0
votes
0answers
15 views

PBW proof proposal

One version of the PBW theorem states: $\omega $:$\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras. I am curious if this is a possible proof for the PBW theorem, part is taken ...
1
vote
0answers
41 views

Prove the Weyl's complete reducibility Theorem on finite-dimensional $\mathfrak{g}-modules$ by Kostant's $\mathfrak{n}$-cohomology result

I've met an exercise in Kumar's book ("Kac-Moody Groups, their Flag Varieties and Representation Theory", Chapter III, page 89, Ex. 3.2. E, (1) & (2)). But I have no idea about its proof. Any ...
0
votes
0answers
27 views

Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ ...
10
votes
2answers
172 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
0
votes
0answers
24 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 ...
0
votes
0answers
21 views

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 $\lambda(h_i)$? ...
3
votes
1answer
39 views

Question on unitary representation of non-compact simple Lie groups

The following is an exercise appearing page 148 in Knapp's book, representation theory of semisimple groups. Let $G$ be a connected linear non-compact Lie group with simple Lie algebra $\mathfrak g$. ...
2
votes
0answers
25 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) ...
0
votes
0answers
21 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) = ...
1
vote
1answer
34 views

Adjoint Lie algebra homomorphism

I have a problem deriving the adjoint action $ad_X(Y)=XY-YX$ from the adjoint transformation of the group on the Lie algebra. Background: The adjoint action of the Lie algebra on itself is given by ...
0
votes
0answers
31 views

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 ...
2
votes
0answers
30 views

Angles between adjacent roots in a reduced root system.

Let $R$ be a reduced root system. ($R$ is a finite set spanning $V$, $\alpha \in R \rightarrow -k\alpha \in R$ iff $k=1$, $s_{\alpha}(R)=R$, $s_{\alpha}(\beta)-\beta=k\alpha$ whit $k$ integer). ...
2
votes
0answers
18 views

Finding a basis and weight space for $L = so_6(\mathbb{C})= \{x \in End(\mathbb{C}^6)|^txS + Sx = 0 \}$

The question: Let $S = \left(\begin{array}{cc} 0 & I_3 \\ I_3 & 0 \end{array}\right)$ and let $$L = so_6(\mathbb{C})= \{x \in End(\mathbb{C}^6)|^txS + Sx = 0 \}$$ 1) Find a basis for $L$ ...
1
vote
1answer
34 views

Confusion regarding PBW theorem

I was reading up Humphrey's Introduction to Lie Algebras and Representation Theory and have a confusion regarding a consequence of PBW. First some notations: Let $L$ be a Lie algebra over ...
0
votes
1answer
17 views

Representation of $\mathfrak{sl}_2(\mathbb{C})$ corresponding to Lie algebra representation

We have a representation $R$ of a Lie group $\mathrm{SL}_2(\mathbb{C})$ in the space of polynomials $\mathbb{C}[x,y]$ such that $R\begin{pmatrix} a & b \\ c & ...
0
votes
0answers
18 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 ...
1
vote
0answers
32 views

Derived series of a Lie algebra

I've been studying semisimple Lie algebras and solvability and was wondering if someone could explain to me the meaning of the derived series of a Lie algebra L and this part: $$L^{(1)}=[LL]$$ I don't ...
0
votes
0answers
30 views

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 ...
0
votes
0answers
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? ...
6
votes
2answers
173 views

Lie Groups/Lie algebras to algebraic groups

I am reading some lie groups/lie algebras on my own.. I am using Brian Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction I was checking for some other references on ...
0
votes
0answers
24 views

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 ...
2
votes
1answer
48 views

What is the explicit formula for classical r-matrices?

It is said that classical r-matrices are those satisfy the classical Yang-Baxter equation $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0$, where $r \in \mathfrak{g} \otimes \mathfrak{g}$. ...
1
vote
1answer
44 views

How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$?

Let $U$ be the positive unipotent radical of $SL_n$ and $\mathfrak{n}$ the Lie algebra of $U$. How to show that $\mathcal{O}_q[U]$ is isomorphic to $U_q(\mathfrak{n})$? Here $\mathcal{O}_q[U]$ is the ...
2
votes
1answer
51 views

Differentiating a representation

I'm reading the paper Presenting Schur algebras as quotients of the universal enveloping algebra of $\mathfrak{gl_2}$. It describes a representation of the group algebra ...
2
votes
1answer
29 views

Some beginner facts on representaions of $\mathfrak{sl}_3(\mathbb{C})$

Beginning to learn about representations of $\mathfrak{sl}_3(\mathbb{C})$. One starts with a subspace $$\mathfrak{h}=\{\begin{pmatrix} a_1 & 0 & 0\\ 0 &a_2& 0\\ 0 & 0 & a_3\\ ...
1
vote
1answer
17 views

A property of simple three-dimensional Lie algebras

I am reading a solution of a problem to classify $3$-dimensional simple Lie algebras. First they prove that there exists $H$ such that $[H,X]=\alpha X$ for some $X\ne 0$ and $\alpha\ne 0$. Then they ...
1
vote
2answers
31 views

Decomposition of symmetric powers of $\mathrm{sl}_2$ representations

Let $\mathfrak{g}$ be the Lie algebra $\mathrm{sl}_2(\mathbb{C})$. There is a classification of irreducible representations of $\mathfrak{g}$: each of them is defined by the only natural number $n$, ...
1
vote
1answer
30 views

Question about weights of $\mathfrak{sl}_2 \mathbf{C}$

On p. 148 of Fulton and Harris' book "Representation Theory: A First Course", they write that "Moreover, by the same token, the $V_\alpha$ that appear must form an unbroken string of numbers of the ...
5
votes
2answers
57 views

Action of $H$ in representations of $\mathrm{sl}_2$

Let $X,Y,H$ be the standard base for the Lie algebra $\mathrm{sl}_2({\mathbb{C}})$, i.e. $H=\begin{pmatrix} 1 & 0\\ 0 &-1\end{pmatrix}$, $X=\begin{pmatrix} 0 & 1\\ 0 & 0\end{pmatrix}$, ...
1
vote
0answers
39 views

References to Lie algebras representaions

Could you give me a reference to a brief introduction to representations of Lie algebras, especially $\mathrm{sl}_2(\mathbb{C})$. I mean some basic Verma modules, Weyl groups etc.
1
vote
0answers
32 views

Adjoint Representation of Lorentz Group

I'm thinking about the image under the adjoint representation $\mathrm{Ad}$ of the proper (identity connected component) Lorentz group $SO^+(1,3)$. Since this group has a trivial centre (it contains, ...
5
votes
2answers
66 views

Representations of the Special Orthogonal Group in Three Dimensions.

This will perhaps be an unenlightening question, but here I go. Hopefully someone can varify my thoughts. $\\$ Considering Lie Group Theory and Representation Theory, for the case of the $SO(3)$, ...
1
vote
0answers
25 views

What is a Complete Set of Weights of a Representation of a Lie Subalgebra?

In relation to Lie Group and Lie Algebra theory, I am studying about the weights of representations. I have come across the terminology "a complete string of weights" in my lecture course, but it is ...
0
votes
1answer
33 views

Representations of two-dimensional Lie algebra

It is widely known that there is only one $2$-dimensional non-abelian Lie algebra: it can be generated by two vectors $e_1$ and $e_2$ such that $[e_1,e_2]=e_1$. Let us lenote it by $L$. The question ...
5
votes
1answer
42 views

Weight space of a representation of ${\frak sl}(2,\mathbb C)$

Suppose $(\pi,V)$ is a finite representation of $SU(2)$. Then there's an induced representation $(\pi_*,V)$ of the complexified ${\frak su}^\mathbb C(2) = {\frak sl}(2,\mathbb C)$. Show that the ...
2
votes
1answer
42 views

An $\mathrm{Ad}$-invariant inner product that agrees with the trace

Let $\mathfrak{g}$ be a real semisimple Lie algebra. Then, we have an obvious $\mathrm{Ad}$-invariant inner product (I don't care about positive definiteness) on $\mathfrak{g}$, namely the Killing ...
2
votes
0answers
38 views

Representations of Nilpotent Lie Algebras

Let $\mathfrak{g}$ be a rational, nilpotent Lie algebra. Then its adjoint representation will consist of elements which are nilpotent matrices over rationals. But this representation generally is not ...
5
votes
1answer
41 views

Can the equality $e^{-tY}Me^{tY} = e^{tX}M $ be shown by showing it only to 1st order? (Lie representations)

We have that A and B belong to different representations of the same Lie group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations. $$A = ...
1
vote
1answer
31 views

Obtaining representations of $G$ from $\mathrm{Lie}(G)$.

Suppose $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$, and $\tilde{G}$ is the unique connected, simply connected Lie group whose Lie algebra is $\mathfrak{g}$. Let $C$ be any discrete ...
2
votes
0answers
41 views

A construction of $\mathfrak{e}_8$ in Fulton and Harris

In section $22.4$ of "Representation Theory: A First Course" by Fulton and Harris, the exceptional Lie algebra $\mathfrak{e}_8$ is constructed using a method of Freudenthal. For background, I will ...
1
vote
0answers
24 views

Lie Algebra of Reduced Heisenberg Group Identities

I am having problems trying to understand a statement by Howe in his paper "On the role of the Heisenberg group in harmonic analysis". Here is the setting: Howe defined the (reduced) Heisenber group ...
3
votes
1answer
76 views

Representation theory and particle physics

Are there good books which explain clearly explain the connections between modern particle physics and representation theory of groups and lie algebras?
2
votes
1answer
23 views

Uniqueness of decomposition of $\mathfrak{sl}(2,\mathbb{C})$-modules

By Weyl's Theorem, I know that every $\mathfrak{sl}(2,\mathbb{C})$-module is completely reducible. I'm under the impression that, up to isomorphism, this decomposition is unique, and I would go about ...
0
votes
1answer
37 views

Exponents of a semisimple Lie algebra

I'd like to compute the exponents of a semisimple complex Lie algebra $\mathfrak{g}$. According to http://math.berkeley.edu/~theojf/LieQuantumGroups.pdf proposition 8.1.2.18, this amounts to ...
2
votes
0answers
51 views

What is the Lie algebra of $G=\mathbb{R}$

The question is updated as following. 1. Let $(\Phi,L^2(R))$ be left regular representation of $\mathbb R$ given by $$ \Phi(g)f(x)=f(x-g). $$ It is unitary representation of $\mathbb R$. 2. For ...
0
votes
0answers
16 views

Combining infinitesimal generators of diferent dimensions

I am reading a paper about ways in which you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of $SU(3)\times{}SU(2)\times{}U(1)$. At a certain point, it starts considering ways of getting ...
1
vote
1answer
53 views

Showing that an $\mathfrak{sl}(2,\mathbb{C})$-module is determined by eigenvalues of $h$

This question is essentially exercise 8.4 from the book "Introduction to Lie Algebras" by Erdmann and Wildon: "Suppose that $V$ is a finite-dimensional $\mathfrak{sl}(2,\mathbb{C})$-module. Show that ...
0
votes
0answers
13 views

A canonical map Aut$_{\mathsf{Lie}_R}(\mathfrak{n} \rtimes_\pi \mathfrak{g}) \to$ Aut$_{\mathsf{Lie}_R}(\mathfrak{n})$

Let $\mathfrak{n}$, $\mathfrak{g} \in \mathsf{Lie}_R$ be two Lie algebras over a commutative ring $R$, s.t. $\mathfrak{g}$ acts on $\mathfrak{n}$ as a derivation: $\pi:\mathfrak{g} \to ...
3
votes
1answer
42 views

Inducing highest weight modules

I have a question regarding highest-weight modules: Let be $\mathfrak{g}$ a Lie algebra, $\mathfrak{b}$ a Borel subalgebra, $\mathfrak{h}$ a Cartan subalgebra and $U(\mathfrak{g})$ its universal ...
2
votes
0answers
40 views

Skew polynomial algebra and deformation

Let $R$ be an associative unital $k$-algebra. If $\alpha \in End_k(R)$ and $\delta$ is a $\alpha$-derivation, then one can define the skew polynomial algebra $R[x; \alpha,\delta]$ by letting $ax = x ...