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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Usage and determination of “rank” and “dimension” of groups & representations

Physicist here. I seem to see conflicting statements about the rank of some groups I've come across lately. A paper I'm reading states that $SO(6)$ is rank 3 and therefore its Cartan subalgebra ...
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116 views

Regarding the definition of vector field flow

To make the connection to the Lie derivative, let $t \mapsto \Phi^X_t$ be the 1-parameter group of diffeomorphisms (or flow) generated by the vector field $ X $. The differential $ d\Phi^X_t ...
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183 views

The enveloping algebra of a finite dimensional Lie algebra has no zero divisor

Let $L$ be a complex, finite dimensional Lie algebra. It is well-known that the graded associative algebra of the enveloping algebra $U(L)$ is isomorphic to the symmetric algebra $S(L)$. Therefore ...
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131 views

Integral forms of loop algebras.

The question following is about integral forms for semisimple Lie algebras and loop algebras constructed from them. Let $\frak g$ a finite-dimensional Lie algebra over $\mathbb C$ and $L(\frak ...
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111 views

Question concerning semisimple Lie algebras

I'm currently solving a problem in Fulton's Representation Theory A first course and I'm not sure why a particular result is true. One part of the problem (exercise 14.15 if anyone is interested) ...
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123 views

The Weyl Group of $F_4$

The Weyl Group of $F_4$ is of order $1152=2^{7} \cdot 3^{2}$. By Burnside's theorem the group is solvable. Is there a way to see solvability from the root system? Is it possible to see the order of ...
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344 views

Invariant inner product $\langle\,,\rangle$ on a Lie algebra

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We can use the Killing form to identify $\mathfrak{h}$ and $\mathfrak{h}^*$ ...
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36 views

Coset Space as a Representation of a Lie Algebra

I'm reading through some notes (about the use of Lie groups/algebras in physics) obtained from a friend from a class that took a while back, and I can't quite figure out where one thing regarding some ...
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26 views

Taylor series identity for polynomial using Lie group

The following question is from Kirillov's Introduction to Lie Groups and Lie Algebras, and my attempt is the following: $$\sum_{n\geq ...
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38 views

Why are $\mathfrak{pgl}_n\simeq\mathfrak{sl}_n$ when characteristic does not divide $n$?

Suppose $k$ is some algebraically closed field whose characteristic does not divide $n$. Why can we identify the lie algebras $\mathfrak{pgl}_n\simeq\mathfrak{sl_n}$ of the projective linear group and ...
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47 views

Standard Basis of $SU(2)$--where does the 1/2 come from?

The most common matrix representation of $SU(2)$ is given by $$ \begin{pmatrix} a & b\\ b^* & -a^*\\ \end{pmatrix} $$ where $a,b\in\mathbb{C}$. If we denote real components by the subscript ...
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47 views

The simplest way to present a Lie algebra to a wide audience?

I would like to get suggestions from you as to the best way to present the idea and contents of Lie algebras to a wide public of people with no detailed background in maths. What wlould you explain to ...
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35 views

Can the parameter $t$ in the exponential map $e^{tX}$ be complex?

From a Lie algebra to a Lie group, can the parameter $t$ in the exponential map $t\rightarrow e^{tX}$ be complex? If the Lie algebra is a complex one, this is legal, right?
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30 views

Action of $sl(2,\mathbb{C})$ on Dual of Polynomials does not Exponentiate

Let $V$ be the space of holomorphic polynomial functions in two complex variables $\xi,\eta$ and let $V^\ast$ be its dual space with subspace $W$ of linear functionals of the form $Df(1,0)$ where $D$ ...
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43 views

Is the exponential map of complex spin group surjective?

The complex spin group $Spin(n,C)$ is defined as the double cover of $SO(n,C)$. If the the exponential map is surjective, it will give a parametrization of this Lie group. Is it true for this ...
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15 views

Quotient of the abelian Lie group $(\mathbb{C}, +)$ by a full rank lattice

How can I show that a quotient of the abelian Lie group $(\mathbb{C}, +)$ by a full rank lattice has no faithful finite-dimensional linear representation as a complex Lie group? I was thinking of ...
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26 views

Prove that a subgroup in a Lie group is homogeneous

Let $\mathbb E:=(\mathbb R^4, \cdot)$ be a Carnot group whose Lie algebra is given by $\mathfrak g=V_1\oplus V_2 \oplus V_3$, where $V_1=span\{X_1,X_2\},$ $V_2=span\{X_3\},$ $V_3=span\{X_4\}$, the ...
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42 views

Nonstandard analysis, Lie groups and universal enveloping algebras

The idea of nonstandard analysis is to combine finite quantities with infinitesimals. And, back in the day, Lie algebras were roughly considered the "infinitesimal elements" of Lie groups. Say we want ...
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47 views

Root system of an abelian lie subalgebra.

Let $L$ be a lie algebra and $H$ an abelian subalgebra of $L$ such that each element of $h \in H$ is diagonalizable under the adjoint representation. So there exists a basis of common eigenvectors for ...
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29 views

Affine Kac-Moody Group Isometry of a Manifold

An isometry of a Riemannian manifold is an infinitesimal displacement generated by a Killing vector field $V=\zeta^aV_a=\zeta^aV_a^i\frac{\partial}{\partial x^i}$. If the isometry corresponds to the ...
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79 views

Representation of ${\frak gl}(n, \Bbb R)$ in ${\cal C}^\infty(\Bbb R^n, \Bbb R^n)$.

I'm trying to check that $\rho\colon{\frak gl}(n, \Bbb R) \to {\frak gl}({\cal C}^\infty(\Bbb R^n, \Bbb R^n))$, given by $\rho(A)\colon {\cal C}^\infty(\Bbb R^n, \Bbb R^n)\to {\cal C}^\infty(\Bbb R^n, ...
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30 views

Tensor formula in SU(3) representations

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
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28 views

Left invariant metrics on a Lie group coming from Lie algebras

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. If we have an inner product on $\mathfrak{g}$ then we can create a left invariant metric $d_G$ on $G$ by translations. On the other hand ...
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16 views

Tensor products and decomposition of $SU(3)$ representations

For each finite irreducible representation of Lie algebra $su(3)$ one knows that it is characterized by highest weight $(\lambda_1, \lambda_2)$ with integral entries. In this notation, $(1,0)$ is ...
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Symmetry of Kahler metric on based loop group

The based loop group, $\Omega G$, is known to admit a Kaehler metric, given as \begin{equation} g(X,Y)=2\sum_{k>0}k\textrm{Tr}(X_{-k}Y_k), \end{equation} this is given in page 150 of Segal and ...
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40 views

Outer automorphism group of $\mathfrak{su}(n)$

I'm reading about the special unitary Lie algebras, and seen it said that complex conjugation is not an inner automorphism of $\mathfrak{su}(n)$ for $n>2$. If there an easy way to see this? I ...
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21 views

$24\otimes 24$ notation for representations of $SO(24)$

I was working through a set of string theory notes, and I came across the notation $24 \otimes 24$ to denote a reducible representation of $SO(24)$, but I am not familiar with the notation. I have ...
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20 views

Two different definitions for Lie Algebras for closed subgroup of $GL_n(\mathbb R)$

Let $G$ be a closed subgroup of $GL_n(\mathbb R)$. There are two definitions for $\mathrm{Lie}(G)$ $\mathrm{Lie}(G) = \{ \gamma'(0) : \gamma : (-\epsilon, \epsilon) \rightarrow G \text{ is ...
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31 views

How does a Lie algebra act on a tensor product of L-modules?

What is the $L$-module structure on $V\otimes W$ where $L$ is a Lie-algebra and $V,W$ are $L$-modules? The following question is related but I can't find the definition in there: tensor product of ...
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46 views

Lie Bracket on Lie Algebra vs. Lie derivative on corresponding group orbit tangent vector field

Suppose I have a matrix group, for example the 2-d affine group and two elements of Lie algebra, A and B, expressed as 3x3 matrices. The Lie bracket is [A,B]=AB-BA. Suppose on the other hand I ...
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24 views

Local Lie derivative on $G$-space at the zero of a vector field

Let a Lie group $G$ act on a manifold $M$ and let $X\in Lie(G)$. For now suppose $G=T$ is a torus (but the answer to this question should hold for $G$ abelian). $L_X$ is a vector field on $M$, at a ...
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57 views

Derivatives of solution to Schrödinger equation

Consider the differential equation (Schrödinger, but rewritten to be pleasing to Lie algebraic eyes): $\frac{d U(t)}{dt} = c(t)U(t)$ where $c(t)=a+w(t)b(t)$, $a,b \in \mathfrak{su}(n)$ and $w$ is a ...
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45 views

Relationship between solutions of two matrix differential equations

Given a ($4\times4$ in the important case) matrix differential equation: $\frac{d U_t}{dt}= A_t U_t$ where $U_t \in SU(n)$ and $A_t \in \mathfrak{su}(n)$. What is the relationship between the ...
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60 views

Why doesn't the “naive” scalar product for $SO(n)$ yield something invariant?

By definition, for $SO(n)$ we have $g^T g=1$ for $g \in SO(n)$. Given some vector $v \in V$ and some representation $R: SO(N) \rightarrow \mathrm{Lin}(V)$, the defining condition above tells us ...
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40 views

Question related to the Campbell-Baker-Haussdorf formula

I have two operators $A$ and $B$ such that $[A,B] = C$ $[A,C] = -2A$ $[B,C] = +2B$ and I would like to obtain an expression for $\log(\exp(A+B)\exp(-B)\exp(-A))$. Is it a linear combination of ...
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45 views

Tangent space of coadjoint orbit

Let $\xi \in T_xOx$ be a tangent vector at $x \in O_x :=\{\mathrm{ Ad} _{u}^*(x); u \in G\}$ for $x \in g^*.$ ($g$ is the Lie-Algebra) Then I read that this $\xi$ can be represented as the velocity ...
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List of simple roots in the H-basis for various Lie algebras?

There are four usual bases one can use to express the roots and weights of a given algebra. The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$. The ...
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35 views

Under what conditions is the homology of a dg coalgebra a graded coalgebra?

I'm trying to get a feel for some differential graded (dg) structures. Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct ...
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35 views

Map for roots of a Lie group to roots of a special subalgebra?

For regular subalgebras $h$ of some group's Lie algebra $g$, $$ h \subset h $$ the root system of the subalgebra is a subset of the root system of the original's group algebra. Subalgebras whose ...
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33 views

signature function of Weyl group element in LieArt

I am currently using LieArt Mathematica package for some calculations in Lie algebra, I am wondering if there is a way to know what is the signature of a Weyl group element, it seems the package can ...
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Why are chains topologically analoguous to distributions?

This question is related to my other question here but is different enough that I thought I might ask separately. At the nLab page on rational homotopy theory it is stated that chains are ...
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What is the connection between $\widehat{\mathbb Q G}$ and distributions near the identity of $G$?

I'm studying Quillen's rational homotopy theory and trying to understand this MathOverflow description of Quillen's functor provided by Hiro Lee Tanaka. When discussing connections between how ...
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Computing the Cohomology of Lie groups

In Bredons "Topology and Geometry" [Chapter V, section 12] the following theorem is proven: If $G$ is a compact connected Lie group its $k$-th cohomology $H^k(G,\mathbb{R})$ is isomorphic to the ...
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75 views

Standard cyclic module of sl2

Let $L=\mathfrak{sl}(2, \mathbb{F})$, $B$ a standard Borel subalgebra. I am trying to solve exercise 20.4 from J.E. Humphreys "Introduction to Lie Algebras and Representation Theory", but I am stuck. ...
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Prove that actions commute

I am trying to understand a proof from Kobayashi and Nomizu (foundations of differential geometry, p. 280). Suppose that we have Lie subalgebras $a<b<g$, with $g$ the Lie subalgebra of $SO(n)$ ...
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Lie groups and Lie algebras for matrices

Recently, I stumbled over a few things in very basic Lie group / Lie algebra theory concerning matrix groups. Basically, my question is: Is there a way to canonically understand all the Lie groups ...
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76 views

Flow of a left invariant vector field on a Lie group equipped with left-invariant metric and the group's geodesics

I think the answer to my question is known to many other people, but I'm still getting confused. Let $G$ be a (possibly infinite dimensional also) Lie group and $g$ be its Lie algebra. Consider the ...
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Certain Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$

Is there a lie algebra structure $ [ \;. ] $ on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(\mathbb{S}^{2})$ which is not isomorphic to the standard structures but satisfies the following: ...
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Write down the explicit form of the $15$ Killing vectors of the 5-sphere

I am looking for a way to write down explicitly the $15$ vectors which are generators of $SO(6)$ in polar coordinates on the $5$-sphere. In particular I have the round metric $$g_{\mu\nu} = \left( ...
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52 views

Help! How to derive the result related to Darboux derivative?

First, define Darboux derivative. There is one Lie group $G$ and one manifold $M$. Let $\phi:M\rightarrow G$ be a smooth map. The Darboux derivative $\Delta(\phi):TM \rightarrow M\times \mathfrak g$ ...