first came across root systems in the classification of finite reflection groups. A point group $\Gamma\subseteq\mathrm O(\Bbb R^n)$ is a reflection group if it is generated by reflections at ...
8
votes
Where do root systems arise in mathematics?
The eigenvalue distribution functions of random matrices in different universality classes are determined by the multiplicities of the restricted roots of the corresponding symmetric spaces, see ...
4
votes
Second cohomology group of the contact Lie algebra $K_3$
Yes, it is true. In fact, it is true that $H^i(K_{2n+1},F)=0$ for $0<i\le 2n$. This can be deduced from the theorem of Feigin sketched in
Feigin, B.L. Cohomology of contact Lie algebras.
(Russian) ...
- 16.5k
5
votes
Accepted
Trivial representation of a maximal torus
The multiplicity is positive if and only if $ \lambda $ lies in the root lattice.
More generally, $\mu$ is a weight of the representation $ V_\lambda$ if and only if $ \mu $ lies in the convex hull of ...
- 4,467
0
votes
Does every nilpotent lie in the span of simple root vectors?
No, not every nilpotent element lies in the span of simple root vectors. In general, the Lie algebra of a reductive group (G) decomposes into a direct sum of its semisimple and nilpotent parts under ...
4
votes
Accepted
Does every nilpotent lie in the span of simple root vectors?
Let $\mathfrak g$ have Cartan subalgebra $\mathfrak h$, let $X_\alpha\in\mathfrak g_\alpha$ be a non-zero vector. Then you are looking at nilpotent elements of the form $\sum_{\alpha\in \Theta}X_\...
- 1,897
3
votes
Does every nilpotent lie in the span of simple root vectors?
Actually I think this has to fail for $G_2$. Using fundamental coweights, we can construct a semisimple element that scales the simple root vectors $u_1, u_2$ by arbitrary nonzero constants $c_1, c_2$....
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6
votes
Accepted
Multiplication factors in folding root systems and Lie algebras by automorphisms
Suppose we fold a root system $(\Phi,\Delta)$ to a root system $(\Phi^\sigma,\Delta^\sigma)$. There exists two conventions:
long roots of $(\Phi^\sigma,\Delta^\sigma)$ correspond to multiple roots in ...
- 1,897
5
votes
Accepted
Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations
(Copied from my own comments.) Yes. In fact, if $V_k$ (for $1\leq k\leq n$) denotes the $k$-th fundamental representation in the order of the nodes of the Dynkin diagram, and $V_0$ the trivial ...
- 30.2k
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