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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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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63 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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53 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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136 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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70 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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37 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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39 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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30 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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61 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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178 views

How to prove the lie algebra of $n\times n$ traceless matrices is semi-simple?

The Lie algebra of all the $n \times n$ matrices is not semi-simple. However, if we restrict ourselves to traceless $n\times n$ matrices, we do obtain a semi-simple (in fact, simple) Lie algebra which ...
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183 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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44 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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120 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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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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107 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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71 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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101 views

Complete reducibility of finite-dimensional representations of $\mathfrak{sl}_2(\mathbb{C})$

By Weyl's theorem every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ is completely reducible, because $\mathfrak{sl}_2(\mathbb{C})$ is a (semi) simple Lie algebra. It seems there ...
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57 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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21 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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Does triality survive in product Lie groups?

Look at the following diagrams of Lie groups ...
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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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234 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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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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322 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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Suppose Del is root system. Then at least one simple component of Del is not isomorphic to A1 if and only if there is an embedding A2 to Del.

Suppose Del is root system. Then at least one simple component of Del is not isomorphic to A1 if and only if there is an embedding A2 to Del. where A1 and A2 are simple root system. The idea is There ...
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Free Lie algebras and basis for a subpcae of a special degree

Let X^* be the the set of all words on basis elements of Lie algebra L and F is the vector space spanned by X^*. I do not know how can I define the basis elements and also the number of basis ...
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18 views

A question on Cartan involution

Is there a real(or complex) Lie algebra $L$ for which the set of all involutions is an infinite commutative set but the center of $L$ is finite dimension space?(So the set of all Cartan involution ...
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23 views

Dimension of a weight space which is of weight $0$.

Let $V$ be a module of a Lie algebra $\mathfrak{g}$ and $V_{0}$ be the weight space of $V$ of weight $0$. $$ V_0 = \{ v\in V: h.v = 0, h \in \mathfrak{h} \}, $$ $\mathfrak{h}$ is a Cartan subalgebra ...
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How to compute $\lambda(h_i)$?

Let $\lambda$ be a weight and $h_i = h_{\alpha_i} \in \mathfrak{h}$, $\alpha_i$ is a simple root. $\mathfrak{h}$ is a Cartan subalgebra of a Lie algebra $\mathfrak{g}$. How to compute $\lambda(h_i)$? ...
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Proof for a corollary from PBW theorem

I need to know how we can prove the following corollary : If $x_1, \ldots, x_n$ is a vector space basis for Lie algebra $L$ then a vector space basis for $U(L)$, $U(L)$ is universal enveloping ...
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25 views

$\mathfrak so(V,B)$ as subalgebra and trace if subsets of it.

I'm studying lie algebras, and got stuck on this one: Let $B$ be a bilinear form on a finite-dimensional vector space $V$ over $\mathbb F$. I've seen many books that say that $\mathfrak so(V,B)$ ...
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21 views

Do the Generalized Gell-Mann Matrices form a complete set?

Please bear with me, I'm studying Lie algebras as they are related to quantum mechanics, and most of my group theory knowledge is self-taught. I'm not sure how to prove this seemingly basic result. ...
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19 views

Trace functionals as invariant elements of $R[\mathfrak{g}]$ under $G$

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ and let $G$ be its inner automorphism group. Then $G$ acts on $R[\mathfrak{g}]\cong S(\mathfrak{g}^*)$ via $(\sigma\cdot f)(x) = ...
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21 views

Question on inner product on space of representations of compact Lie groups

Let $K$ be a compact connected Lie group, wiewed as subgroup of unipotent matrices. Let $G=\mathfrak{k}^\mathbb C$ be the complexification with Lie algebra $\mathfrak{g}=\mathfrak{k}\oplus ...
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definition of universal algebra and universal enveloping algebra

The basis of a universal algebra is a function b that takes some algebra elements as values b(i) and satisfies either one of the two equivalent conditions named Outer condition and Inner condition ...
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Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
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35 views

automorphism group of Lie algebras

According to the difinition of automorphism group of lie algebra we must have the following condition: f [a,b]=[f(a),f(b)] for f in Aut (L) and L is a Lie algebra Now I have computed the automorphism ...
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what is the image of an automorphism of lie algebra

Let suppose L be a simple lie algebra over GF(2) , if α be an automorphism of L then for any element of L we must have α[a,b]=[α(a),α(b)]. Now I want to have a clear understanding of image of α. Since ...
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Character of a symmetric square

Let $V$ be a representation of $\mathfrak{sl}_2(\mathbb{C})$. As far as I am concerned a character of $V$ is a Laurent polynomial $\sum_{k\in\mathbb{Z}}d_k\cdot t^k$, where $d_k$ is the dimension of ...
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17 views

Questions about an action of $U(\mathfrak{g})$.

Let $\mathfrak{g}$ be a Lie algebra and $U(\mathfrak{g})$ its universal envoloping algebra. Let $G$ be the Lie group of $\mathfrak{g}$ and $U$, $B^{-}$ the upper unipotent subgroup and lower Borel ...
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21 views

What is $d\mu$ of $\mu:T^*\mathbb{C}^n\rightarrow \mathfrak{gl}_n^*$?

This is an elementary question. Let $\mu:T^*\mathbb{C}^n \longrightarrow \mathfrak{gl}_n^*$ be the moment map given by $(x,y)\mapsto xy$. Then concretely, what is the differential $d\mu$ of $\mu$? ...
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Why center acts by scalars?

Let $\mathfrak{g}$ be a Lie algebra. Let $U(\mathfrak{g})$ be its universal enveloping algebra. Let $Z(U(\mathfrak{g}))$ be the center of $U(\mathfrak{g})$. Let $V$ be an irreducible representation of ...
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21 views

Real or complex representation

How can one know for a given algebra $\frak{g}$ if a specific representation is real or complex? For example if $\frak{g}=so(10)$ how can one know that the representation $\underline{16}$ is complex? ...
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25 views

Universal enveloping algebra of $sl_2$

I need prove that any element of $U(sl_2)$ can be represented by linear combination of elements $e^i h^j C^k$, where $C=ef+fe+\dfrac{h^2}{2}$. $e=\begin{pmatrix} 0 && 1 \\ 0 && 0 ...
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For a nilpotent Lie subalgebra, $\mathfrak{h}$, is $ad(\mathfrak{h})$ simultaneously diagonalizable if each $ad(H)$ is diagonalizable?

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{h}\subseteq \mathfrak{g}$ be a nilpotent subalgebra such that for every $H \in \mathfrak{h}$, the adjoint map $ad(H): \mathfrak{g} \rightarrow ...
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52 views

Which linear combinations of simple roots are roots?

An answer to the following question should be well known to any specialist on Lie theory. Since I don't have time to go through textbooks, I post it here. Let $\Delta$ be a root system, $\Delta^+$ ...
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11 views

How to prove that $U(\mathfrak{h})$ is isomorphic to $\mathcal{O}(\mathfrak{h}^*)$.

Let $\mathfrak{h}$ be a Cartan subalgebra of a Lie group $G$. It is said that $U(\mathfrak{h})$ is isomorphic to $\mathcal{O}(\mathfrak{h}^*)$. Here $\mathcal{O}(\mathfrak{h}^*)$ is the ring of ...
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13 views

Restricted Universal Enveloping Algebras

Is there example of restricted universal enveloping algebra $uL$ of the $p$-Lie algebra $L$ over field $k$ of characteristic $p > 0$ such that $L$ hasn't nonzero $p$-algebraic elements and global ...
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13 views

How to find next M.Hall's word question

Given a Hall word. How do I write the next one?