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Questions tagged [lie-algebras]

For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
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536 views

Abelian Cartan subalgebras

If a Lie algebra is semisimple or reductive, its Cartan subalgebras are Abelian, and their elements semisimple. Are there non-reductive algebras with Abelian Cartan subalgebras all of whose ...
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Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
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235 views

Why are parabolic subgroups called “parabolic” subgroups?

I used to think that things called "parabolic" must have something to do with parabolas or their defining quadratic equations. In fact, terms like parabolic coordinate, parabolic partial differential ...
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Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\...
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Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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Subgroups of $\text{Spin}(7)$.

I am interested in subgroups of $\text{Spin}(7)$ and identifying certain elementary properties that they have. Specifically relating to commutativity. Let me apologise at this point for my ignorance ...
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How to calculate the Lie algebra of a neural network?

Define $F$ as the standard multi-layer feed-forward perceptron: \begin{equation} F(\mathbf{x}) = \Theta( W_1 \circ \Theta( W_2 \circ .... W_L(\mathbf{x}))) \end{equation} where $\Theta$ is the sigmoid ...
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Motivation?: Lie algebra and algebraic group Cohomology

This is just an apriori question to get a motivational heuristic idea: If an algebraic group G (more generally, G an affine group scheme), is connected over an algebraically closed base-field k. ...
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Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
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Complexification of compact connected Lie groups: do these curves have the same tangent vector?

I'm trying to understand the complexification of Lie groups from page $207$ here and I'm having trouble with a computation. Assume $A, B$ are hermitian metrices, and $k$ is a unitary matrix. I want ...
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Solving a matrix differential equation

I am trying to solve: $\frac{d U_t}{dt} = Tr(G^{\dagger}U_t)G - Tr(U_t^{\dagger}G)U_t G^{\dagger} U_t$ Where $U_t \in SU(4)$ and $G \in SU(4)$ is given and constant. Is it possible to solve this ...
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Generating function for characters of representations

One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ ...
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333 views

Root Systems of Lie Groups.

Let $G$ be a compact Lie group assumed to be a subgroup of $U(n)$. Also, let $T$ be a maximal torus of G. Then there exists a basis $\{v_1, \ldots ,v_d\}$ of the Lie algebra of $G$, $\mathfrak{g}$, ...
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What is the relation between the determinant of the adjoint representation, and volume or eigenvalues?

I'm relatively new to Lie Groups, and finding in a text the following statement, as it is supposed to be obvious. I would love some reference or context to either of these two (probably related) ...
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How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
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Geodesics on Homogeneous Spaces

Consider a homogeneous space $G/\text{Stab}_p \cong M$ where $G$ is a compact Lie group active transitively on $M$ (a compact manifold). If $F$ is a Finsler Metric on $G$ which pushes forward ...
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Is every complex Lie algebra a complexification?

I'm wondering if every finite-dimensional complex Lie algebra can be written as a complexification of a real Lie algebra. At the vector space level, clearly every $\mathbb{C}^n$ is a complexification ...
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728 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) \...
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338 views

Non-degenerate bilinear forms of Lie algebra with a degenerate Killing form

Definition: A Lie algebra is defined by: $$ [e_a,e_b]={f_{ab}}^ce_c $$ The Killing form is $$ g_{ab}=-{f_{ac}}^d {f_{bd}}^c $$ Set-Up: The type of Lie algebra of our interests (found out during a ...
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Application of SU(2) in physics

How can we interpret the representations of SU(2) and $\mathfrak{su}(2)$ in physics? I have studied a lot of mathematics including representation theory and differential geometry, so I understand SU(...
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Generators of so(7)

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is isomorphic to $\mathfrak{so}(7)$. Can ...
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A subspace is invariant by the Lie group if it is invariant by the Lie algebra

Let $G$ be a connected Lie group and $$\varphi:G\to \mathrm{GL}(V)$$ a representation on a finite dimensional real vector space $V$. Let $$\psi:\mathfrak{g}\to\mathrm{End}(V)$$ be the associated Lie ...
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Coproduct of Lie algebras

Fix a commutative ring $k$ and look at the category of Lie algebras over $k$. How do coproducts in that category look like? Notice that what is usually called the "direct sum" of Lie algebras is not ...
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duality for (co)homology of Lie algebras

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. What is the relationship between $H_k(\mathfrak{g};R)$, $H_{n-k}(\mathfrak{g};R)$, $H^...
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Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
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Are the two definitions of the BGG category equivalent?

Let $\mathfrak{g}$ be a finite dimensional complex semisimple lie algebra, the BGG category $\mathcal{O}$ is defined as the set of $\mathfrak{g}-$ module $M$ such that $M$ is finitely generated; $M$ ...
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236 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let $G$ be an affine group scheme, and $\mathrm{Dist}(G)$ its hyperalgebra. I am wondering what is the relationship between $\mathrm{Dist}$(G) and $G$ interms of Cohomology? Is there a cohomology ...
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Weyl group of a non-symmetrizable Generalized Cartan Matrix

Let $A$ be a generalized Cartan matrix on the index set $I$. Define the Weyl group of $A$ as the Coxeter group on the basis $I$ with $m(i,j)=2,3,4,6,\infty$ according to whether $A_{ij} A_{ji}$ is $0,...
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Irreducible representations of $\mathfrak{sl}_3\mathbb{C}$

I am working through the exercises in Fulton and Harris's Representation Theory, and am stuck on two on page 189. Let $\text{Sym}^2V$ denote the second symmetric power of the standard 3-dimensional ...
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Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with $\lambda(H_\alpha)...
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Pullback of a 3-form to SU2

I have a left invariant 3-form, $\sigma$ on an simply connected Lie group, $G$ whose value at the identity is $\sigma=\langle[x,y],z\rangle$, where $\langle\cdot,\cdot\rangle$ denotes an invariant ...
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An exercise in Serre's Lie algebra book

Let $k$ be a commutative ring. Prove that a Lie $k$-algebra $\mathfrak{g} = 0$ iff $U\mathfrak{g} = k$. Use the adjoint representaion. Here is my attempt at it: The only non-trivial statement is ...
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Where does the symplectic structure on coadjoint orbits of Lie groups on their Lie algebras come from?

I have read in several places that if $\Omega$ is the coadjoint orbit of $\zeta \in \mathfrak{g}^*$, the map from $G \to \Omega$ that sends $g \mapsto Ad^*(g)(\zeta)$ gives a surjection, and taking ...
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Finding the sub-algebras (up to conjugation) of $\frak{sl}_2(\mathbb{C})$

I am trying to find the connected subgroups of the Lie Group $SL_2(\mathbb{C})$ up to conjugation. My thought was to find the sub-algebras of the Lie Algebra $\frak{sl}_2(\mathbb{C})$, and then ...
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Automorphisms of root system and its action on quotient of weight lattice

Let $\Phi$ be a root system in $E=\mathbb{R}^l$, $\Delta=\{\alpha_1,\cdots,\alpha_l\}$ be a base of $\Phi$. Let $\langle\alpha,\beta\rangle:=\frac{2(\alpha,\beta)}{(\beta,\beta)}$ for $\alpha,\beta\...
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When are the roots of a Lie algebra the differences of the weights?

In books about group theory written for physicists, there's a strange procedure used to find the roots of a Lie algebra. The steps are: Write down the fundamental representation. This is the '...
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Beilinson-Bernstein localization, equivariant modules

I have a question regarding the equivariance in the Beilinson-Bernstein localization. Let $G$ be an simply connected algebraic group over a field of charateristic $0$ and $K$ a closed subgroup of $G$ ...
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Direct sum of injective modules is injective.

By the Bass-Papp Theorem, for a unital ring $R$, any direct sum of injective left $R$-modules is injective if and only if $R$ is left Noetherian. I would like to restrict my consideration to an ...
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Is there a relationship between the trace and the Clifford/geometric product?

In what follows, let $V=\mathbb{R}^n$ (although the following probably applies also to a larger number of finite-dimensional spaces). We assume throughout that we have made a choice for an inner ...
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Real representations of Lie groups

I am looking for a reference that discusses the representation theory of Lie groups, especially the classical groups, on real vector spaces. It seems that almost every text I've looked at restricts ...
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309 views

Orthosymplectic Lie Superalgebra

I am trying to work out a presentation for the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2n)$. I am following Musson's book "Lie Superalgebras and Enveloping Algebras". From what I ...
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237 views

Connection between Lie Theory and differential Galois theory.

I would like to know if there exists a relationship between Lie theory ( Lie groups and Lie algebras ) and Linear differential equations through Differential Galois theory. Thanks in advance for your ...
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226 views

Classification of irreducible (g,K)-modules for other g than sl2

Harish Chandra showed how to associate to an admissible representation $(\pi,V)$ of a real semisimple Lie group $G$ the so-called Harish-Chandra module $V_K$ of $K$-finite vectors in $V$. This is a $(\...
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Does solvability of Lie algebra have useful application in study of PDEs?

If certain Lie algebra is solvable then what difference this algebra would create in application point view for PDEs ? For example, in case of ODE of fourth order admitting three dimensional solvable ...
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Multiplicities in weight diagram of representations of $\mathfrak{sl}(3,\mathbb{C})$

In the weight diagram of an irreducible (finite dimensional, complex) representation of $\mathfrak{sl}(3,\mathbb{C})$, there are 'rings' of weights in the shapes of triangles or hexagons. Is there an ...
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120 views

Radical of an infinite dimensional Lie Algebra

I am studying Lie Algebras and I just encountered the notion of radical $R$ of a finite dimensional Lie algebra $L$ over a field $F$, the maximal solvable ideal. Since the dimension of $L$ is finite, ...
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How to understand Weyl chambers?

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane ...
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cohomology of general linear group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $\mathrm{GL}_n(\mathbb{Z}_2)$ be the group consisting of all $n\times n$ matrices with entries in $\mathbb{Z}_2$ with non-zero determinant. What is the ...
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Existence of product $\ast$ in a Lie algebra so that $[X,Y]=X*Y-Y*X$

I've been studying particle physics, and studying Lie algebra using physics text book doesn't give me enough information, so I'm asking my question here. Given a Lie algeba $\mathcal{A}$ where ...