# Questions tagged [semisimple-lie-algebras]

A simple Lie algebra is non-abelian Lie algebra with no nontrivial ideals. A *semisimple Lie algebra* is a Lie algebra which is the direct sum of simple Lie algebras. This tag is for questions about semisimple Lie algebras, including their classification and correspondent to root systems and Dynkin diagrams.

385 questions
Filter by
Sorted by
Tagged with
40 views

### Mapping between vectors in irreducible Sp-representation

Let $V$ be the standard Sp-representation with symplectic basis $\{ a_i, b_i \}$. I believe the vector $(b_1 \wedge b_2) \otimes (b_1 \wedge b_2 \wedge a_3 \wedge a_4)$ lies inside the irreducible ...
• 326
26 views

### Nilpotent Lie algebra decomposed into direct sum of root spaces for a torus

In this article Solvable complete lie algebras. I written by Meng Dao Ji and Zhu Lin Sheng. They use a decomposition using root systems, but when I searched about the Root systems I found that it is ...
• 283
1 vote
51 views

### Finding inverse to inner automorphism (Humphreys' Lie Algebra book)

This is basically an algebra question. In Humphreys' book on Lie algebras he states that one can find the inverse to $\exp \delta$, where $\delta$ is a nilpotent derivation - say $\delta^k=0$, by ...
• 343
37 views

### Relation between linear representation and "induced adjoint representation" of Lie algebra?

Consider a representation $\rho \colon \mathfrak{g} \mapsto \mathrm{End}(V)$ of a Lie algebra $\mathfrak{g}$ on a vector space $V$. What can we say about the induced representation on the space of ...
• 303
58 views

### Real forms of a solvable Lie algebra

Consider the Lie algebra $\mathfrak{r}_2=\mathfrak{aff}(\mathbb{C})$ of the group of affine maps of $\mathbb{C}$, and let $\mathfrak{g}=\mathfrak{r}_2 \oplus \mathfrak{r}_2$. I am interested in ...
• 21
67 views

### Adjacent roots in root systems

In Fulton and Harris : while dealing with rank $2$ root systems, authors say that the angle between two roots must be the same for any pair of adjacent roots in a two-dimensional root system. In two ...
• 751
32 views

### $k \alpha$ is a root implies $k = \pm 1$.

This is a doubt about proof given in section 21.1, Fulton and Harris. Authors consider the representation $$i = \oplus_k g_{k\alpha}$$ of the Lie algebra $s_{\alpha} \cong sl_2 \mathbb C.$ I ...
• 751
31 views

### Fulton-Harris Proposition C.17 clarification

There's a line in the proof of this proposition C.17 in Fulton-Harris that I don't understand. For completeness,the statement of the proposition is: Let $\mathfrak{g}$ be a semisimple Lie subalgebra ...
• 4,794
1 vote
34 views

### How can Lie algebra cohomology be nontrivial for a semisimple algebra?

Let $\mathfrak{g}$ be a semisimple Lie algebra over an (algebraically closed) field $k$ of characteristic zero. I am going by the definitions in Weibel, chapter 7. Here's my logic: A finite-...
• 699
50 views

### What can be said of general representations of solvable Lie algebras?

I'm curious about the representation theory of solvable Lie algebras. Consider the following quote from Fulton & Harris: [To] study the representation theory of an arbitrary Lie algebra, we have ...
• 6,672
1 vote
62 views

### How to be certain one has found all possible real forms of a semi-simple Lie algebra?

This is sort of a follow up question to this question on Satake-Tits diagrams. In the question, user Callum comments that for, $\mathfrak{so}(n, \mathbb{C})$, the real forms (potentially with double ...
• 821
68 views

• 2,005
144 views

• 326
74 views

### Moving between $Sp_{2n}(\mathbb{C})$ reps and $SL_{n}(\mathbb{C})$ reps

Say I have some irreducible $Sp_{2n}(\mathbb{C})$ representation, such as $\Gamma_{0,1,0,1}$. Consider the subgroup of $Sp_{2n}(\mathbb{C})$ isomorphic to $SL_n(\mathbb{C})$, consisting of matrices of ...
• 326
1 vote
180 views

### Is it true that for a semisimple Lie algebra $\mathfrak{g}$, $[\mathfrak{g},\mathfrak{g}^e]\subset[e,\mathfrak{g}]$ for $e$ a regular nilpotent?

Let $\mathfrak{g}$ be a finite dimensonnal semisimple Lie algebra, let $e \in \mathfrak{g}$ be a regular nilpotent element ie. an element whose adjoint endomorphism is nilpotent and of maximal rank. ...
• 91
101 views

• 326
98 views

• 2,005
1 vote
81 views

### Help using LiE computer package? [closed]

I'm trying to decompose some tensor products of semi-simple Lie algebra representations into irreducibles. I've read great things about Lie but I get a "502 Bad Gateway" when I use the ...
• 326
1 vote
102 views

### Are the weights of an irreducible representation of a complex semisimple algebra induced by the weights of a representation of $SL(N)$ for some $N$?

If we consider the weights of a complex irreducible representation $V$ of a complex semisimple Lie algebra $\mathfrak{g}$, with respect to a chosen Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$, ...
• 5,375