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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highest weight of adjoint represesentation

Let $\mathfrak{g} = \mathfrak{gl}(3,\mathbb{C})$ and let $\mathfrak{h}$ be the subalgebra of $\mathfrak{g}$ consisting of diagonal matricies. Then for $1 \leq i \leq n$, let $\epsilon_i \in ...
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39 views

Matrix representation of a 6-dimensional Lie algebra

The question is about the matrix representation of the following 6-dimensional Lie algebra, with 6 generators $t_1,t_2,t_3,t_4,t_5,t_6$. This Lie algebra is nilpotent, non-abelian, non-reductive and ...
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19 views

Lie element in a non-commutative algebra?

While I am reading a paper, there is a weird, at least for myself, notion I have never seen: let $R:=\mathbb{Q}_\ell\{\{X,Y\}\}$ be a $\mathbb{Q}_\ell$-algebra of formal power series in non-commuting ...
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18 views

How to compute the center of the universal enveloping algebra of a simple Lie algebra

Given a simple Lie algebra $\mathfrak g$ over $\mathbb C$. Then the center $Z(\mathfrak g)$ of the universal enveloping algebra of $\mathfrak g$ is a polynomial algebra in $l$ generators, where $l = ...
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25 views

How to compute Casimir elements of $g \otimes g$?

Let $g$ be a Lie algebra. How to compute Casimir elements of $g \otimes g$? I am asking this question because in the book a guide to quantum groups, page 80, there is an equation $r_{12} + r_{21}=t$, ...
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17 views

Criterion for semi-simplicity of Lie algebra generated by vector fields

Suppose I have a finite collection of smooth vector fields $V:=\{V_1,...,V_k\}$ on a smooth manifold $M$. Moreover suppose that the Lie algebra $g$, generated by $V$ (where the Lie bracket is defined ...
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50 views

Lie algebra of a finite group

I'm trying to find the normalizer of the Pauli group $G_n$ (as a subgroup of $SU(2^n)$) utilizing Lie algebras, as is done in a reference to find the normalizer of the Heisenberg group $HW(n)$. There, ...
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27 views

Special linear Lie algebra.

When I am reading the Notes on Lie algebra by Hans Samelson, there is a sentence: The standard skew-symmetric (exterior) form $det[X, Y ] = x_1y_2−x_2y_1$ on $\mathbb{C}^2$ is invariant under ...
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17 views

sl_2 triple and comparing two nilpotent orbits in a Lie algebra

I've been working on a question on local Galois deformation theory. It eventually boils down to the following questions on nilpotent elements in a complex semisimple Lie algebra $\mathfrak g$ which is ...
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34 views

Lie algebras over non-algebraically closed field

Could someone recommend a book on Lie algebras over a not-necessarily algebraically closed field? I am particularly interested in the representation theory, so it should contain all the usual results ...
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10 views

Arrive to the group law in exponential coordinates using the vector fields expressed in exponential coordinates

I need an help with the following question. I have this definition for Engels group $\mathbb E$: it is the only connected and simply connected Lie group that has the Engels algebra $\mathfrak g$ as ...
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42 views

Advantages and disadvantages of defining Lie bracket via right invariant vs. left invariant vector fields

I was just wondering what are the advantages and disadvantages of the two conventions used for defining Lie brackets? For example, if we use right invariant vector fields as the convention for ...
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21 views

Adjoint action on Lie algebra su(2) ($A \in SU(2), X \in \mathfrak{su(2)} \Rightarrow AXA^{-1}\in \mathfrak{su(2)}$)

I am trying to understand ho $SU(2)/\{\pm I\} \cong SO(3)$ (see: how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$) but i am not sure about the adjoint action. In especially, as I understand, the adjoint ...
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28 views

Lie functor produces an antihomomorphism in Lavendhomme's synthetic differential geometry text?

Classically the Lie functor maps a Lie group homomorphism to a Lie algebra homomorphism. But in Proposition 15 on page 249 in Basic Concepts of Synthetic Differential Geometry, Lavendhomme states that ...
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21 views

Cartan type of real forms of semisimple Lie algebras

In lists of Cartan's classification of the real forms of semisimple complex Lie algebras, the (isomorphism classes of) real forms of the families of classical complex semisimple Liealgebras A,B,C,D ...
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13 views

Flag varieties from the representation of a solveable Lie algebra

I've been reading Lie Algebras, and I've come across this problem: "Let $\mathfrak{g}$ be a solveable Lie Algebra over $\mathbb{R}$. $V$ a vector space over $\mathbb{R}$, and $\rho$ a representation ...
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23 views

Showing $\ker(\phi)$ is a $\mathfrak{g}$-invariant subspace of $V$.

Let $\phi:V\to W$ be a linear map between the irreducible $\mathfrak{g}$-modules $V,W$. I want to show that $\ker(\phi)$ is a $\mathfrak{g}$-invariant subspace of $V$ and $\text{im}(\phi)$ is a ...
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7 views

Proof verification: $\mathfrak{g}$-module $V$ is irreducible if and only if $V\subseteq \mathfrak{g}(v)$ for each nonzero $v\in V$.

I want to prove that: The $\mathfrak{g}$-module $V$ is irreducible if and only if $V\subseteq \mathfrak{g}(v)$ for each nonzero $v\in V$. Since $\mathfrak{g}(v)$ for any non-zero $v\in V$ generates ...
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22 views

Is there a general formula for the following Lie algebra quantity?

Consider the generators of $SO(n)$, written as $M_{\mu\nu} = - M_{\nu\mu}$ and they satisfy $$ \left[ M_{\mu\nu} , M_{\rho\sigma} \right] = i \left( \eta_{\nu\rho} M_{\mu\sigma} + \eta_{\mu\sigma} ...
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29 views

Map to a Lie group as exponential of a map to the Lie algebra

Let $U$ be an open subset of $\mathbb{R}^n$, $G$ a Lie group and $f:U\rightarrow G$ a smooth surjective map. Under which conditions there is a smooth function $\phi:U\rightarrow\mathfrak{g}$ such ...
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28 views

Eigenvalues of skew-symmetric matrix $(x_i-x_j)_{ij}$

Given a set of positive reals $x_1,\ldots, x_n$, construct a skew-symmetric matrix $A=(a_{ij})=x_i-x_j$, in matrix form, ...
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36 views

In what sense do roots span a vector space?

If I am in two dimensional space, the meaning I have for the span is the usual one from linear algebra. But I do not know what it means to say the roots in a root system, R, span the inner product ...
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26 views

Projective covers for simple Lie algebras in characteristic zero

I want to use the projective cover for the trivial module over a finite-dimensional simple Lie algebra in characteristic zero. Is there a reference that I can quote to assert its existence?
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16 views

Deriving SO(3) multiplication rule from so(3) commutation rules

I've heard it said that the commutation relations of the generators of a Lie algebra determine the multiplication laws of the Lie group elements. I would like to prove this statement for $SO(3)$. I ...
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29 views

Understanding Cartan clasification

In class we defined Cartan subalgebra h (of g) as maximally abelian subalgebra containing only ad diagonalizable elements. ad is adjoint map $ad_{H_i}(E) =[H_i, E]$. I have a couple of questions ...
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5 views

How to determining the roots and step operators of a L(SO(3)

I've come across a question in my "Lie Group and Lie Algebras for Physicists" course that asks me to determine the a basis for the Cartan subalgebra of $L(SO(3))$ and "hence find the roots and write ...
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22 views

sympletic form of the hyperboloid

I'm working with the Hyperboloid of two leefs $H_2$ as a coadjoint orbit of $\mathfrak{sl}^*(2, R)$. I know that $H_2$ is a symplectic manifold by the following theorem: Given a Lie group G and ...
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14 views

Deriving the structure constants of the SO(n) group

The commutation relations for the $\mathfrak{so(n)}$ Lie algebra is: $$([A_{ij},A_{mn}])_{st} = -i(A_{j[m}\delta_{n]i}-A_{i[m}\delta_{n]j})_{st}$$ where the generators $(A_{ab})_{st}$ of the ...
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25 views

The Levi-Civita connection on $S^3$ and $SU(2)$

The fundamental theorem of Riemannian geometry implies that there is a unique symmetric (i.e., $\Gamma^{a}_{bc}=\Gamma^{a}_{cb}$, using a coordinate basis) connection on the three-sphere, $S^3$ which ...
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8 views

How to derive second order casimir invariant for u(N)

I am told that for semi-simple lie algebra, second order casimir invariant is defined to be: $I_2=g^{\mu\nu}\rho(e_\mu)\rho(e_\nu)$, where summation over repeated indices is used. $g^{\mu\nu}$ is the ...
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8 views

How to find a highest weight vector in a tensor product of two representations of $sl_2$.

Let $V_\lambda, V_{\mu}$ be two representations of $sl_2$. How to find a highest weight vector in a tensor product of two representations of $sl_2$? Thank you very much.
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13 views

Index of Hessian of Composition of maps

Given a Lie group $G$ and a pair of smooth maps: $f:\mathbb{R}^n \rightarrow G \\ g: G \rightarrow [0,1]$ with $g$ possessing a single global optima, but potentially many saddle points. consider ...
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24 views

A chain between two positive roots having same length

Let $\Phi$ be an irreducible root system and $\Phi^+$ a system of positive roots. Denote by $\Delta=\{\beta_1, \beta_2,\ldots, \beta_n\}$ the corresponding base. My question concerns Jyrki's answer ...
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27 views

Intersection of a family of closed Lie subgroups

If I have a Lie group $G$, and $\{H_{\alpha}\}_{\alpha\in A}$ is a family of closed Lie subgroups with Lie algebra $\{\mathfrak{h}_{\alpha}\}_{\alpha\in A}$, it's easy to see that $\bigcap_{\alpha} ...
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26 views

rank of exterior derivative

As i am sitting here reading some lecture notes on lie algebras I found myself getting stock because of the word "rank". As I understand rank, it's just the dimension of the image of a linear map and ...
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12 views

Understanding Weights and Roots

I'm refering to this book called Semi-Simple lie algebras in Particle Phsics by Cahn for understanding weights and roots as given by our instructor. It has a definition of weights on Pg.33 which is ...
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19 views

Find the dimension of the real Lie algebra su(n)={A∈sln(C)|A+A∗ =0}, A∗ =A(conjugate transpose)

Find the dimension of the real Lie algebra su(n)={A∈sln(C)|A+A∗ =0}, A∗ =At. Also have to Show that the map A∗ =At. C⊗R su(n)→sln(C), z⊗A􏰀→zA su(n)={A∈sln(C)|A+A∗ =0}, is an isomorphism (of complex ...
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20 views

Find a stratification of a Carnot group

My problem is how to find the stratification in an Carnot group (see here pag. 3). Let's make an example. Let $\mathbb R^3$ be endowed with a composition law $*$ that makes $(\mathbb R^3, *)$ a Lie ...
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24 views

Can a point have a nontrivial isometry group?

This question is extremely related to this other question. In fact, a positive answer here directly implies a positive answer there. However, since it is a mathematically different question I decided ...
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31 views

One parameter subgroups

I am trying to solve a image registration problem where my deformations are one parameter subgroups of diffeo. i.e., solution to the equation; $ \partial \varphi(x,t) = v(\varphi(x,t)). $ The ...
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25 views

Continuous complex finite dimensional irreducible representation of $GL_n( \mathbb C )$

What are all the continous finite dimensional irreducible representation of $GL_n( \mathbb C )$? I tried the following since the continuous irreducible representations of $GL_n( \mathbb C )$ are in ...
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23 views

Finite-Dimensional Representations of The Classical Groups in Tensor Spaces: Invariant Theory

I. When we study finite-dimensional irreducible representations over the space of general tensors (e.g.,Chapter 13, Group Theory in Physics by Wu-Ki Tung), is it enough to obtain all the ...
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Are Whittaker functions for lie algebras the same as Whittaker functions for corresponding Lie groups?

Some papers call some Whittaker function the Whittker function for some Lie group $G$. Some other papers call some Whittaker function the Whittker function for some Lie algebra $g$. Is the Whittaker ...
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24 views

How is a Sim3 lie group used to project a 3D point?

I have a $Sim3$ transformation with scale, rotation and translation that describe the camera to world transformation. How can I use it to transform a point in world coordinates to camera coordinates? ...
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15 views

From Generators of Lie Groups to Representations

Howard Georgi in his book on Lie algebras mentions a very interesting formula $$\dfrac{\partial}{\partial a_b}e^{ia_aX_a} = \int_{0}^{1} ds \ e^{isa_aX_a}(iX_b)e^{i(1-s)a_cX_c}.$$ How can one ...
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10 views

Showing the commutivity of a Casimir operator with generators

Suppose we have a semi simple Lie algebra. By using the commutation relations for the generators, prove that by defining a quadratic Casimir operator $C_2 = g^{ab}T_a T_b$, we have that $[C_2, ...
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14 views

structure of the normalizer of an abelian lie algebra

I don't know much about Lie-algebra. The following question might be obvious (I find it on one Serre's paper, if my statement is wrong, please point it out). Let $V$ be a 2-dim vector space over the ...
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14 views

Lie bracket on $\Gamma(TM\oplus (M\times \mathfrak{g}))$?

Let $M$ be a smooth manifold and $(\mathfrak{g}, [\cdot, \cdot])$ be a $\mathbb R$-Lie algebra. How can I introduce a Lie bracket on $\Gamma(TM\oplus (M\times \mathfrak{g}))$? Above $TM\oplus ...
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18 views

explicit special unitary matricies

The general form of a 2 by 2 special unitary matrix is well known. Are the general forms of 3x3 and 4x4 special unitary matrices possible to explicitly write down in a similar way?
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14 views

How to show that a finite-dimensional irreducible $o(3,\Bbb C)$-module is contained in a $gl(n)$ module?

For each pair of integers $i,j$ satisfying $1\leq i,j\leq n$, I want to show that there is an $o(3)$ subalgebra of $gl(n)$ and I am not sure how exactly to show this. We know that the Lie algebra ...