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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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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 ...
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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 ...
Mary Maths's user avatar
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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 ...
raynea's user avatar
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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 ...
Another User's user avatar
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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 ...
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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 ...
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$k \alpha$ 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 ...
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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 ...
subrosar's user avatar
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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-...
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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 ...
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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 ...
Craig's user avatar
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Weyl dimension formula and the adjoint representation of G2 [closed]

I'm having some problems understanding the Weyl dimension formula for G2. From the following post: How to use Weyl dimension formula? I understand that the Weyl formula for G2 takes the form: $$\dim V(...
Francisco's user avatar
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Kernel of map between Lie algebra representations

Consider the $\mathfrak{sl}_7(\mathbb{C})$ representation $\Gamma_{1,0,0,0,0,0,0,0} \otimes \Gamma_{0,0,1,0,0,0,0} \subset \wedge^2 \wedge^3 (V \oplus V^*)$, where $V$ is the standard representation ...
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Problem 1.7 in Humphreys' Lie algebra book

So, according to Humphreys:$\mathfrak{gl}_n$ is all $n \times n$ matrices, $\mathfrak{sl}_n$ is $n\times n$ matrices with sum of diagonal elements (trace) equal to zero and $\mathfrak{s}_n$ is set of ...
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Semisimple Lie algebras: general features of ladder operators and towers of vectors [closed]

I would like to ask what are the most general results and constraints on the representations of finite-dimensional semisimple Lie algebras; I know that one can group the generators into Cartan ...
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How to derive explicit matrix representations for the non-diagonal generators of a simple Lie algebra?

Given a semi simple Lie algebra's Dynkin diagram/Cartan matrix one can easily find out the weights of any particular representation. The weights of the defining representation are enough to give the ...
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What is the exact definition of the "index" which matches the determinant of the Cartan matrix and how to prove the equivalence?

In this answer, it is claimed that the determinant of a Cartan matrix is same as its "index" of the root lattice in the weight lattice. Firstly, what is the equation which describes the ...
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When do we include singlets while branching simple algebras to regular subalgebras?

According to Cahn's book [1] (pg. 148), given an irreducible representation (irrep) of a simple algebra of rank $r$, to construct the irreps of its regular subalgebras we can consider an extended ...
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Explicit differential equation of the global character of UIR's of SL(2,R)?

The semisimple Lie algebra $\mathfrak{g}:=sl(2,\mathbb{R})$ has the Casimir operator $$\Omega:=(1/2)h^2+ef+fe$$ with respect to the standard basis $(h,e,f)$. For an irreducible unitary representation $...
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Decomposition of Adjoint Group of Semisimple Lie Group

I have trouble understanding the beginning sentence of section $3$ of a paper by C.Moore: https://www.jstor.org/stable/2373052, which I quoted below: Suppose now that $G$ is a (connected) semi-simple ...
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Verifying that a vector is a highest weight vector

I'm trying to show that $(e_1 \otimes e_2 \otimes e_3 - e_3 \otimes e_1 \otimes e_2) \otimes e_n^*$ is a highest weight vector for the irreducible submodule of $V^{\otimes 3} \otimes V^*$ with highest ...
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Conjugating of root subspaces by exponents

$\renewcommand{\al}{\alpha} \newcommand{\bra}[1]{\left<#1\right>} \newcommand{\brc}[1]{\left(#1\right)} \newcommand{\Aut}[2]{\operatorname{Aut}_{#1}\brc{#2}} \renewcommand{\C}{\mathbb{C}} \...
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Conjugation of a Cartan subalgebra by an exponent

$\renewcommand{\al}{\alpha} \newcommand{\bra}[1]{\left<#1\right>} \newcommand{\brc}[1]{\left(#1\right)} \newcommand{\Aut}[2]{\operatorname{Aut}_{#1}\brc{#2}} \renewcommand{\C}{\mathbb{C}} \...
Dmitry's user avatar
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Why does the semisimple (s) and nilpotent (n) component of an element x=n+s of a cartan subalgebra commute with y as soon as y commutes with x.

I do not understand one line in the proof of the following theorem. Within the proof of d) it's stated that, "if $y \in \mathfrak{h}$, then y commutes with x and hence also with s and n." I ...
Lauren3829's user avatar
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Cartan subalgebras of reductive subalgebras

Let $\mathfrak{g}_1\subseteq\mathfrak{g}_2$ be an inclusion of reductive Lie algebras $\mathfrak{g}_1$ and $\mathfrak{g}_2$ over $\mathbb{C}$. Is it always possible to find Cartan subalgebras $\...
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Highest weight representations of $sl(2;\mathbb{C})$ and relation to spin

I’m studying representation theory of semisimple Lie algebras through highest weights. We have seen that finite dimensional irreducible representations are in 1 to 1 correspondence with dominant ...
Jorge Ortiz's user avatar
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Being an integral weight can be checked on a base.

Let $\Phi$ be a root system and $\Delta$ a base for a finite dimensional semi-simple complex Lie algebra. For $\lambda \in H^*$ we say that $\lambda $ is integral if $\frac{2(\lambda,\alpha)}{(\alpha,\...
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Weyl chambers are connected components

Let L be a semisimple finite dimensional complex Lie algebra and $\Phi$ its root system, $E=\text{span}_{\mathbb{R}} \{\alpha\in \Phi\}$. Denote for $\alpha\in \Phi$ the hyperplane orthogonal to $\...
Adronic's user avatar
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In a fssc Lie algebra, if for some x the killing form of x with itself is non zero then x is ad-diagonalizable

I am currently studying Lie algebras and stumbled upon this little lemma in my lecture notes. The proof given there is insufficient I believe. The proof there is based on choosing an "orthonormal&...
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Verma modules as representations of the Borel

Let $T \subset B \subset G$ be a reductive group, a Borel subgroup and a maximal torus over a field of characteristic $0$, with respective Lie algebras $\mathfrak{b} \subset \mathfrak{g}$. For a ...
Martin Ortiz's user avatar
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1 answer
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Tensor product of irreducible representation of $\mathfrak{sl_3}$

Let $\omega_1$ and $\omega_1$ be fundatmentl weights and $V(\lambda)$ be the unique irreducible representations of highest weight $\lambda$ of $\mathfrak{sl_3}$. I want to decompose, for $m>n$, $V(...
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Is the Lie-subalgebra generated by the root-spaces of a semisimple Lie-algebra semisimple?

Let $\mathfrak{g}$ be a finite-dimensional real semisimple Lie algebra and $\Sigma$ its root system (not necessarily reduced). Let $\alpha \in \Sigma$. Then $$ \mathfrak{l} := ( \mathfrak{g}_\alpha \...
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2 votes
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Semisimple Lie groups with finite center

Before the problem, here goes the background: let $G$ be a semisimple Lie group with Lie algebra $\mathfrak{g}$. Consider $X_s$ an semisimple element of $\mathfrak{g}$ such that every eigenvalue of $...
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Levi subgroups of exceptional (complex) Lie groups (reference request)

I'm trying to do some explicit computations for which I need to know the maximal Levi subgroups of exceptional complex Lie groups $\mathrm{F}_4=\mathrm{F}_4(\mathbb{C})$. Of course from the Dynkin ...
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How to find restriction matrix for branching?

Yesterday I asked this question. Let $G = Sp_{2g}(\mathbb{C})$ and $H$ be the subgroup of $G$ generated by matrices of the form $$\begin{pmatrix} M&0\\ 0&(M^{t})^{-1} \end{pmatrix}$$ with $...
Chase's user avatar
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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 ...
Chase's user avatar
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1 answer
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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. ...
Hugo MTV's user avatar
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Two alternative descriptions of $\mathrm{so}_{2n}(\mathbb{C})$

It seems that there are two ways to define the Lie algebra $\mathrm{so}_{2n}(\mathbb{C})$. The first one is $\mathrm{so}_{2n}(\mathbb{C})_{(1)}:=\{M \in \mathrm{gl}_{2n}(\mathbb{C}) \ | \ M + M^t = 0\}...
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Simple modules of $\mathcal{g}\otimes\mathbb{C}[x]$ for a lie algebra $\mathcal{g}$

Let $\mathcal{g}$ be a simple, finite-dimensional complex Lie algebra, and let $\mathcal{M}$ be a representative system of finite-dimensional simple $\mathcal{g}$-modules (up to isomorphism). ...
john_psl1298's user avatar
2 votes
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Vectors belonging to each component of decomposition of semi-simple Lie group representation

Say that $W$ is a $SL_4(\mathbb{Q})$ representation that decomposes into irreducibles (using the notation of Fulton-Harris) $$W = \Gamma_{a,b,c} \oplus \Gamma_{d,e,f}$$ Now say that I have a vector $v ...
Chase's user avatar
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$G$-equivariant map between Lie group representations

Say $f: A \rightarrow B$ is a $G$-equivariant map between finite dimensional $G$-representations, for some semi-simple Lie group $G$. If I know: $B$ decomposes into irreducibles as $B = B_1 \oplus ......
Chase's user avatar
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0 answers
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Process for decomposing tensor products of Lie algebra representations into irreducibles?

I'm trying to decompose tensor products of semisimple Lie group/algebra representations into direct sums of irreducible representations. I know this question has been asked many times before, with ...
Chase's user avatar
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3 votes
1 answer
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Other direction of Weyl's theorem on semisimple Lie-algebras

Weyl's theorem states Let $\mathfrak{g}$ be a finite dimensional semisimple Lie-algebra over an algebraically closed field with characteristic 0. Then all finite dimensional representations are ...
Flynn Fehre's user avatar
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Do all maximally noncompact Cartan subalgebras in a real semisimple Lie algebra arise from split toral subalgebras?

Let $\mathfrak{g}$ be a real, semisimple Lie algebra and $\mathfrak{g}_{\mathbb{C}} = \mathfrak{g} + i\mathfrak{g}$ its complexification. Following Knapp, a Cartan subalgebra of $\mathfrak{g}_{\mathbb{...
Strichcoder's user avatar
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1 vote
1 answer
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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 ...
Chase's user avatar
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1 vote
1 answer
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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 $\mathfrak{g}$, ...
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Kernel of an irreducible representation on the direct sum of two semisimple Lie algebras

Suppose $L=L_1\oplus L_2$, where $L_1,L_2$ are finite-dimensional complex semisimple Lie algebras. Suppose $V$ is a finite-dimensional irreducible representation of $L$ (through $\rho:L\to \mathfrak{...
barbatos233's user avatar
2 votes
1 answer
68 views

Generators and relations for $\mathfrak{sl}(N, \mathbb{C}[t])$

The Lie algebra $\mathfrak{sl}(N, \mathbb{C}[t])$ can be thought of as matrix valued polynomials. As a vector space $$\mathfrak{sl}(N, \mathbb{C}[t]) = \bigoplus_{k =0}^{\infty}\mathfrak{sl}(N, \...
Rodion  Zaytsev's user avatar
1 vote
1 answer
52 views

Adjoint of Translation Functors in Category O

Let $\mu,\lambda\in\mathfrak{h}^*$ with $\mu-\lambda\in \Lambda$ and let $pr_\mu$ and $pr_\lambda$ be natural projections of category $O$ onto blocks $O_{\chi_\mu}$ and $O_{\chi_\lambda}$ respectively....
Eric's user avatar
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4 votes
1 answer
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Root space decomposition of Lie subgroups

Let $\varphi \colon G \to \tilde{G}$ be a group homomorphism between two real semisimple Lie groups. For example, $\varphi$ could be an inclusion of a subgroup $G \subseteq \tilde{G}$. Let $\mathfrak{...
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