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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A question on the nilradical and the nilpotent ideal of $\mathfrak{p}\subseteq \mathfrak{gl}_n(\mathbb{C})$

Let $\mathfrak{p}\subseteq \mathfrak{gl}_n(\mathbb{C})$ be a parabolic algebra of a parabolic group $P\subseteq GL_n(\mathbb{C})$. What is the difference among the nilradical of $\mathfrak{p}$, the ...
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3answers
142 views

How to prove that $B^\vee$ is a base for coroots?

Let $\Phi$ be a root system in a real inner product space $E$. Define $\alpha^\vee = \frac{2\alpha}{(\alpha, \alpha)}$. Then the set $\Phi^\vee = \{\alpha^\vee: \alpha \in \Phi \}$ is also a root ...
3
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1answer
76 views

How to deal with multiple representations of quaternions

I'm using a quaternion to represent the orientation in a kalman filter. My algorithm works fine until I rotate "upside down". I think this is because there seems to be multiple ways to represent the ...
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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 ...
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1answer
45 views

Can someone tell me books or papers on subalgebras of $\operatorname{SL}(3)$?

I hope to find the smallest subalgebra of $\operatorname{SL}(3)$ that contain the matrix $$\begin{pmatrix} 0 & a & 0\\0 & 0 & b\\c & d & 0 \end{pmatrix}$$ Are there any ...
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1answer
55 views

Vector Space of Lie Algebra

Lie algebra $ \mathfrak{g} $ for a Lie group $ \mathcal{G}$ is closed under commutation. Also, the elements of Lie Algebra form a Linear Vector Space(LVS). Firstly, when is it allowed to define an ...
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1answer
40 views

For an element $x$ in an algebraic group $G$, why do we have $\mathscr{L}(C_G(x))\subset\mathfrak{c}_{\mathfrak{g}}(x)$?

I'm reading Humphreys' Linear Algebraic Groups, trying to understand the following argument found on the top of pg. 76. Let $G$ be an algebraic group over some field $k$, with $x\in G$. Let ...
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1answer
80 views

What are the properties of this Poisson algebra?

I have the following (real) quantities (which are from a Classical Mechanics problem): $$A_1=\frac 1 4(x^2 +p_x^2-y^2-p_y^2 ) \quad A_2=\frac 1 2(x y +p_x p_y)$$ $$A_3=\frac 1 2(x p_y - y p_x )$$ ...
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1answer
20 views

Difference between the pairings $\text{Tr}(xy)$ and $\text{Tr}(x^t y)$

Let $\mathfrak{g}$ be the tangent space to $GL_n(\mathbb{C})$ at the identity. What is the difference between the two maps? Any subtle geometric or algebraic difference between the two pairings $$ ...
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35 views

Ideals in the unitary group

What would be examples of one-dimensional ideals in the lie algebra of the unitary group? Moreover, how would one show that it is in the tangent space of the center of the unitary group and that the ...
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1answer
24 views

Induced automorphism on a tangent bundle

I had a pretty simple question but was having trouble finding the answer anywhere. If I have an orthogonal matrix $A: \mathbb{R}^n \to \mathbb{R}^n$, it should induce an automorphism on the tangent ...
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1answer
28 views

Given tangent space of submanifold of Lie group, is it possible to recover the submanifold?

I have computed the tangent space of certain submanifolds (the unstable manifolds) of a Lie group at a particular point. I know that the exponential map lets us move between the Lie algebra and the ...
3
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1answer
79 views

Inequality of Frobenius norm for skew matrices

Let $A$ be a complex skew-symmetric $n \times n$ matrix, that is, $A^T = -A$. Denote by $\|\cdot\|_F$ the Frobenius norm, that is, $\|B\|_F^2 = \text{trace}(B^*B)$. I would like to prove that $$ ...
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2answers
71 views

Tensoring with induced representation

On J. Humphreys' book "Representations of Semisimple Lie Algebras in the BGG Category O", Theorem 3.6, a Tensor Identity is quoted: $$ (U(\mathfrak{g}) \otimes _{U(\mathfrak{b})} L) \otimes M\simeq ...
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0answers
37 views

Basis of Witt algebra

The Witt algebra $W(n,m)$ is defined as the set of element $\{\sum f_j D_j$ such that $ f_j ∈ A(n,m)\}$ with usual Lie bracket. I am a bit confused about basis for $W(n,m)$? What is the meaning of ...
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1answer
18 views

Uniqueness of the Lie brackets in the quotient space of a Lie algebra

Suppose I have a Lie algebra $\mathfrak g$ which is an ideal of $\mathfrak a$. Then I consider the quotient set $\mathfrak g / \mathfrak a$ which is the set of all equivalence relations of $\mathfrak ...
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1answer
20 views

Space of tangents of a matrix group G?

Given a smooth path A(t) through the identity in any matrix group G, how would one prove that the smooth path through any g in G, is of the form gA(t)? It is clear that gA(t) is differentiable and ...
2
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1answer
48 views

simple Lie groups

A Lie group is a group which is a smooth manifold such that the multiplication and inversion are smooth. When does a Lie group become simple? What is the difference between simple and semi-simple Lie ...
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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 ...
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1answer
85 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
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1answer
80 views

Cauchy gave 1st example of a Lie algebra in 1847 & exterior product in 1853‽

I read in PDF pg. 5 of this that Cauchy gave the first example of a Lie algebra in 1847: It also claims that he invented the exterior product in 1853. Does anyone have references for this?
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0answers
22 views

Bilinear form on the space of smooth complex valued functions.

Let $G$ be a Lie group and $h$ be the Hermitian bilinear form on smooth complex valued functions then how can we define bilinear form on the space of smooth complex valued functions.
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37 views

A question on Lie Algebra

This is an exercise from 'Lie Algebras in Particle Physics' by Howard Georgi, Ex.6.B. Suppose that the raising lowering operators of some Lie Algebra satisfy: $[Eα,Eβ]=NE(α+β)$, where the $α,β$ are ...
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1answer
116 views

Lie algebra (su(1,1)) from legendre polynomials; question regarding http://arxiv.org/abs/1205.6353

Apologies if this question is a duplicate. OK, so my question heavily involves the paper http://arxiv.org/abs/1205.6353 which nicely details the Lie algebra su(1,1) coming from the Laguerre $L_n$, ...
3
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1answer
197 views

Baker–Campbell–Hausdorff formula for [exp(x),exp(y)]

Can someone provide a explicit (the first priority with leading orders, then the secondary consider as complete as possible, or) expansion like Baker–Campbell–Hausdorff formula for the commutator: ...
2
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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 ...
2
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1answer
41 views

Real orthogonal Lie algebra isomorphic to Clifford bivectors

I'm studying Clifford algebras on this moment, and I frequently find the statement $$\left(\mathbb{R}_m^{(2)},[\cdot,\cdot]\right) \cong \mathfrak{so}_{\mathbb{R}}(m)$$ stating that the bivectors of a ...
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1answer
32 views

Question on lifting the Weyl group into the group of inner automorphisms of $\mathfrak{g}$

I'm looking for some clarification of a statement that I found in Kac and Peterson's paper (112 realizations of the basic representation of the loop group of $E_8$). Let $\mathfrak{g}$ be a simple ...
0
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1answer
246 views

Heisenberg XXX spin model

Let $\pi$ be the standard representation of $sl_2(\mathbb{C})$ on $\mathbb{C}^2$. Let $p_1,p_2,p_3$ the three Pauli matrices. Define $S^a:=\frac{1}{2}\pi(p_a)$. What does such matrices looks like?
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1answer
46 views

Computing Lie algebra of a subgroup

I will like to know how does one compute the Lie algebra of an abstractly given subgroup of a Lie group? Specifically, let $G = \mathrm{SO} ( n + 1, 1 )$ and consider the flow $$ g_t = ...
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0answers
63 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
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0answers
32 views

Integration on associated vector bundle

Let $G$ be a compact lie group and $\mathfrak{g}$ be its Lie algebra then we can construct the integral on $G\times \mathfrak{g}$ by $$\int_G\int_{\mathfrak{g}}f(x,Y)dxdY$$ Where $x\in G$ and $Y\in ...
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1answer
20 views

Help with proof in Humphreys (2)

Lemma: If $\mathfrak{k}$ is a subalgebra of $\mathfrak{g}$ that contains an Engel subalgebra, then $\mathfrak{k}$ is self-normalizing. Proof: Suppose $\mathfrak{k}\supset \mathfrak{g}_0(ad\; x)$ for ...
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1answer
21 views

Is the radical of a Lie algebra preserved by any of its derivations?

Let $\mathfrak{g}$ be a finite dimensional complex Lie algebra. A derivation $D: \mathfrak{g} \rightarrow \mathfrak{g}$ is $\mathbb{C}$-linear map $\mathfrak{g} \rightarrow \mathfrak{g}$ such that for ...
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1answer
61 views

How to find Casimir Operators and their degree.

Consider the quite general problem of computing all Casimir Operators of a given Lie Algebra $\mathfrak{g}$. How does one proceed, in general? And how is possible to compute the degree of a given ...
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26 views

Why $\widehat{G^{\mathbb{C}}}$ can be identified with the space of highest weights

Let $G$ be a compact connected Lie group and $G^{\mathbb{C}}$ be the complexification of Lie group $G$ and we denote $\widehat{G^{\mathbb{C}}}$ the set of isomorphism classes of irreducible rational ...
0
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1answer
20 views

Show that $(X^*)^*=\epsilon\epsilon 'X$

A finite dimensional vector space $V$ with a non-degenerate form (,) s.t. $(u,v)=\epsilon (v,u) \forall u,v\in V$ is called a quadratic space of type $\epsilon$. Let $V$ be a quadratic space of type ...
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1answer
36 views

Why $X^* X\in \mathfrak{g}(V)$.

A finite dimensional vector space $V$ with a non-degenerate form (,) s.t. $(u,v)=\epsilon (v,u) \forall u,v\in V$ is called a quadratic space of type $\epsilon$. Let $V$ be a quadratic space of type ...
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1answer
66 views

Tangent space of $\mathfrak{ so}(3)$ Lie algebra

Very basic question and the terminology makes it difficult to find a reference. I just know the basics of differential geometry but my question is simple. Is the tangent space at the point ...
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0answers
63 views

Structure constants of Lie algebra from system of linear equations

The Jacobi identity in terms of the structure constants $c_{p,q}^r$ of an $N$-dimensional Lie algebra with $p,q,r=1,\ldots,N$ reads $$ J_{i,j,k}^l \equiv c_{i,j}^m \, c_{k,m}^l + ...
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1answer
69 views

A question about differential forms on Lie groups

Let $G$ is a Lie group and $\mathfrak{g}$ is its Lie algebra and $\mathfrak{g}^{\mathbb{C}}$ be its complexification. Also assume that $\mathfrak{h}\subset \mathfrak{g}^{\mathbb{C}}$ be its ...
3
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1answer
94 views

Weyl Character Formula to find $M_\lambda(\mu)$

In Introductory Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the ...
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29 views

Weyl Character Formula to find $M_\lambda(\mu)$

In Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the dimensions of the ...
1
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2answers
50 views

Simple Lie algebra is a matrix algebra?

Wedderburn's Theorem. Let $A$ be a simple finite $k$-algebra. Then $A$ is a matrix algebra over a finite $k$-algebra $K$ which is a skew field. (Here matrix algebra means $A=M_n(K)$ for some $n$.) ...
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0answers
36 views

Equivalent representations of $\mathfrak{sl}_2$

Hello I have a question about the equivalence of two representations of the Lie algebra $\mathfrak{sl}_2$. The first representation is $(ad,\mathfrak{sl}_2)$ the adjoint representation with map ...
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0answers
34 views

A question in the proof that the weight of a finite dimensional module is W-invariant

Recently I'm reading Humphrey's book "Introduction to Lie algebra and representation theory", section 21 on the finite dimensional module of a semisimple lie algebra, and I have a question here which ...
2
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0answers
77 views

Invariant bilinear forms on Lie algebras

Consider a (compact) Lie group $H$ that acts on its Lie algebra $\mathfrak h$ in the usual way, $x\mapsto gxg^{-1}$ for any $x\in\mathfrak h$ and $g\in H$. Suppose we are given a real symmetric ...
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1answer
142 views

SO(2) group generator Lie Algebra

For the $2 \times 2$ orthogonal group of matrices which for the $SO(2)$ group, there is only one free parameter in the group element and hence only one generator for the group. Which is, $$ X_g = ...
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Show that $\langle[U,X],V\rangle = -\langle U,[V,X]\rangle$ for bi-invariant metric in Lie group

I know that $\langle U,V \rangle = \langle dR_{x_{t(e)}}U, dR_{x_{t(e)}}V \rangle$ and $\langle U,V \rangle = \langle dL_{x_{t(e)}}U, dL_{x_{t(e)}}V \rangle$ because it is bi-invariant. How do I ...
5
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3answers
118 views

Is the center of the universal enveloping algebra generated by the center of the lie algebra?

Let $\mathfrak{g}$ be a Lie algebra over a field $k$, and let $U(\mathfrak{g})$ be its universal enveloping algebra. $\mathfrak{g}$ is canonically embedded in $U(\mathfrak{g})$; identify it with its ...