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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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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16 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
16 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 ...
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1answer
45 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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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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68 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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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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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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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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45 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 $L_n$, ...
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69 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: ...
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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 ...
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34 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
25 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 ...
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1answer
158 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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43 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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59 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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29 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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Help with proof in Fulton and Harris

On page 488, let $H$ be any element of $\mathfrak{g}$ such that the generalized null space $\mathfrak{g}_0(H)$ has minimal dimension. Then consider $X\in\mathfrak{g}_0(H)$, and the decomposition $X = ...
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1answer
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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
19 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
42 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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20 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 ...
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19 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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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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47 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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51 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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64 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 ...
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1answer
63 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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27 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 ...
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46 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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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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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 ...
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64 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
54 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 ...
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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 ...
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2answers
86 views

Equivalence of Two Lorentz Groups

How can I prove that $O(3;1)$ and $O(1;3)$ are the same group?
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1answer
35 views

Dimension of the weight space in stadard cyclic L-modules

Let $\lambda \in H^*$ be the irreducible standard cyclic module $V(\lambda)$ of weight $\lambda$ of a semisimple Lie algebra $L$. What are all the possible ways to determine : 1) Which $V(\lambda)$ ...
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1answer
36 views

Adjoint representation of $SL(2,\mathbb{R})$

Let $G$ be a Lie group. The adjoint representation $\text{Ad}:G \rightarrow \text{GL}(T_e G)$is given by $$ \text{Ad}(x):=T_e \mathcal{C}_x, $$ where $\mathcal{C}_x(y)=xyx^{-1}$. Now suppose that ...
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1answer
80 views

Generators of Translation - Lie Algebra [duplicate]

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
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3answers
195 views

Translations in two dimensions - Group theory

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
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1answer
65 views

intuitive interpretation of Lie algebra

As you know, the isomorphism between $SO(2)$ and $e^{i\theta}$ allows an intuitive visualization of the Lie algebra $\mathfrak{so}(2)$ as the line $ti$. I wanted to know if there was a similar ...
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How does the lie algebra capture compactness of the lie group?

This is a soft question. Let $V,\rho$ be a representation of the lie algebra $\mathfrak{so}_3(\mathbb{R})$. Then if I understand everything right, $V$ is necessarily completely reducible, because the ...
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Would the transformation of a differential equation obey the same algebra?

I've found that the algebra of this differential equation $$\frac{d^2y}{dz^2}-(3z^2+\gamma)\frac{dy}{dz}+(cz+\alpha)y=0$$ is in $sl(2)$ because it is possible to use the generators of the $sl(2)$ ...
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1answer
41 views

Complete reducibility of a field extension of an lie algebra representation

Let $\mathfrak{g}$ be a lie algebra over a field $k$ with characterstic $0$ and $k\subset k'$ a finite field extension. Suppose $\mathfrak{g}\otimes k'$ has the property, that all finite dimensional ...
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1answer
35 views

Question on Left-Invariant Vector Fields

Let $G$ be a Lie group, and $\xi \in T_{e}G$ a tangent vector at the identity. Given a function $f \in C^{\infty}(G)$, verify that $ g \rightarrow ((\ell_{g})_{*}\xi)f$ is a $C^{\infty}$ function on ...
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Fulton-Harris Lemma 3.35

In the proof of Lemma 3.35 in Fulton--Harris, Representation Theory, it is claimed that the identification $H(\phi^2(x),y)=H(x, \phi^2(y))$ implies that $\lambda$ is a positive real ($\phi^2$ is known ...
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Explicit Weyl group invariant polynomials

Quoting this post, "Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset\mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to g. Let ...
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41 views

Kernel of homomorphism on unit circle S1

Let $f : S^1 \to S^1$ be defined such that $f(z) = z^2$, where $z$ is a complex number. It's easy to check that this is a homomorphism on $S^1$. However, how would you find the kernel and the coset ...