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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Cayley-Hamilton type decomposition of SL(3,R) matrices

Given an element $\lambda = \theta_a T_a$ of SL(3,R) Lie algebra, where $T_a$s are the generators and $\theta_a$s are parameters, is there a general formula to determine the coefficients A,B and C ...
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Elements of $gl(2l+1,\mathbb{C}): x^tS =- Sx$, How they are found and Erdman exercise 4.2

On page 130 of Erdman's book "Introduction to Lie Algebras" we have: Let $L = gl_S(2l+1,\mathbb{C})$ for $l \geq 1$ where $S = \left(\begin{array}{cc} 1 & 0 & 0 \\ 0 & 0 & I_3 \\ 0 ...
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25 views

Convolution and Characters

I am confused about the purpose of the Formal Character, character functions, and the convolution in representation theory of Lie algebras. Is the Character function different than just the Character? ...
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44 views

A criteria for a subalgebra of M(n,C) being M(n,C)

Suppose $S$ is a subalgebra of the matrix algebra $M_n(\mathbb{C})$. If for any vector $v$ and $w$ in $\mathbb{C}$, there always exists a matrix $A$ in $S$, depending on $v$ and $w$ of course, which ...
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Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
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25 views

Isometries of S^3 and some Lie algebras

By considering $S^3$ as the group of unit quaternions, and letting it act on itself from both the left and right, one can get an isomorphism $SO(4)\cong (S^3\times S^3)/C_2$, where the $C_2$ subgroup ...
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36 views

How can we compute a Lie bracket for powers of elements of given lie algebra?

Let $L$ be a lie algebra over finite field, for $ x,y$ in $L$ I want to solve the following bracket: $[yx^k,x]=?$ How can we describe that in the format of $[...[y,x],x],...,x]=[y,x]_i$ ($i-times$)
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47 views

Factorization of Compact Lie Algebras into Irreducible Ideals

I have read in some lecture notes on Lie theory that any compact Lie algebra $\mathfrak{g}$ can be factored as a direct sum of of irreducible ideals for the $\mathrm{ad}$ representation. That is, ...
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47 views

Confusion in Lie algebra notes

I'm self-studying through these notes, and I ran into a roadblock on the page 38, chapter $sl(2)$ and its irreducible representations. Right after defining $U(sl(2)) \otimes_{U(b^+)} \mathbb C$ ...
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Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
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Visualizing Lie algebra of SO(3)

Let $SO(3)$ be the Lie group of 3D rotations. Rotation about z-axis by an angle $\phi$ is represented in standard basis by this matrix: $$ \begin{pmatrix} \cos \phi & -\sin\phi & 0 \\ ...
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Lie algebra: If ad(g) is solvable then g solvable?

I'm trying to prove that if the image of the adjoint representation of a Lie algebra g is solvable then g is solvable, ie, if for some n (ad(g))^(n) = 0 then there exists a m such that g^(m) = 0 My ...
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24 views

Considerations for moving a function inside or outside of an integral

Excluding the possibility that $A(t)$ is the limit of a sequence, are there any special considerations I should be concerned with regarding the following assertion: Let $A(t)$ be an $n\times n$ ...
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39 views

Complex conjugation of positive roots

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...
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20 views

Multiple Cartan sub-algebras

How is it that for a Semi-simple Lie Algebra there is not one Cartan Sub-Algebra? I assume since there are multiple CSA's of a SS Lie algebra that must mean some of the ss elements of the Lie ...
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34 views

Diagonalizabilty of ad(adjoint map)?

let $\mathsf{g}$ be a finite dimensional lie algebra and $\xi\in\mathsf{g}$. Under which conditions the adjoint map $ad_\xi :\mathsf{g}\longrightarrow \mathsf{g}$ is diagonalizable? what about ...
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19 views

Closed Connected Subgroup of $SO(5)$

I was reading a paper in which a part of it they want to classify the closed connect subgroups of $SO(5)$. What they write is this: Let $G^0$ be a closed connected subgroup of $SO(5)$. Let $T$ be a ...
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46 views

Don't understand Levi decomposition theorem

Levi decomposition theorem states that any finite-dimensional real Lie algebra $L$ is the semidirect product of a solvable ideal and a semisimple subalgebra. I don't understand this since to me it ...
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22 views

Lattices in Lie Algebras

I am having a little confusion with the different types of lattices involved with Lie algebras. Root system: represented as euclidian vector arrows. However I have seen the same arrangement with ...
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30 views

regular representation of algebras

Let suppose we have universal enveloping algebra, what is the meaning of the notion of the right regular representation of that? How can we determine the right regular representation of universal ...
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Intuition behind PBW

The PBW theorem states: $\omega:\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras. Where $\mathfrak {S} $ is the symmetric tensor algebra of a Lie algebra $ L $. Where $\mathfrak ...
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How to obtain a Lie algebra homomorphism from a Lie group homomorphism

In class we learn a theorem tells us one can cook up a Lie algebra from a Lie group: If $f: G\to H$ is a homomorphism of a Lie group then $T_I f: T_I G\to T_I H$ is a homomorphism of Lie algebra. ...
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27 views

Semi-simple Irreducible Representations

I am studying the Representation Theory of Lie Algebras and came across this dilema. When can the representations of semi-simple Lie algebras be irreducible? I thought Weyl's theorem said this ...
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Eigenvalues of ad (Adjoint action) in semisimple lie algebra?

Suppose $V=V_0\oplus V_1$ be a $Z_2$-graded semi-simple lie algebra and, $\xi\in V_1$. The maps $ad_\xi \circ ad_\xi :V_0\longrightarrow V_0$ and $ad_\xi \circ ad_\xi :V_1\longrightarrow V_1$ are ...
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Construction of the simply connected Lie group of a given Lie algebra

Given a finite dimensional real Lie algebra $\mathfrak{g}$, I am trying to obtain a concrete realization of its simply connected Lie group $G$, with $\mathrm{Lie}(G) \cong \mathfrak{g}$. Let us ...
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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 ...
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58 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
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53 views

SAGE vs. Mathematica for Lie algebras / groups?

What math software is better for working with Lie algebras and Lie groups, SAGE or Mathematica?
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14 views

On the construction of the Verma module

My question is about the construction of Verma module of a lie algebra $L$, there is one step in the construction which I do not quite understand. Let $L=N_-\oplus H\oplus N_+$ be the triangular ...
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32 views

absolutely irreducible module

If L be lie algebra over F ( F is a field), I want to know what is the definition of absolutely irreducible FL-module? I have confused of several key words related to irreducible modules!? Is there ...
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Standard set of Generators

A standard set of generators for a semisimple Lie algebra $ L $ is defined as: {${x_\alpha}, {y_\alpha}, {h_\alpha} $} Where: $ x_\alpha \in L_\alpha, $ $ y_\alpha \in L_{-\alpha}, $ $ ...
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31 views

Implied relationships between Lie groups and Lie algebras.

Suppose $\mathcal{L}$ is a finite-dimensional Lie algebra, and $\mathcal{G} = e^{\mathcal{L}}$ is it's compact, connected Lie group. Given a closed sub-algebra $\mathcal{L}' \subset \mathcal{L}$, it ...
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Codimension of $\ker $ $\alpha $

Can someone explain why the codimension of $\ker $ $\alpha $ is $1$ in $ H $, with complement $ Fh_\alpha $? Is this because $ h_\alpha $ when $ \alpha $ is simple is part of the dual basis to ...
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Dual spaces: Roots and Cartan subalgebra

Can someone show that the roots and the Cartan subalgebra are dual vector spaces? I don't see how simple roots acting on non-corresponding indices of a Cartan basis produce 0 and a simple root ...
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How to calculate the Maurer-Cartan form in the adjoint representation?

While I am reading a paper, I come across a difficulty. Here, we have a Lie group and we know its Lie algebra defined as $[G_a,G_b]=f_{ab}^{\phantom{ab}c}G_c$ with $G_a\in\mathfrak g$. Under the ...
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Do involutions suffice to find reflected vectors in a reflection group representation?

Consider a reflection group $W$ acting by isometries on a Euclidean space $V$. I want to understand the union of $(-1)$-eigenspaces for this action, the set $$\{v \in V : \exists w \in W\ (w\cdot v = ...
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Convergence of Baker-Cambpbell-Hausdorf for compact groups

It is well known that the Baker-Campbell-Hausdorf formula doesn't need to converge for general elements of a Lie algebra, resp. for matrices with norms larger then 1. On the other side, if $G$ is a ...
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How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...
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56 views

Associative Lie algebra without Jacobi identity

1) Is there a name for associative Lie algebra that does not require Jacobi identity to hold? 2) Can such algebra exist, and if it does exist, can this algebra contain infinitely many elements? 3) ...
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two Roots questions

Just two questions on roots... 1) Can the length of roots only be defined relatively? And does length only come about because of the dot product and cartan integers? 2) This might be a weird ...
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how can we prove commutation formula?

Can anyone help me to prove the following proposition: Definition: Let F denote any field and suppose that A ia a vector space over F. If f be a bilinear mapping on A×A → A. the pair (A,f) is referred ...
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48 views

Closure of a Fundamental Weyl Chamber

Can someone explain what a "closure" of a Fundamental Weyl Chamber means? I assume it is related to an algebraic closure, but I don't see how. In addition, how does the Weyl group act on it and why ...
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39 views

killing form and the dot product

When going from talking about roots as functionals to talking about roots as vectors in a Euclidian space (root system), does the killing form become the dot product? Are the killing form and dot ...
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How to create a commutative lie algebra from commutative ring?

How does one create a commutative lie algebra (lie algebra is inherently anti-commutative, so this is added restriction) from commutative ring?
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How to visualise the Killing form of a Lie algebra

Given a Lie algebra $\mathfrak{g}$, we can define its Killing form $$K(x,y) = \mathrm{Tr}(ad_x\circ ad_y)$$for $x, y\in \mathfrak g$. Whilst I understand that the Cartan decomposition $$\mathfrak g ...
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Killing form and Roots

I know that the roots of a Lie Algebra are functionals such that if $\alpha$ is a root and $h \in \mathfrak h$ is an element of the Cartan subalgebra, then $\alpha(h)$ is an eigenvalue. I'm looking ...
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Root space decomposition

Regarding the direct sum of vector spaces/algebras, the dimensions of the parts of the sum should equal the whole. With the root decomp, the cartan sub algebra seems to have a dimension as high as the ...
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Campbell Baker Hausdorff formula for exp(X+Y)exp(X-Y)

Original Question: Given $X,Y\in \mathfrak g\mathfrak l_{\mathbb R}(n)$, and the CBH formula for $\exp(X)\exp(Y)$ (wiki), what is the corresponding formula for $\exp(X+Y)\exp(X-Y)$? The main ...
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Toral sub algebra

It seems to me, I could be wrong, that the toral sub algebra goes against the following rules: For a semisimple Lie algebra: If the killing form is nondegenerate the Lie algebra is semi simple-> the ...
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Root space question

Do the roots of a root space decomposition have a kernel? Since it is the duel space to the cartan subalgebra ,evaluation of the roots on a non-equal index cartan basis element should be zero. Thanks ...