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.

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Calculation of Dynkin Index issue

Following Onishchik's "Topology of Transitive Transformation Groups" section 3.10 the Dynkin index, $j_{\rho}$, of a Lie group homomorphism $\rho: H \rightarrow G$ of connected simple ...
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Prove Harish-Chandra's theorem on central characters with algebraic-geometrical observation and the linkage relations among weights

For representations of semisimple Lie algebras $L$, we have the following theorem on central characters: Theorem (Harish-Chandra): Let $\lambda, \mu \in H^{\ast}$. Then $\chi_{\lambda} = \chi_{\mu}$ ...
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Exercise 24.12 in Humphrey's Lie algebra book : Application of Steinberg's formula

I'm working on the following exercise (Exercise 24.12) in Humphrey's book Introduction to Lie algebras and representation theory. My question is how to solve the following exercise: Exercise: Deduce ...
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Irreducible finite-dimensional real $\mathfrak{sl}_2(\mathbb R)$-representations

It is easy to show that every irreducible finite-dimensional real $\mathfrak{sl}_2(\mathbb C)$-representation is a weight module. The operator commonly denoted as $h$ in $\mathfrak{sl}_2$ has an ...
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Is the Lie algebra so(2,6) semi-simple?

I'd like to know whether the Lie algebra so(2,6) is semi-simple. I know that I could compute the Killing form for that. But is there an easier way?
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A question about roots and root spaces of semisimple Lie algebras

In the book "Lie Groups: Beyond and Introduction" by Anthony Knapp, the author describes roots of a semisimple (A Lie algebra without any non-zero solvable ideal) Lie algebra as elements of ...
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Softwares to determine semi-simple types of Lie algebras generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of matrices

I wish to determine the type of Lie algebra generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of square matrices with irrational elements. I've been using GAP ...
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A description of the $A_n$-root system in terms of $\mathbb{R}^{n}$

The root system $(V,\Delta)$ of type $A$ is usually defined as a subset of $\mathbb{R}^{n+1}$: $$\{ e_i - e_j ;\, 1 \leq i,j \leq n+1,\, i \neq j\}.$$ However these vectors span an $n$-dimensional ...
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Character formula for real representations

For an irreducible representation of a complex semisimple Lie algebra the Weyl character formula is well known. For an irreducible real representation of a real semisimple Lie algebra is there a Weyl ...
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Scaling behaviour and volumes in coadjoint orbits

Let $\mathfrak{g}$ be a semisimple real Lie algebra and $G$ its associated adjoint Lie group. If we set $\kappa$ to be its Killing form, and $\theta$ to be some Cartan involution, we get a scalar ...
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The action of the longest element on weights

I am doing some computations related to some work I do in Lie theory and I need to compute the result of the action $$w_0 \omega_{i},$$ where $w_0$ denotes the longest element in the Weyl group $W$ of ...
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Does every nilpotent element in a real Lie algebra lie in some minimal parabolic Lie algebra?

Definitions: Let $\mathfrak{g}$ be a finite-dimensional, real, semisimple Lie algebra with some Cartan involution $\theta$. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the usual Cartan ...
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On a simple real Lie algebra whose complexification is not simple

I wish to prove the following result: Theorem: Let $\mathfrak{h}$ be a real simple Lie algebra and $\mathfrak{h}_{\mathbb{C}}$ be its complexification, which is not simple. Then, there is a complex ...
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What Does It mean to Solve the Lie Algebra

I have found the Lie Symmetries for a particular PDE, and have constructed the commutator table. The symmetries are as follows: \begin{align} X_1 &= x \frac{\partial}{\partial x} + 3t \frac{\...
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Decompose weights of same multiplicity into different Weyl group orbits

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$. Let $\lambda\in\mathfrak{h}^*$ be a dominant integral ...
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Computing the rank of the Lie Algebra of the symplectic group

I was unsure if I should post this to the physics forum since mathematician often tend to think differently than physicists, but this is a claim about Lie algebras so I thought this forum might be ...
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How to generate an associative algebra given a Lie subalgebra

I am new to the study of Lie algebras and particularly algebraic Lie algebras. In the paper Chevalley, Claude, Algebraic Lie algebras, Ann. Math. (2) 48, 91-100 (1947). ZBL0032.25202. the author ...