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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5
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2answers
225 views

Subgroups of $\Bbb{R}^n$ that are closed and discrete

I am trying to prove that every closed discrete subgroup of $\Bbb{R}^n$ under addition is a free abelian group of finite rank. I have tried to do this by induction on the dimension $n$. Base ...
8
votes
2answers
367 views

Significance of the following Matrix?

I am unfamiliar with advanced Matrix theory (nor am I a mathematician), so please bear with me. Is there anything significant about the following Matrix structure? Are there any special symmetries or ...
12
votes
1answer
809 views

Exercise 6.5 in Humphrey's Book on Lie Algebras

I am trying to solve Exercise 6.5 part 4 in James Humphreys' Introduction to Lie Algebras and Representation Theory. I added the (homework) tag because my question is about an exercise, but this is ...
4
votes
2answers
150 views

If $\exp(itH) A \exp(-itH) = A$ for all $t$, do $A$ and $H$ commute?

Let $H$ be a self-adjoint $n \times n$ matrix with complex entries. $H$ gives rise to a continuous 1-parameter group of unitaries $t \mapsto U_t = \exp(itH) : \mathbb{R} \to U(n)$. Let $A$ be some ...
2
votes
0answers
63 views

ADE type root lattice

Let $\Phi$ be a root system of ADE type, $L$ is the corresponding root lattice, show that $\Phi=\{\alpha\in L:(\alpha,\alpha)=2\}$, where $(,)$ is the normalized non-degenerate symmetric bilinear form ...
5
votes
1answer
346 views

When do all derivations come from automorphisms?

Let $k$ be a field (algebraically closed for simplicity) and let $A$ be an $n$-dimensional algebra over $k$ (not necessarily commutative or even associative). The group $G=\mbox{Aut}(A)$ is an ...
-1
votes
2answers
113 views

Questions about $su(2)$. [closed]

Edit: In physics, it seems that people usually study $su(2)$ but not only $sl(2)$? Why people study $su(2)$ but not only $sl(2)$?
0
votes
1answer
61 views

notation of super Lie algebras

The classic super Lie algebras are of type A: $\operatorname{sl}(m+1 \mid n+1)$, $m\neq n$, $\operatorname{psl}(n+1 \mid n+1)$, type B: $\operatorname{osp}(2m+1 \mid 2n)$, type C: ...
4
votes
1answer
55 views

The differential of $\psi: GL_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ which sends $g\mapsto gAg^{-1}$

Suppose $\psi: GL_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ defined by sending $$ g\mapsto gAg^{-1}. $$ Then why is it that $d\psi:T_eGL_2(\mathbb{C})=M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ is ...
2
votes
2answers
91 views

Computing the differential of the map $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$

Let $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ be the map $$(B_1,B_2)\mapsto [B_1,B_2]$$ which takes two $2\times 2$ matrices to its Lie bracket. Then why does ...
7
votes
1answer
785 views

Some questions about representations of $SO(6)$

I would like to know the proof/explanation for the following three properties of the representation of $SO(6)$, What is the importance of symmetric traceless tensors of arbitrary rank w.r.t $SO(6)$ ...
4
votes
1answer
511 views

The longest word in Weyl group and positive roots.

How to write down a reduced decomposition of the longest word in a Weyl group? For example, how to write down a reduced decomposition of the longest word in type B3 Weyl group? For a decomposition of ...
4
votes
1answer
551 views

Fundamental and the anti-fundamental representation of $U(n)$

I guess that conventionally one thinks of the fundamental representation and the anti-fundamental representation of $U(n)$ as the complex $n-$dimensional representation and its complex conjugate. ...
4
votes
1answer
615 views

Lie derivative: concrete example for linear Lie group

I am trying to understand the notion (and notation) of the Lie derivative on a general manifold by trying to convert the notation the concrete example of the Lie group O(n). Let $X,Y$ be smooth ...
1
vote
1answer
297 views

Kernel of adjoint of Lie algebra

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. The adjoint representation of the Lie algebra $\mathfrak{g}$ is defined as: $$ \text{ad: } \mathfrak{g} \rightarrow ...
5
votes
1answer
199 views

Lie algebra of a Lie subgroup

Let $G$ be a Lie group and $H$ a Lie subgroup of $G$, i.e. a subgroup in the group theoretic sense and an immersive submanifold. Let $\mathfrak{g}$ and $\mathfrak{h}$ be the associated Lie algebras. ...
2
votes
3answers
806 views

$so(4)=su(2)× su(2)$ contradiction

Modified question There was $sl(2,\mathbb R)$ used instead of $su(2)$ in the previous version. Thanks to MattE for pointing it out. I have seen it claimed many times that $so(4,\mathbb R)=su(2)\times ...
3
votes
1answer
542 views

left-invariant vector field: counterexample

Let $G$ be a Lie group, $L_g$ the left-translation on this group with differential $d L_g$. A vector field $X$ on $G$ is called left-invariant if $$ X \circ L_g = d L_g \circ X \quad \forall g \in ...
1
vote
0answers
144 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 ...
1
vote
1answer
62 views

Question about Lie superalgebra.

What are the generators and relations for the Lie superalgebra $\mathfrak{psu}(2, 2 | 4)$? Thank you very much.
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0answers
59 views

Understanding quotients $\mu^{-1}(0)/G$

This question is related to this link: Geometric difference between two actions of $GL_n(\mathbb{C})$ on $G\times \mathfrak{g}^*$ Further analyzing Scenerio 1: Let $G=GL_n(\mathbb{C})$ act on $G$ by ...
1
vote
2answers
274 views

Antisymmetric powers of $SO(n)$ representation.

I am particularly interested for $SO(3)$. Let us say that I start with the natural/defining $3$-real-dimensional vectorial representation of $SO(3)$ and I choose the generator of rotation in the ...
3
votes
2answers
154 views

What are the generators and relations for type $B_3$ Weyl group?

What are the explicit generators and relations for type $B_3$ Weyl group? Thank you very much. Edit: type $B_3$ Weyl group $G$ is $(\mathbb{Z}/2\mathbb{Z})^{3} \rtimes S_3$, so the order of $G$ is ...
2
votes
1answer
127 views

Showing that the universal enveloping algebra of some $\mathfrak{g}$ is isomorphic to $\mathbb{C}[x_i,\partial/\partial x_i]$

The universal enveloping algebra $U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ is defined to be $$ \dfrac{\mathbb{C}\oplus\mathfrak{g}\oplus ( \mathfrak{g}\otimes ...
2
votes
0answers
252 views

Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
4
votes
1answer
146 views

restricted lie algebras definition

Jacobson (Lie algebras, p.187) defines what is meant by a restricted Lie algebra: Def4: A restricted Lie algebra, $L$, of characteristic $p\not = 0$ is a Lie algebra of characteristic $p$ in which ...
3
votes
1answer
143 views

Lie algebra representation induced from homomorphism between spin group and SO(n,n)

Consider the spin group, we know it is a double cover with the map: $\rho: Spin(n,n)\longrightarrow SO(n,n)$ s.t $\rho(x)(v)= xvx^{-1}$ where $v$ is an element of 2n dimensional vector space V and ...
3
votes
1answer
373 views

Exponentiation of Center of Lie Algebra

Let $G$ be a Lie Group, and $g$ its Lie Algebra. Show that the subgroup generated by exponentiating the center of $g$ generates the connected component of $Z(G)$, the center of $G$. Source: ...
2
votes
1answer
328 views

$GL_n(k)$ (General linear group over a algebraically closed field) as a affine variety?

In the context of linar algebraic groups, I read in my notes from the lecture that's already some while ago that $GL_n(k)$ is an algebraic variety because $GL_n=D(\det)$, $ \det \in k [ (X_{ij})_{i,j} ...
2
votes
1answer
185 views

Highest or positive weights (or roots)

Let $T= (\mathbb{C}^*)^2$ be embedded in $GL_2$ along its diagonal entries, and suppose $T$ acts on $M_2(\mathbb{C})$ via conjugation. Denote $\chi_i(g)=z_i$ where $$ g = \left( \begin{array}{cc} z_1 ...
4
votes
0answers
434 views

Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.

I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question. To start with we work with the $\mathbb{Q}$ version of ...
2
votes
1answer
278 views

Two Lie algebras associated to $GL(n,\mathbb{C})$

I have elementary questions about Lie groups and their associated Lie algebras. Let $G=GL(n,\mathbb{C})$. Then associated to this Lie group is the Lie algebra $M_n(\mathbb{C})$ with the commutator ...
6
votes
3answers
699 views

Physical interpretation of the Lie Bracket

I've come accross this physical interpretation for $ [X,Y] $ which I don't understand : Follow $X$ for some time $\epsilon$; Follow $Y$ for $\epsilon$; Follow -X for $\epsilon$; Follow -Y for ...
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0answers
94 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 ...
7
votes
1answer
234 views

Moment map of the action of $\operatorname{SO}(3)$ on the sphere

The moment map of the action of $\operatorname{SO}(3)$ on the sphere can be thought of as inclusion from $S^2$ into $\mathbb R^3$ by identifying $\mathfrak{so}(3)$ (the Lie algebra of ...
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0answers
38 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 $M_n(k)$ ...
4
votes
2answers
545 views

Application of Harish-Chandra theorem

Let $\mathfrak{g}$ be a semisimple finite dimensional Lie algebra and $V_\lambda$, resp. $V_\mu$ its finite dimensional highest weight modules with highest weights $\lambda$, resp. $\mu$. Let ...
0
votes
2answers
539 views

How to use Weyl dimension formula?

Let $V(\lambda)$ be a highest weight module of a semi-simple Lie algebra with highest weight $\lambda$. The Weyl dimension formula is $\dim V(\lambda) = \frac{\prod_{\alpha>0} (\lambda+\rho, ...
4
votes
1answer
95 views

Highest weight of a trivial representation of a Lie algebra.

Let $g$ be a Lie algebra. $\mathbb{C}$ is a trivial representation of $g$. What is the highest weight of $\mathbb{C}$? I think the weight is the function $f: \mathfrak{h} \to \mathbb{C}$ such that ...
2
votes
1answer
124 views

$\mathfrak{h}_1,\mathfrak{h}_2$ Cartan subalgebras with $\mathfrak{h}_1\cap\mathfrak{h}_2=0$

Let $\mathfrak{g}$ be a finite dimensional simple Lie Algebra over an algebraically closed field $K$. I'm having trouble to show that always exists Cartan subalgebras $\mathfrak{h}_1,\mathfrak{h}_2$ ...
2
votes
1answer
84 views

Weights of a locally finite-dimensional module

Let $\frak g$ be a complex finite-dimensional simple Lie algebra and $V$ be a $\frak g$-module with weights bounded by above by some fixed weight and suppose that $V$ is locally finite-dimensional. ...
8
votes
1answer
232 views

Schur -Weyl duality for $sl_2$ and $S_n$

$V$ is an $m$ dimensional vector space having a structure of $sl_2(\mathbb{C})$-module, where $sl_2(\mathbb{C})$ is the Lie algebra of the Lie group $SL_2(\mathbb{C})$. The symmetric group $S_n$ acts ...
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0answers
42 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 $$a^{-1} |n\rangle = \frac{1}{\sqrt{n+1}} |n+1\rangle$$ ...
1
vote
1answer
62 views

#(cardinality of irreps Lie algebras) > #(irreps of ssociative algebras). Proof?

I know that irreducible representations of associative $*$-algebras are fairly restricted: any $*$-algebra $A$ is isomorphic to a finite sum of simple algebras ...
3
votes
2answers
206 views

Lie algebras from differentiation

I noticed that the characterizations of the Lie algebras of matrix Lie groups can be obtained by differentiation. For example: $$O(n) = XX^t = \mathbb{1} \implies \mathfrak{o}(n) = X + X^t = ...
5
votes
0answers
115 views

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 ...
1
vote
0answers
31 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 ...
0
votes
1answer
530 views

Vector space generated by the tensor products of pauli matrices

Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices: \begin{equation} ...
2
votes
1answer
406 views

Eigenvalues of ad (Adjoint action)

The Question: Let $A$ be an $n \times n$ matrix with distinct eigenvalues $\lambda_{1},...,\lambda_{n}$. Show that ad$_{A}$ acting on the space of all $n \times n$ complex matrices has $n^{2}$ ...
4
votes
1answer
381 views

Why is the Lie algebra corresponding to group SO(3) called o(3)?

How is Lie algebra named? Why is it $\mathfrak{su}(2)$ for group $SU(2)$, but $\mathfrak{o}(3)$ for group $SO(3)$? What does the "$\mathfrak{s}$" in algebra $\mathfrak{su}(2)$ mean?