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Results tagged with lie-algebras
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user 125941
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
10
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3
answers
1k
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Hopf structure on the universal enveloping of a super Lie algebra
The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the …
- 4,713
4
votes
1
answer
242
views
Decomposing tensor powers of the fundamental representation of exceptional Lie algebras
For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, th …
- 24.2k
6
votes
2
answers
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Non-faithful irreducible representations of simple Lie groups
For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group.
Mor …
- 2,821
25
votes
1
answer
699
views
The de Rham complex of the octonionic projective spaces
The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic …
- 188k
5
votes
3
answers
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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 W …
- 27.9k
4
votes
2
answers
403
views
GAP versus SageMath for branching to Lie subgroups
Which computer package is better, GAP or SageMath, for
decomposing an irreducible representation of a (simple) Lie group
$G$ into representations of a Lie subgroup. I am most interested when
branchi …
- 2,449
4
votes
1
answer
202
views
Branching to Levi subgroups in SAGE and the circle action
In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:
http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_sy …
- 8,597
4
votes
Accepted
Weyl's Branching Rule for $SU(N)$-Setting
The question is answered on page 385 of the classical Zhelobenko book
Compact Lie groups and their representations
for the more general case of $SU(n+m)/SU(n) \times SU(m)$.
- 835
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votes
1
answer
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Young tableaux for exceptional Lie algebras
Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series.
Does su …
- 16.5k