Tagged Questions

1
vote
0answers
24 views

Representation of Complexification of Lie Algebra

Is the following obvious? I think it is, but wanted to make sure before an exam tomorrow! "There is a bijection between the complex representations of a real Lie algebra and the complex ...
3
votes
0answers
59 views

Weights set spans

Definition Let $T$ be a torus and $R: G \to GL(V)$ a representation. $R(T)$ is a collection of commuting matrices and therefore can be simultaniously diagonalized. For a character $\lambda \in ...
2
votes
1answer
43 views

First Homology Group and Abelianization

On the Wolfram Mathworld article for Commutator Subgroup, it states that the first homology group is the abelianization, $$H_{1}(G) = G \big/ [G,G]$$ which totally blows my mind because I've only seen ...
4
votes
0answers
30 views

Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
2
votes
0answers
30 views

Character of half-spin representation

Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum $$\sum x_1^{\pm ...
1
vote
1answer
42 views

Isomorphisms of the Lorentz group and algebra

I'm trying to read a few books on QFT and some seem to say the Lorentz algebra obeys $\mathfrak{so}(1,3)\otimes \mathbb{C} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ while others say ...
3
votes
1answer
89 views

Decomposing tensor product of lie algebra representations

I'm given a lie algebra representation $\pi$ of some semi-simple algebra and that it decomposes into a sum of irreducible representations. What technique should I use to show the decomposition of ...
0
votes
0answers
17 views

concerning coadjoint representation

Let $\xi $ be the vector field on $\frak{g}^*$ (dual of Lie algebra) which correspond to element $X$ of the Lie algebra $\frak{g}$. Then why have we $\xi(F)=K_*(X)F$ where here $K=Ad^*(g)$ is ...
0
votes
1answer
40 views

Classifying all rank 2 and 3 root systems

I am working with the representation theory of complex simple Lie algebras, and have a question: It is intuitively clear that the root systems $A_1\times A_1$, $A_2$, $B_2$, and $G_2$ comprise all ...
1
vote
0answers
21 views

Why is the dual space of Cartan subalgebra an irreducible representation of Weyl group

it is proposition 14.31 in Fulton-Harris book. The proof goes like this. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and assume $\mathfrak{z}\subseteq\mathfrak{h}^*$ were preserved ...
1
vote
1answer
29 views

Why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?

Let $G=SO(n,R)$ be a Lie group and $\mathbb{g}$ its lie algebra. Take $X\in \mathbb{g}$. Then why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?
0
votes
0answers
94 views

Real representations of Lie algebra $\mathfrak{so}(3)$

How does one construct a real-valued, $n$-dimensional representation of the three generators of the Lie algebra $\mathfrak{so}(3)$ for a given value of $n$?
0
votes
0answers
20 views

coadjoint representation

For the definition of coadjoint representation $K$ we have $K(g):\mapsto p_V(gFg^{-1})$ where $V\subset Mat_n(\mathbb{R})$ and $Mat_n(\mathbb{R})=V\oplus \mathfrak{g}^\perp $. Here $g\in G$ and ...
7
votes
0answers
82 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
1
vote
1answer
65 views

A quesion in Fulton & Harris book “representation theory a first course”

In Section 11.2 A little plethysm, it discusses the tensor product of two different representations of $sl_2\mathbb{C}$. It says "If $V=\bigoplus V_{\alpha}$ and $W=\bigoplus W_{\beta}$ then ...
0
votes
0answers
30 views

Subalgebra generated by Cartan subalgebra and root spaces

Let $\alpha_1,...,\alpha_k$ be the roots of a semisimple Lie algebra $\mathfrak{g}$ and $\mathfrak{g}_{\alpha_i}\subset \mathfrak{g}$ the corresponding root spaces. Then the subalgebra of ...
1
vote
1answer
52 views

Embedding of $PGL_n\mathbb{C}$ and friends

I would like the find an embedding/faithful representation from the projective linear group $PGL_n\mathbb{C}\to GL_m\mathbb{C}$ for some $m$, and likewise for the other projective groups ...
1
vote
0answers
70 views

Exterior and symmetric powers of $\mathfrak{sl}(4,\mathbb{C})$ representation

I am taking a course on representation theory, and going through Lecture 15 of Fulton and Harris's Representation Theory. One of the topics we're currently covering is the example of ...
1
vote
0answers
51 views

Projection map $\text{Sym}^2(\text{Sym}^3V)\to \text{Sym}^2V$ viewed as a Hessian

Exercises 11.21 and 11.22 in Fulton's Representation Theory are the following: Let $V$ be the standard representation of $\mathfrak{sl}_2\mathbb{C}$. The projection map from ...
1
vote
0answers
64 views

If $\mathfrak{g}$ admits a decomposition then it is semisimple

I want to show: If $\mathfrak{g}$ is a Lie algebra that has an abelian subalgebra $\mathfrak{h}$ such that $\mathfrak{g}$ has a Cartan decomposition $\mathfrak{g}=\mathfrak{h}\oplus(\bigoplus_{\alpha ...
3
votes
1answer
65 views

Decomposition of semisimple Lie algebra via its roots

Exercise 14.33 in Fulton and Harris's Representation Theory claims that If the roots of a semisimple Lie algebra lie in a collection of mutually orthogonal subsets, one sees that the Lie algebra ...
3
votes
0answers
89 views

Irreducible representations of $\mathfrak{sl}_3\mathbb{C}$

I am working through the exercises in Fulton and Harris's Representation Theory, and am stuck on two on page 189. Let $\text{Sym}^2V$ denote the second symmetric power of the standard 3-dimensional ...
5
votes
1answer
152 views

Irreducibility and weights of a representation

For some reason I can't get a good hold of those topics (I'm reading Brian C. Hall's Lie Groups, Lie algebras and Representations. So it's matrices only). I'll try to narrow it a bit more: ...
0
votes
0answers
25 views

The simplicity of $\bigwedge^i \mathbb{C}^{n+1}$ as a representation of $\mathfrak{sl_{n+1}}$ and its weight vectors

I want to show that $\bigwedge^i \mathbb{C}^{n+1}$ is a simple representation for $\mathfrak{sl}(n+1,\mathbb{C})$ for each $1\le i \le n+1$ but I'm already stuck at determining the weight vectors. So ...
1
vote
1answer
40 views

Representations of semisimple Lie algebra

Let $L$ a Lie algebra and $V$ a representation of $L$. We define $$ V^{L}:= \{ v \in V \, | \, xv=0, \, \forall x \in L \} $$ and $V_{L}:=V/LV$. Let $\pi: L \rightarrow V_{L}$ be the quotient ...
6
votes
0answers
79 views

The center of a simply connected semisimple Lie group

I am learning about Lie groups, and I have the following basic question: Every Lie group $G$ has a (unique) universal covering group $ \bar G $ that is simply connected, and such that the covering ...
2
votes
1answer
112 views

Conjugate Representations

Are there any general results on when conjugate representations of a real Lie algebra are equivalent? I'm inclined to say that they are often not, but this is merely going on my case by case ...
1
vote
1answer
41 views

When can you build up all representations from the fundamental and antifundamental ones?

Under what conditions can you determine all representations of a Lie algebra from the fundamental and antifundamental ones using just the tensor product, direct sum and Clebsch-Gordan decomposition? I ...
1
vote
1answer
81 views

How can the Cartan-Weyl basis of su(2) be a basis if it does not consist of antihermitian operators?

Consider a Lie algebra. The ladder operators (i.e. root vectors, or eigenvectors of the Cartan subalgebra with respect to the adjoint representation) form a handy basis of the algebra called a ...
1
vote
0answers
59 views

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 $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
2
votes
1answer
63 views

Conjugate Representations of Lie Algebra of Lorentz Group

I'm trying to understand the Lie algebra of the Lorentz group and am almost there, but am stuck at the final hurdle! It's easy to prove that $$\frak ...
4
votes
2answers
114 views

Representations of Direct Sum of Lie Algebras

I'm trying to prove the following. Let $\frak{g}$ and $\frak{h}$ be (semisimple) Lie algebras. Then every representation $d$ of $\frak{g}\oplus\frak{h}$ is the tensor product of representations $d^1$ ...
1
vote
1answer
83 views

Lie Algebra of the Lorentz Group $SO(1,3)^{\uparrow}$

I'm trying to get my head around the Lie algebra of the Lorentz group once and for all, but have got tied up in knots. Where is my error in the following? The universal covering group of the Lorentz ...
5
votes
1answer
105 views

Invariant Inner Product on Lie Algebra

Let $G$ be a Lie group, $\frak{g}$ its Lie algebra. Suppose $\mathcal{D}$ a representation of $G$ on $V$, $d$ the associated Lie algebra representation. Suppose $V$ is endowed with an inner product. ...
2
votes
1answer
29 views

Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
5
votes
2answers
107 views

Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
2
votes
1answer
35 views

Representation of non-Abelian, dimension 2 Lie algebra

Let $k$ be a field and $\mathfrak{g}=kx\oplus ky$ with $[x,y]=y$. Show that $\rho(x)=t\,\frac{d}{dt}$ and $\rho(y)=t\cdot$ (mult. by $t$) define a representation $\rho:\mathfrak{g}\to ...
4
votes
2answers
102 views

Classification of irreducible representations via Casimirs

Physicists almost always label irreducible representations via Casimirs (e.g., characterizing the irreducible representations of $SO(3)$ by spin). I've been looking far and wide to see the general ...
0
votes
0answers
49 views

Lie group reps induced by Lie algebra reps

Let $G$ be a Lie group and $\mathfrak g$ its Lie algebra. Suppose that $\rho_\mathfrak{g}$ is a representation of $g$ on a vector space $V$. Is it true that the mapping $\rho$ from the identity ...
2
votes
1answer
55 views

Existence of a Lie subgroup

Let $G=SU(k)\times T^1$, $S$ a subgroup of the center of $SU(k)$ ($Z(SU(k)\cong \mathbb{Z}_k$) and $\eta$ a homomorphism from $S$ into $T^1$. Suppose $(S, \eta)$ denotes the subgroup of $G$ contains ...
3
votes
0answers
65 views

Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with ...
1
vote
0answers
30 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 ...
1
vote
2answers
83 views

Action of $\mathfrak{sl}({V})$ in tensor spaces

what is the natural action of $\mathfrak{sl}({V})$ in tensor spaces ?
3
votes
2answers
74 views

Orthonormal basis of Cartan subalgebra relative to Killing form

I'm trying to understand a step in a proof: Let $\mathfrak{g}$ be semi-simple (finite dimensional) Lie-algebra over $\mathbb{C}$, $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra and let ...
3
votes
1answer
81 views

Invariant inner products on infite-dimensional representations

Let $G$ be a compact group and let $V$ be it's continuous representation. It is well known that if $V$ is finite-dimensional, then there is an $G$-invariant inner product on $V$. I haven't found a ...
1
vote
1answer
43 views

Getting a derivation from a path of automorphisms of an algebra

These representation theory notes leave the following claim to the reader: Recall that a derivation on an algebra is a map $d$ such that $d(ab)=d(a)b+ad(b)$. If $A$ is a finite-dimensional algebra ...
2
votes
1answer
227 views

Finding All Irreducible Representations of $SO(3)$

I've read that one may prove that all irreducible representations of $SO(3)$ are tensor product representations of the fundamental representation (or tensor product representations of the spin 1/2 ...
0
votes
1answer
60 views

Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?

At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint? (..a related query: ...
1
vote
1answer
51 views

Indecomposable L-module

I have the following exercice which I have be trying to solve: Let L be a Lie algebra and $r:L\rightarrow gl_3(F)$ a representation of L such that $im(r)=t_3(F)$ (the upper triangular matrices). Show ...
2
votes
0answers
81 views

When is the adjoint representation self-dual?

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...

1 2 3