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)

1
vote
1answer
22 views

Coadjoint representation $Ad^*$ of the Heisenberg group

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
0
votes
0answers
19 views

The $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group)

I did not understand why, the $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group and $\mathcal{g}*$ is the dual space of its Lie algebra $\mathcal g$) are given by i) The ...
0
votes
1answer
30 views

Why do linear splitting maps of Lie Algebra central extensions induce cocycles?

If we consider a central extension $\mathfrak h$ of a Lie algebra $\mathfrak{g}$ by the abelian $\mathfrak a$: $$0 \longrightarrow \mathfrak a \longrightarrow \mathfrak h ...
2
votes
0answers
12 views

Find basis $\beta$ of $M$ such that $\phi(H)$ sits inside diagonal matrices, $\phi(S)$ sits inside upper triangular matrices w.r.t. basis $\beta$

Following question appeared in my Lie Algebra exam,but unfortunately i could not solve this question. Let $L$ be a semi simple complex Lie Algebra and $ \phi:L \to gl(M)$ be a finite dimensional ...
1
vote
1answer
31 views

The inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$.

I am reading the book. On page 80, there is a concept the inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$. Here $\mathfrak{h}$ is a Cartan subalgebra of a Lie algebra ...
0
votes
0answers
18 views

Notation in Belavin-Drinfeld's classification of solutions to classical Yang-Baxter equations.

I am reading the paper, on page 6, equation (3.5), there is a notation $(1 \otimes \alpha)r_0$, where $r_0 \in g \otimes g$, $g$ is a semisimple Lie algebra, $\alpha$ is a root. For example, suppose ...
2
votes
1answer
56 views

When are orbits maximal integral manifold

If G is a Lie group acting on a manifold M through $\Psi$, one can argue that orbit of any $p\in M$ is an integral submanifold of the generators of group action. Roughly the proof is : 1) Fixing p ...
0
votes
1answer
25 views

Root spaces for a Cartan subalgebra of $\frak{sl}(2,\mathbb{C})$

I'm having trouble computing this. If I take the Cartan subalgebra generated by $$\left(\begin{matrix}0&1\\-1&0\end{matrix}\right)$$ then which are the two eigenspaces for the nonzero roots?
0
votes
0answers
16 views

What is the relation between solutions of classical Yang-Baxter equations and solutions of modified Yang-Baxter equations.

Let $g$ be a Lie algebra. The classical Yang-Baxter equation (CYBE) is: $$ [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. $$ The modified classical Yang-Baxter equation (MCYBE) is: $$ ...
1
vote
0answers
34 views

Lie Algebra of $\mathrm{SO}(2)$ and $\mathrm{O}(2)$ are the SAME - why?

If $G$ is a Lie Group (with identity element of $e$), then my definition of the Lie Algebra $\mathfrak{g}$ of $G$ is the tangent space of $G$ at $e$, so that $\mathfrak{g} = T_{e}G$. The Lie Algebra ...
1
vote
1answer
32 views

Cartan subalgebras and Jordan Normal form

I'm stuck with Kac notes on Introduction to Lie Algebras. I logically understand all the definitions and everything is fine but I can't understand what's the thinking behind it. So I'm not asking for ...
0
votes
0answers
24 views

How to show that $(\Lambda^2(g))^g = H^2(g)$?

Let $g$ be a semisimple Lie algebra and $\Lambda^2(g) = g \wedge g \subset g \otimes g$ the exterior square of $g$. Consider the adjoint action of g on $g \wedge g$ and let $$(\Lambda^2(g))^g = \{x ...
5
votes
1answer
139 views

Composition series for Verma modules.

Let $L$ a Lie Algebra. I need prove that that every Verma module $\Delta(\lambda)$ admits a composition series, i.e a series of submodules with simple factors. I found a proof that is quite short in ...
2
votes
1answer
48 views

What is the definition of a weight of a Lie Algebra?

Sorry for what is probably a stupid question, but our lecturer did not define weights (just told us to find the "standard definition") and I have some questions to do concerning them. Wikipedia ...
0
votes
0answers
12 views

Inner automorphisms on the Lorentzgroup

Consider the Lie algebra at the neutral element of the Lorentzgroup $O_1(n)$, which is $\mathfrak{g}_0=\{ X\in R^{n\times n}|X^T\eta+\eta X=0 \}$ and its complexification ...
1
vote
0answers
29 views

Application of Weyl's Theorem

Recall that Weyl's theorem says that any finite dimensional representation of semi simple Lie algebra is completely reducible. I'm trying some examples to understand this theorem properly. Since $\bf ...
1
vote
1answer
43 views

Conformal Killing vectors and Spheres

Consider $S^n\cong SO(n+1)/SO(n)$. Thus we have an homogeneous space, whose isometry group is $SO(n+1)$. We have the round metric (the tensor preserved by the isometry group) and the generators of ...
1
vote
0answers
22 views

When are stabilizers of the coadjoint action equal?

Let $G$ be a compact connected Lie group with Lie algebra ${\frak g}$. For $\lambda\in{\frak g}^*$ let $$G_\lambda=\{g\in G:{\rm Ad}_g^*\lambda=\lambda\},$$ i.e. $G_\lambda$ is the stabilizer of ...
4
votes
1answer
73 views

Relation of $G$-invariants and $g$ -invariants.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $H$ a Hopf algebra and $M$ an $H$-module. By definition, $m \in M$ is called invariant if $x.m=\epsilon(x)m$, $\forall x \in H$, where $\epsilon: H ...
1
vote
1answer
46 views

Do we have $(g \wedge g)^g = 0$?

Let $g$ be a simple Lie algebra. Let $(g \wedge g)^g = \{a \wedge b \in g \wedge g: x.(a \wedge b) = [x,a] \wedge b + a \wedge [x,b] = 0\}$ be the set of $g$ invariants under the adjoint action. Do ...
0
votes
0answers
20 views

Finite-dimensional complex representations of nilpotent Lie algebras over a subfield of $\Bbb{C}$

From Lie's theorem we know that the complex representations of any solvable Lie algebra $\mathfrak{g}$ over a subfield of $\Bbb{C}$ are such that there exists a basis in which the representation ...
1
vote
0answers
59 views

Tits algebras of E_6

The general construction of Tits algebras of algebraic groups can be found in Knus, Merkurjev, Rost, Tignol - The book of involutions § 27 For every projective, homogeneous G variety $X:=G/P$, with ...
1
vote
1answer
30 views

Lie Algebra Homomorphisms for Lie Subgroups

Let $G$ be a Liegroup and $H$ a Lie subgroup of $G$. Then we find a Liegroup homomorphism $i \colon H \to G$ and the induced map $i_* \colon \mathfrak{h} \to \mathfrak{g}$ between the corresponding ...
8
votes
1answer
170 views

Proof of the Isomorphism between: $SL(2,\mathbb R) \times SL(2, \mathbb R) \cong SO^+(2,2)$

I want to do a proof that $SL(2,\mathbb R)\times SL(2, \mathbb R) \cong SO^+(2,2)$. My idea was to use the same Argument as in this Question. So I wanted to begin with the Basis of the Lie algebra ...
2
votes
1answer
34 views

What are the $g$-invariants of $g \otimes g \otimes g$ under adjoint representaion?

Let $g$ be a Lie algebra. Consider the adjoint action $g \times g \otimes g \otimes g \to g \otimes g \otimes g$ given by \begin{align} x.(a \otimes b \otimes c) = [x, a] \otimes b \otimes c + a ...
0
votes
0answers
21 views

The sum of two r-matrices.

Let $g$ be a Lie algebra. Suppose that $r \in g \wedge g$ satisfy the condition: $[[r, r]] = [r_{12}, r_{13}]+[r_{12}, r_{23}] + [r_{13}, r_{23}]$ is a non-zero unique, up to scalar multiple, ...
0
votes
0answers
22 views

The smallest Lie subalgebra contains a subset

Let $\mathfrak g$ be a Lie algebra and let $S$ be a subset of $\mathfrak g$. How to show that the Lie subalgebra generated by $S$ consists of all linear combinations of the elements $[s_m, s_{m−1}, ...
0
votes
0answers
24 views

diagonal subalgebras of classical Lie algebras

How to use the Invariance Lemma to prove that the diagonal subalgebra of classical Lie algebras are self-normalizing? When $\operatorname{char}\ F=0$? (Invariance Lemma) Assume that $F$ has ...
0
votes
1answer
26 views

Maximal Lie subalgebra of $sl_n$

How to find the maximal solvable Lie subalgebra of $\mathfrak sl(n,\mathbb R)$? Maybe the invariance lemma is the key!
3
votes
2answers
76 views

Finding the basis of $\mathfrak{so}(2,2)$ (Lie-Algebra of $SO(2,2)$)

I want to calculate the Basis of the Lie-Algebra $\mathfrak{so}(2,2)$. My idea was, to use a similar Argument as in this Question. The $SO(2,2)$ is defined by: $$ SO(2,2) := \left\{ X \in ...
1
vote
1answer
33 views

Lie bracket of a semidirect product

I'm trying to solve problem 1.12 of chapter 1 from Duistermaat & Kolk' Lie groups. In the exercise you have a Lie group $G$ and a finite-dimensional vector space $V$, and a homomorphism ...
2
votes
1answer
33 views

Exact sequence for the Spin group

I read that the $Spin(n)$ is the double cover of $SO(n)$ such that the following sequence is exact $$ 1 \to \mathbb{Z}_2 \to Spin(n) \to SO(n) \to 1 $$ My first question is what information does this ...
0
votes
1answer
63 views

Representation of the lie algebra of a simply connected algebraic group $G$ induces a representation of the group itself

Let $G$ be a simply connected algebraic group over $C$. We know that a representation of an algebraic group $$\phi : G \to GL(V)$$ induces a representation of its lie algebra (taking the ...
1
vote
1answer
57 views

Are the $C$-points of a simply connected algbraic group simply connected?

Let $G$ be a simply connected algebraic group defined over $C$. Note : I see the definition of simply connected as "Every isogeny to $G$ is an isomorphism" as given in Hochschild's "Basic Theory of ...
2
votes
1answer
30 views

Commutator formula in $U(g)$.

Let $g$ be a simple Lie algebra and $U(g)$ the universal enveloping algebra. Let $a,b,c,d \in U(g)$. Then I think that we have $[a \otimes b, c \otimes d] = [a,c] \otimes bd + ca \otimes [b,d]$. Is ...
0
votes
0answers
24 views

Good reference on the parametrization of SU(3) and SU(N)

For the 2-dimensional $SU(2)$ matrices, there is a fairly general parametrization formulation: $s_2=\begin{bmatrix} e^{i\alpha}cos(\theta) & -e^{-i\beta}sin(\theta) \\ ...
0
votes
1answer
19 views

Lie algebra with a nilpotent quotient

If $g$ is a finite dimensional Lie algebra. If $h$ is an ideal of $g$ with $g/h$ is nilpotent and $ad_x|_h$ is nilpotent for all $x\in g$. How to show that $g$ is nilpotent?
2
votes
1answer
24 views

Can the last non-zero term in the central series of an indecomposable nilpotent Lie algebra be smaller than the center?

Let $L$ be an indecomposable nilpotent Lie algebra (finite dimensional and over $\mathbb{C}$). Is it possible for the last non-zero term of the central series to be strictly smaller than the center? ...
2
votes
1answer
39 views

Spheres as Homogeneous Spaces

Any odd dimensional sphere $S^{2n+1}$ can be expressed as an homogenous space of $SU(n)$ by $S^{2n+1} \simeq SU(n)/SU(n-1)$. Any even dimensional sphere $S^{2n}$ sphere can be expressed as an ...
2
votes
1answer
23 views

Right unidimensional extension of Heisenberg Algebra

I'm reading a book on Mathemathical Physics and speaking of the Quantum harmonic oscillator says (this is a translation in english, hope is right): The commutation relations between operators are: ...
1
vote
1answer
24 views

Nilpotent 4-dim Lie algebra

What is an example of a nilpotent Lie algebra $\cal g$ with dimension four and: $dim\ \mathcal g'=1$, $dim \ \mathcal g'=2$, $dim \ \mathcal g'=3$? Is there a way to construct them?
0
votes
1answer
18 views

3-dim Lie algebra with two commutative elements

Let $\cal g$ be a Lie algebra and let $a,b,c\in \cal g$ be such that $ab=ba$ and $[a,b]=c\not =0$. Let $\mathcal h=span\ \{a,b,c\}$. How to prove that $\mathcal h$ is isomorphic to the strictly upper ...
0
votes
1answer
22 views

What role does abstract Jordan decomposition play in the study of semisimple Lie algebras?

I have seen the notion of the abstract Jordan decomposition be given for semisimple Lie algebras over an algebraically closed field of char = 0 in a number of places, a few theorems proved about it ...
1
vote
1answer
36 views

The center of a nilpotent Lie algebra intersects each ideal

If $\cal h$ is a nonzero ideal in a nilpotent Lie algebra $\cal g$. How to prove that $\mathcal h\cap Z(\mathcal g)\not =0$, where $Z(\mathcal g)$ is the center of $\mathcal g$?
2
votes
1answer
31 views

If L is nilpotent then $K\cap L^n \not=0$

Let $K$ be a proper ideal of a nilpotent Lie algebra $L$. If the nilpotency class of $L$ is $n$ (i.e $L^n\not = 0, L^{n+1}=0$). Is it correct that $K\cap L^n \not=0$?
1
vote
1answer
43 views

Whitehead's lemma (Lie algebras) for reductive Lie algebras.

I move the question here. Whitehead's lemma (Lie algebras) is: Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero, $V$ a finite-dimensional module over it and $f$: ...
1
vote
1answer
16 views

Selfnormalizing sub-algebra and direct sum decomposition

I got the following setting: Consider the decomposition $L=H+\sum_{\alpha\in \phi}L_{\alpha}$, where the sum is a direct sum, the $L_{\alpha}$ are the root spaces and $H$ is nilpotent (because its ...
2
votes
1answer
36 views

The center of nilpotent Lie algebra

Let $L$ be a non-abelian nilpotent Lie algebra and $Z(L)$ be its center. Is it possible for $Z(L)$ to be a maximal ideal in $L$? An attempt: Since $L$ is not abelian then there exists $x\in L, ...
0
votes
0answers
18 views

The Lie algebra quotient of a maximal ideal

If $\mathcal g$ is a Lie algebra, and $Z(\mathcal g)$ is its centralizer. Suppose that $Z(\mathcal g)$ is a maximal ideal of $\mathcal g$. Is it correct that $\mathcal g/Z(\mathcal g)$ is of dimension ...
0
votes
2answers
26 views

codimension of the derived algebra of a nilpotent Lie algebra

Let $\mathcal g$ be a nilpotent Lie algebra with $dim>1$. Why it is impossible that $codim \ \mathcal g'=1$?