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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Conjugate Representations for $\mathfrak{sl}(2,\mathbb{C})$

Let $\mathfrak{sl}(2,\mathbb{C})$ be the complex Lie algebra of $SL(2,\mathbb{C})$ and $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ be its realification; that is $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
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337 views

jacobian involving SO(3) exponential map: $\log(R \exp(m))$

I would like to compute the 3 × 3 Jacobian of $$ \log(R \exp(m)) $$ with respect to the 3-vector $m$, evaluated at $m=0$. In the above, $\exp$ is the exponential map from so(3) to SO(3), $\log$ is ...
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143 views

Root system of a Lie Algebra

Could anybody help me to solve this problem with roots system? Be $\Phi$ an irreducible root system. $\Phi^{+}$ a choice of positives roots in $\Phi$. Prove that if $(\alpha,\beta)\ge0$ $\forall ...
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65 views

Invariants of representation theory of Lie groups

How to compute the determinant of a representation of an element of the special linear group? How do I argue that it doesn't change? (@Marek: @rschwieb: Yes well, given one represenation (with ...
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43 views

Lie Derivative in Projective Hilbert Space

In considering a projective Hilbert space, $P(H)$, for linear maps (tensors) of vectors in the space, $A^{a}_{b}v_{a}=u_b$, is there a natural definition for the Lie Derivative for such linear maps? ...
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124 views

Submanifold of a Lie group - tangent space

Let $G$ be a compact Lie group and $H, H' \leq G$ Lie subgroups. Consider the set $M = H' \cdot H = \{h\cdot h' \ \vert \ h \in H, h' \in H'\}$. Is it possible to describe explicitly the tangent space ...
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74 views

Is there any Lie algebra that is not constructed from an associative algebra

I see in Wikipeida that every Lie algebra is either constructed from an associative algebra by defining: $[x,y]=xy-yx$, or a subalgebra of a Lie algebra thus constructed. Where can I find a proof? ...
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Followup question in Brian Hall's Lie Groups and Algebras.

In ex 9, page 60, he writes down that in order to prove that each invertible matrix $A$ can be written as $A=e^X$, where $X\in M_{n\times n}$, one need to use the fact that if $A$ is unipotent then ...
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55 views

Basics of Lie 2-algebras?

Could somebody (simply) explain the basics foundations of Lie 2-algebras, and some basic practical applications ? For instance, does it exist a 3-map (equivalent to the 2-map commutator for Lie ...
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154 views

The universal enveloping algebra of a loop algebra as a quotient of the free associative algebra.

Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra and set by $\tilde{\mathfrak{g}}:=\mathfrak{g}\otimes_{\mathbb C} \mathbb{C}[t,t^{-1}]$ its loop algebra. How to express the ...
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104 views

$1$-parameter subgroups in $GL_n(\mathbb{C})$

I came across this link on planetmath and a few facts on that link are confusing me. According to planetmath, any $1$-parameter subgroup in $GL_n(\mathbb{C})$ arises from the exponential map. That ...
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39 views

The nonexistence of nontrivial solvable series in $M_n(k)$

I am a bit confused about semisimple Lie algebras. For the sake of simplicity, let's take $\mathfrak{g}=M_n(k)$ where $k=\bar{k}$. According to Wiki, $M_n(k)$ is solvable if the radical of $M_n(k)$ ...
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45 views

Inverse boson operator realization of $\mathfrak{so}(3)$

This is actually a homework problem. The inverse boson operators $a^{-1}$ and $\left(a^\dagger\right)^{-1}$ are defined as $$a^{-1} |n\rangle = \frac{1}{\sqrt{n+1}} |n+1\rangle$$ ...
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32 views

set of roots satisfying a minimal condition related to the induced Killing form

Let $\mathfrak{g}$ a finite-dimensional complex simple Lie algebra with Cartan subalgebra $\frak h$. Let denote $(\cdot,\cdot)$ the non-degenerate bilinear form on $\frak h^*$ induced by the Killing ...
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72 views

Question about root space

Let $\mathfrak{g}$ be a Lie algebra and consider $\operatorname{Rad}(\mathfrak{g})$, the radical of $\mathfrak{g}$, that is, the sum of all solvable ideals in $\mathfrak{g}$. Suppose that we have the ...
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202 views

$SU(2)$ is a covering space of $SO(3)$.

The method of topology is very clear.Then there's a question asking to use adjoint representation of lie group $SU(2)$ $(\operatorname{adj}:SU(2)\to GL(su(2)))$to prove this. I can't solve this .
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57 views

$G_1$-Scalar factors for Clebsch-Gordan coefficients for $ U(n)$

when evaluating the $G_1$ scalar factors for CGC's of $U(n)$ it seems that some of the factors are undefined. The explicit formula for the evaluation of the scalar factors is Eq. (6) in 18.2.8 of N.J. ...
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127 views

First-order derivatives in differential forms calculus

Let $d$ denote the Cartan differential, and let $\delta$ denote the codifferential. The underlying domain is not important for what follows. The canonical generalization of the Laplace-operator ...
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43 views

Subspace of a homogeneous space.

Given two homegenous spaces $\frac{G}{H}$, $\frac{A}{B}$ with $A\subset B$ is there a way to prove that $\frac{A}{B}\subset \frac{G}{H}$ ie that $B\subset A\cap H$? In particular I would like to ...
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132 views

PBW Theorem applied to graded Lie algebras

Fix a $\mathbb Z_+^n$-graded Lie algebra ${\frak a}=\oplus_{r \in\mathbb Z_+^n}^{} {\frak a}[r]$ such that ${\frak g}:={\frak a}[0]$ is a finite-dimensional semisimple Lie algebra over the complex ...
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84 views

Cartan subalgebras of a loop algebra.

For an algebraically closed field $\mathbb F$ of characteristic zero, a finite-dimensional Lie algebra $\frak G$ has a Cartan subalgebra and these subalgebras are conjugated in a certain sense. Let ...
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67 views

The form of a subgroup of $GL(n,K)$ when the derived group is of certain form

The famous Lie-Kolchin theorem in the theory of algebraic groups states: Let $G$ be a connected solvable subgroup of $GL(V)$, $0 \neq V$ finite dimensional. Then $G$ has a common eigenvector in ...
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23 views

When does $C_G(s) \times Cl_G(s)s^{-1}$ equal $G$

I have read on James E. Humphreys' Linear Algberaic Groups If $G$ is an algebraic subgroup contained in $GL(n,K)$, and $s$ is a semisimple element of $G$, then $\mathfrak{g}$ has the ...
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83 views

Does triality survive in product Lie groups?

Look at the following diagrams of Lie groups ...
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106 views

Finite order automorphisms of Lie algebras

Let $\Gamma$ be a Dynkin diagram automorphism of diagram type $A_{2n}$ and let $\sigma$ be a non-trivial finite order automorphism of $\Gamma$. Let $g$ the Lie algebra associated to $\Gamma$ and ...
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339 views

How to show that the structure constant of SU(3) is invariant?

So suppose $f_{ijk}$ is the antisymmetric structure constant of SU(3), and $D^8_{ij}(g)$ is the matrices of 8-dimensional adjoint representation of SU(3), then how to show that ...
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146 views

Do the classical Lie algebras all satisfy $XM + MX^T = 0$?

I'm working on a homework assignment in which part of the question statement says that each of the classical Lie algebras can be described as the set of all matrices $X \in gl(n,\mathbb{C})$ ...
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364 views

Length of root strings

Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra $g$ of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the length of an $a$-string through $b$ is at most ...
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14 views

$\mathfrak{sl}_2$ has the root lattice of type $A_1$.

Let $L$ be a Lie algebra over $\mathbb{Z}$ constructed from a root lattice $R$. It is well-known that if $R=A_1$, then $L \cong \mathfrak{sl}_2$ and this is widely used example in many books on Lie ...
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27 views

Finite dimensional algebraic representation of $SL_2(\mathbb{C})$

I heard that for each $n\in \mathbb{N}$, there is the unique algebraic irreducible representation of $SL_2(\mathbb{C})$ with dimension $n$ over $\mathbb{C}$. Would you let me know what is such ...
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Irreducible Representation of $sl(3,C)$

We know that the the roots of $\mathbb{g} = sl(3,C)$ under the adjoint action are given by $L_i - L_j$ where $L_i (diag(a_1, a_2, a_3))=a_i$ for $i = 1,2,3$. If $V$ is any irreducible representation ...
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26 views

Non-Lie character of Leibniz algebra

Let $J$ be the largest ideal of Leibniz algebra $L$ which denotes the non-Lie character of $L$. Is it possible to write $L=L_{Lie}\cap J$? We know that $L_{Lie}= L/J$. I am going to give the following ...
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22 views

Topologies of partially exponentiated lie algebras, especially in regard to $SU(2)$

Consider the fundamental respresentation of $\mathfrak{su}(2)$ given in terms of the Pauli matrices as $\mathfrak{su}(2) = \langle ...
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how to show that the representation of $SL(2, \mathbb{C})$ is holomorphic

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to ...
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15 views

Split Lie algebra extensions?

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras. A Lie algebra extension is a short exact sequence $$0\longrightarrow \mathfrak{h}\stackrel{\jmath}{\longrightarrow} ...
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22 views

Is $V \otimes V$ a $g \otimes g$-module?

Let $g$ be a Lie algebra and $V$ a $g$ module. Then $V \otimes V$ is a $g$ module under the action $X.(x \otimes y) = X.x \otimes y+x \otimes X.y$, $x, y \in V$, $X \in \mathfrak{g}$. Is $V \otimes V$ ...
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14 views

Centralizer of Maximal Toral Subalgebra of $L$

I am trying to understand the proof of the following result from Humphreys Lie Algebra book: Let $L$ be a semisimple complex Lie Algebra.Let $H$ be Maximal toral subalgebra of $H$,Let $C_L(H)$ ...
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18 views

Abstract Lie algebra

For six dimensional Lie algebra with non-zero Lie brackets defined as follow: $[e_{1}, e_{3}] = -e_{1}, [e_{1}, e_{6}] = -e_{2}, [e_{2}, e_{3}] = -e_{2}, [e_{2}, e_{4}] = e_{1}, [e_{2}, e_{5}] = ...
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Relationship between Casimir and index of a representation of a Lie algebra.

In several QFT textbooks (namely, those of Peskin and Shroeder and of Schwartz) there is presented an identity for representations of Lie algebras, $$ d(R) C_2(R) = T(R) d(G),$$ where $d(R)$ is the ...
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10 views

How many cyclic conjugate class can $su(2)$ has

In the lie algebra of $su(2)$, we can easily find out three linear independent elements $t_1,t_2,t_3\in su(2)$ such that $[t_1,t_2]=t_3,[t_3,t_1]=t_2,[t_2,t_3]=t_1$ and this relation is preserved ...
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8 views

The Killing form of elements in the Cartan subalgebra

Let $L$ be a Lie algebra and $H\oplus\bigoplus_{\alpha\in\Phi}L_\alpha$ be its Cartan decomposition. Now if $x\in L_\alpha$ and $y\in L_\beta$ and $\alpha+\beta\not=0$ then the Killing form ...
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Calculating the Killing form of the classical algebras

I was reading in the book Parabolic Geometries (p.170-172). But I didn't get how he obtained the Killing form from the roots of each simple Lie algebra. For example: In case $sl(n,\mathbb C)$, given ...
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38 views

Compact Lie group $G$ with Lie algebra $\frak g$ satisfying $gZg^{-1}=-Z$ for $Z\in\frak g$ and $g\in G$

Let $G$ be a compact Lie group with Lie algebra $\frak g$. Are there known conditions on $G$ guaranteeing the following property: $$ \hbox{For each $Z\in\frak g$ there exists an element $g\in G$ ...
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16 views

Lie subalgebra generated by elements of $L_{\alpha}$ and $ L_{-\alpha}$

Let $L$ be a Lie algebra with four roots $\alpha,-\alpha,\beta,-\beta$. Let $K$ be the subalgebra generated by $L_\alpha,L_{-\alpha},L_\beta,L_{-\beta}$. Is $K=\{[L_\alpha,L_{-\alpha}]\oplus ...
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23 views

Pushforward of a Matrix Lie Algebra Bracket

For a matrix Lie group we know that the left inv. push forward is given by $$ (L_g)_* X = g X \quad\quad | X\in \mathfrak{g}, g \in G $$ With Lie bracket the commutator $$ [X,Y]_\mathfrak{g} = XY-YX ...
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22 views

Lie algebra of $SL_2(\mathbb{R})$ and show $\exp(X)=I+X$ where $I \in SL_2(\mathbb{R}) $ and $X \in sl_2(\mathbb{R})$

I am doing an undergraduate course on Representation Theory and am trying to solve these consecutive questions. The first two I am ok with (I just included them for context), but I could do with some ...
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29 views

Existence of a canonical map of quadratic forms

For $X=\mathbb C^N\oplus \mathbb C^N$ equipped with a real structure $J^2=1$ and symplectic structure $S$ satisfying $$J(z_1,z_2)=(\bar z_2,\bar z_1),~~~~~S(z_1,z_2)=(z_1,-z_2)$$ we see that $X$ has ...
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16 views

Does $r \in \Lambda^2 g$ imply that $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] \in \Lambda^3 g$?

Let $g$ be a Lie algebra. Does $r \in \Lambda^2 g$ imply that $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] \in \Lambda^3 g$? Thank you very much.
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33 views

Vector Fields and basis

Let $M$ be a differentiable manifold; $p \in M$ ; $\sigma$ be a chart at $p$ with $\sigma(p)= (x^{i}),i=1,2,\cdots n$. $T_{p}(M)$ the tangent space at $p$ has basis $\{ \frac{\partial}{\partial ...
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15 views

why $ t\oplus {\sqrt{-1}t}$ is a Cartan subalgebra of $\mathfrak{g}$?

Let $G$ be the complexification of a connected, simply-connected compact Lie group $K$ and $\mathfrak{g}$ be a Lie algebra of $G$. If $t$ is a maximal abelian subalgebra of the Lie algebra ...