For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

learn more… | top users | synonyms (1)

2
votes
1answer
94 views

Does an ideal of finite codimension in a finitely generated algebra have always to be finitely generated?

I have been reading a book on Lie Algebras ("Álgebras de Lie" by San Martin) and there is this exercise in the chapter on universal enveloping algebras with a claim that I can not prove: Suppose ...
1
vote
0answers
16 views

Hermit reciprocity, $\mathfrak{sl}_2(\mathbb{C})$

Let $V$ be the standard $2$-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$. Hermit reciprocity states that $S^n(S^mV)\simeq S^m(S^nV)$. Can anybody give me a hint to prove it or give a ...
2
votes
0answers
29 views

Weyl Group of Parabolic subgroups

Let $G=SL(n,\mathbb R)$ with Lie algebra $\mathfrak{g}=\mathfrak{sl}(n,\mathbb R)$. The classical minimal parabolic subgroup $B$ consists of the upper triangular matrices. The parabolic subgroups $P$ ...
0
votes
0answers
14 views

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 ...
0
votes
1answer
13 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
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 ...
1
vote
0answers
31 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
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 $\mu$? ...
0
votes
0answers
28 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? ...
0
votes
0answers
24 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 ...
6
votes
2answers
166 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 ...
3
votes
0answers
31 views

Compact Lie algebras and Lie groups

A simple or semisimple Lie algebra is said to be compact if the $\mathrm{Tr}\left \{ T^\mathrm{adj}_{a}, T^\mathrm{adj}_{b}\right \}$ is positive definite where $T^\mathrm{Adj}_{a}$ are the generators ...
2
votes
1answer
28 views

Computing a differential on a derivation

Let $\varphi:G\to G'$ be a morphism of algebraic groups over an algebraically closed field $k$, so that $d\varphi:\mathscr{L}(G)\to\mathscr{L}(G')$ is a morphism of Lie algebras. Here I view ...
3
votes
1answer
118 views

Which subgroup of $\mathrm{SL}(2,\mathbb{C})$ is this?

I am looking into sub-algebras of $\mathfrak{sl}_2(\mathbb{C})$ and the subgroups of $\mathrm{SL}(2,\mathbb{C})$ they generate. The basis of $\mathfrak{sl}_2(\mathbb{C})$ I am using consists of 3 ...
0
votes
0answers
20 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 ...
0
votes
0answers
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, $\Delta^+$ ...
0
votes
0answers
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 ...
0
votes
1answer
22 views

What is the Lie group of $\mathfrak{h}$?

Let $\mathfrak{h}$ be a Cartan subalgebra of a Lie group $G$. What is the Lie group of $\mathfrak{h}$? By definition, the Lie group of $\mathfrak{h}$ consisting of elements of the form $e^{h}$, $h \in ...
0
votes
0answers
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 ...
2
votes
0answers
19 views

Linearly independent skew symmetric complex matrices having the least eigenvalues

Question: Let $A$, $B$ be two $5 \times 5$ (or $7 \times 7$) skew-symmetric complex matrices (i.e. $A^t = -A$), and suppose that $$ \forall t,s \in \mathbb{C}, \quad M(t,s):=(tA+sB)^*(tA+sB) \text{ ...
8
votes
1answer
240 views

Proving that there exists a saturated set with given highest weight

This is an question about an exercise in Humphreys book on Lie algebras. First of all a bunch of definitions and notation, see §13 in Humphreys for details. Let $\Phi$ be a root system, $\Delta$ a ...
0
votes
0answers
13 views

How to find next M.Hall's word question

Given a Hall word. How do I write the next one?
1
vote
0answers
31 views

Lie bracket as defining element for transformations

Why is it precisely the Lie bracket that encodes the information about a given transformation? A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given ...
0
votes
0answers
19 views

Is adjoint map invertible?

I've already studied the group of automorphisms of a simple lie algebra on a finite field, but according to the definition of an adjoint representation of a Lie algebra, can we claim an adjoint map is ...
2
votes
1answer
42 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
0answers
44 views

Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
1
vote
1answer
31 views

Lie group and generated ideals

I have this question in my textbook, and I can't seem to solve it on my own: Let $P \subset GL(n,\mathbb{C})$ be a subgroup as following: $P$ consists of all matrices in block ...
2
votes
0answers
17 views

Does “Spherical Symmetry” as defined in General Relativity imply a Foliation of Spheres?

In Carroll's "spacetime and geometry" he defines a spherical symmetrical spacetime as a spacetime $(M,g)$ for which there exists a Lie algebra homomorphism between the Lie algebra of a subset of the ...
1
vote
1answer
33 views

Lie groups. How to show that the group operations are smooth.

$N:=\{g\in GL(n,R) : g_{ij}=0 \forall j>i , g_{ii}=1 ∀i\}$. For this matrix group, how can we show that it is a Lie group? I am at the beginning of the subject of Lie groups so I can not ...
3
votes
1answer
41 views

Why are these algebras non-isomorphic?

I am trying to classificate 3-dimensional complex Lie algebras, and this is the first place where i got stuck. Consider a 3-dimensional vector space with basis {$x$, $y$, $z$}. Now i have managed to ...
2
votes
1answer
36 views

Is $sp(4)$ a subalgebra of $su(5)$?

Is $sp(4)$ a subalgebra of $su(5)$? And how can I prove/disprove this? I know already that it cannot be a regular maximal subgroup of $su(5)$ since the Dynkin diagram (which has two roots of unequal ...
1
vote
1answer
42 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
47 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 ...
1
vote
1answer
61 views

Lie groups, maps and the Weyl group

If I have a map of simple Lie groups $H \to G$, do I get a map of Weyl groups $W_H \to W_G$? If $H$ is the semisimple component of a parabolic subgroup then we can clearly get this (see Ivan's answer ...
2
votes
2answers
37 views

If HP=PH+P for H,P n×n complex matrices, must H be diagonalizable?

If $F$ is a field of characteristic zero, $H,P$ are $n\times n$ matrices over $F$, $0 \neq \alpha \in F$, and $HP=PH+\alpha P$, then must the minimal polynomial of $H$ be square-free and must $P$ ...
2
votes
1answer
28 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
29 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
0answers
17 views

How to classify all $\theta$-stable Cartan sub algebras?

Let $G$ be a linear connected semisimple Lie group, $\mathfrak g$ its Lie algebra. With respect to the Cartan involution $$ \theta:X\mapsto -\overline{X}^t, $$ one has $\mathfrak{g}=\mathfrak{k}\oplus ...
1
vote
1answer
28 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 ...
0
votes
1answer
29 views

Question to Roger Carter's “Lie Algebras of Finite and Affine Type”

In the proof of Proposition 7.31 in Roger Carter's Lie Algebras of Finite and Affine Type, Carter notes that the sets $H_\mu$ and $H_\alpha$ are distinct. Can someone find a good argument why that is ...
1
vote
1answer
35 views

Dynkin Diagram $SU(n)$

The goal is to give the Dynkin diagram of $SU(n)$. One can show that the complexification of the Lie algebra $\mathfrak{g}$ of $G$ is given by $\mathfrak{G}_{\mathbb{C}}=\mathfrak{sl}(n,\mathbb{C})$ ...
2
votes
2answers
66 views

Naive question about the group $SU(n)$?

As usual, let $SU(n)$ represent the set of all the $n\times n$ unitary matrices with determinant $1$. It's easy to show that any matrix $U$ takes the form $U=e^{iA}$ ($A$ is a $n\times n$ traceless ...
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. $H=\begin{pmatrix} 1 & 0\\ 0 &-1\end{pmatrix}$, $X=\begin{pmatrix} 0 & 1\\ 0 & 0\end{pmatrix}$, ...
3
votes
1answer
34 views

A maximal subalgebra of $E_6$ !?

I'm puzzeled by the following sentence in one of Baez's posts: The Lie algebra $E_6$ has a subalgebra of maximal rank isomorphic to $\mathfrak{so}(10)\oplus \mathfrak{u}(1)$. However, I thought ...
1
vote
0answers
38 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.
4
votes
1answer
59 views

The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
4
votes
2answers
69 views

Is there a definition of a dual Lie algebra?

Let $L$ be a Lie algebra. For vector spaces, modules, Banach spaces, etc. we have the notion of a dual. Question: Is it possible to define naturally a Lie algebra $L^*$ that is in some sense dual to ...
2
votes
0answers
32 views

Where does the name “toral” come from?

Where does the name "toral" come from in "toral subalgebra"? I know a little (very little) Lie groups theory, so I guess it could be related to a Lie group whose Lie algebra is the toral one. Is ...