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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Proving a Jacobi-like identity with four elements in a Lie algebra

Let ${\frak g}$ be a Lie algebra. Then $$ [[[X,Y],Z],W] +[[[Y,X],W],Z]+[[[Z,W],X],Y]+[[[W,Z],Y],X]=0 $$for all $X,Y,Z,W \in {\frak g}$. I started using the Jacobi identity four times to get: \...
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13 views

Lie Algebra: Dimension of a one-parameter group (dimension of orbit)?

I am reading a book about Applications of Lie Groups to Differential Equations by Peter Olver. Lets say we have a PDE with $p$ independent variables and $q$ dependent variables In Chapter 3.1 (page ...
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1answer
52 views

Finding lie algebra of a group by using exp map and tangent space [duplicate]

I'm studying Lie groups and I am in trouble with finding lie algebras of the classical groups. How can I calculate $\mathfrak{sp}(n,\mathbb{C})$ or $\mathfrak{so}(n,\mathbb{R})$ using exp map and ...
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25 views

What does the set of dominant integral elements in a Cartan sub algebra look like?

I'm reading about the theorem of the highest weight: Irreducible finite dimensional representations of a complex semisimple Lie algebra (with a fixed Cartan sub algebra, $\frak{h}$ and choice of ...
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18 views

The smallest $m \in \mathbb{N}$ such that $b(n, \mathbb{C})$ is soluble.

Say I have the Lie algebra $L = b(n, \mathbb{C})$, the set of all $n \times n$ matrices with entries in $\mathbb{C}$ that are upper triangular with the standard Lie bracket (the commutator $[A, B] = ...
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57 views

A soft question on Gauge Equivalence in Integrable Systems

I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations (...
2
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1answer
33 views

Endomorphism vector bundle isomorphic to the adjoint bundle of its frame bundle?

Could somebody help me to prove the following isomorphism (in particular what is the isomorphism)? \begin{equation} End(\xi) \cong ad(E_{\xi}) = E_{\xi} \times_{GL(n,\mathbb{R})} \text{Mat}_n(\...
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0answers
29 views

Is the BGG category $\mathcal{O}$ a Serre subcategory of $\mathfrak{g}$-mod? [duplicate]

Let $\mathcal{O}$ be the BGG category for a be a finite-dimensional, semi-simple complex Lie algebra $\mathfrak{g}$. Let $\mathfrak{g}$-mod be the category of all $\mathfrak{g}$-modules. Is the BGG ...
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0answers
21 views

Expression of the Laplacian of the reduced Heisenberg group?

Let $\mathbb C^n$ be the n-dimensional complex field endowed with a positive definite hermitian form $H(z,w)$. The corresponding symplectic form is $E(z,w)= \Im (H(z,w))$, where $\Im $ denotes the ...
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34 views

Matrix exponential between Lie algebra and Lie group (help with a proof)

Theorem 3.42 in Hall's Lie Groups, Lie Algebras and Representations is a key step towards proving that the matrix exponential maps a neighbourhood of zero in the Lie algebra to a neighbourhood of the ...
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24 views

How $U_{q}(\mathfrak{sl}_{2})$ becomes the universal enveloping algebra $U(\mathfrak{sl}_{2})$ of $\mathfrak{sl}_{2}$

My question is how $U_{q}(\mathfrak{sl}_{2})$ becomes the universal enveloping algebra $U(\mathfrak{sl}_{2})$ of $\mathfrak{sl}_{2}$ if we set $t=q^h$ and $q$ tends to 1.
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1answer
27 views

Lie Algebra: Optimal system of one-dimensional sub-algebras of the heat equation

This is a follow up question to Invariants of a PDE by Lie Symmetries, as I tried to follow the reasoning from the book Applications of Lie Groups to Differential Equations (Peter J. Olver, Example 3....
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1answer
19 views

Maple: How to use partial differential operators?

I am trying to calculate the commutator $[v,w]=vw-wv$ for given infinitesimals $$v=\dfrac{\partial}{\partial x}$$ and $$w=x\dfrac{\partial}{\partial t}$$ I know how to calculate the commutator by ...
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16 views

Does the trivial character always show up as a weight?

Let $G$ be a linear algebraic group, $T$ a subtorus of $G$ of dimension $\geq 1$. Let $\mathfrak g$ be the Lie algebra of $G$. Then the Ad operator $$\textrm{Ad } : G \rightarrow \textrm{GL}(\...
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1answer
44 views

Connecting the regular representation of $\mathfrak{so}(3)$ and the exterior algebra of $\mathbb{R}^3$

It is well known that the regular representation of $\mathfrak{so}(3)$ is the so-called "cross product" matrix $A(x)$ which follows $A(x)y = x\times y$, and $x,y\in\mathbb{R}^3$, while the cross ...
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18 views

augmentation ideal for universal enveloping algebras

Let $L$ be a restricted Lie algebra with the restricted enveloping algebra $u(L)$ over a field $F$. Let $ω(L)$ denote the augmentation ideal of $u(L)$ which is the kernel of the augmentation map $\...
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1answer
27 views

Existence of Certain Lie Groups

Let $\mathfrak{h}$ be a Lie algebra (not necessarily finite dimensional). Does there necessarily exist a Lie group $G$ such that for the Lie algebra corresponding to $G$, denoted $\mathfrak{g}$, we ...
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1answer
47 views

Ordering on the weight lattice

When given a finite dimensional complex Lie algebra $\mathfrak{g}$ that is also semisimple and a choice of Cartan subalgebra $\mathfrak{h}$ we may talk about its weight lattice $\Lambda_{W} $ in $\...
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1answer
38 views

Meaning of generators in Lie Algebra of PDE

Consider some PDE involving a scalar function $u(x,t)$ with two independent variables $x$ and $t$. Assume that this PDE has a Lie Algebra spanned by the following generators, $X_1=\partial_x,\quad ...
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1answer
15 views

Lowering a non-zero weight vector gives a non-zero vector (representation of $\mathfrak{sl}(2)$)

In Lie algebras we study $\mathfrak{sl}(2)$ (the complex span of the usual matrices $X,Y,H$ where $X$ and $Y$ are the raising and lowering operators respectively). The defining commutator relations ...
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2answers
24 views

Soft question about Lie Groups and 3D rotation

Let $R(\phi, \boldsymbol{n})$ be a member of Lie Group SO(3). According to Wikipedia If $R(\phi, \boldsymbol{n})$ denotes a counter-clockwise 3D rotation through an angle $\phi$ about the axis ...
2
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1answer
38 views

Showing $[L,rad(L)]$ is nilpotent.

Suppose L is a finite dimensional Lie Algebra over an algebraically closed field of characteristic zero. I want to show that $[L,rad(L)]$ is nilpotent. I was given a hint that all operators of the ...
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1answer
17 views

Proof of Jordan Decomposition of derivation in Lie Algebras, 'Acts Diagonalisably'

My understanding is $[x,y] \in L_{\lambda + \mu} \Rightarrow (\delta - (\lambda +\mu )I_L)^m [x,y]=0$ Somehow, the $m$ has changed into a $1$ and I believe that is because of the 'acts ...
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1answer
62 views

Show that $\pi(Z)$ acts as a scalar over $\mathbb{g}$

Let $(\pi, V)$ be a finite dimensional irreducible representation of $\mathbb{g}$ $V$ is a vector space of homogeneous polynomials in 3 variables of degree d over $\mathbb{R}$ $\mathbb{g}=\begin{...
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14 views

Generate Lie-Algebra $su(N)$

Let $A \in \mathbb{C}^{n \times n}$ be a diagonal matrix with $Tr(A)=0$ and all eigenvalues (purely imaginary eigenvalues) are mutually different from each other. Hence, in particular $A \in \mathfrak{...
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1answer
32 views

Sextonion Cayley Table

I've been reading up on the sextonions and was wondering if it would be possible to construct a Cayley table for the split sextonions the same way as one would do so for the split quaternions and ...
5
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1answer
151 views

Left Invariant vector field on SO(3)

I have a Lie group, namely on SO(3), i.e. $SO(3,\mathbb{\mathbb{R}})=\left\{ A\in GL\left(3,\mathbb{R}\right)\mid A^{T}A=\mathbb{1},\,\det\left(A\right)=1\right\}$. I have a Left action $L_g$ and I ...
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1answer
58 views

Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
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2answers
37 views

Angular momentum operators

Suppose we have angular momentum operators $L_1,L_2,L_3$ which satisfy $[L_1,L_2]=iL_3$, $[L_2,L_3]=iL_1$ and $[L_3,L_1]=iL_2$. We can show that the operator $L^2:=L_1^2+L_2^2+L_3^2$ commutes with $...
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2answers
31 views

Bruhat decomposition and the order of the group.

Suppose we have a group $G(q)$ over a finite field $\mathbb{F}_q$. How can the Bruhat decomposition be used in order to calculate the order of $G(q)$? Are there any examples for some particular groups?...
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15 views

$\mathfrak{sl}_2$ has the root lattice of type $A_1$.

Let $L$ be a Lie algebra over $\mathbb{Z}$ constructed from a root lattice $R$. It is well-known that if $R=A_1$, then $L \cong \mathfrak{sl}_2$ and this is widely used example in many books on Lie ...
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33 views

Finite dimensional algebraic representation of $SL_2(\mathbb{C})$

I heard that for each $n\in \mathbb{N}$, there is the unique algebraic irreducible representation of $SL_2(\mathbb{C})$ with dimension $n$ over $\mathbb{C}$. Would you let me know what is such ...
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1answer
32 views

$ [[[A,B],C],D] + [[[B,C],D],A] + [[[C,D],A],B] + [[[D,A],B],C] = 0 $

If A and B are $n \times n$ matrices, define the Lie product $[A,B] = AB-BA$. Exercise 1.37 of the book Basic Linear Algebra by T.S. Blyth and E.F. Robertson asks to prove that $$ (*) \ \ \ \ \ \ \ ...
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1answer
69 views

Prove that the sum of all simple roots is a root

Let $\Delta$ be an indecomposable root system in a real inner product space $E$, and suppose that $\Phi$ is a simple system of roots in $\Delta$, with respect to an ordering of $E$. If $\Phi = \{\...
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1answer
30 views

Functoriality of the adjoint representation

Just a simply question. I came across the following statement which is used for deriving Weyl's integral formula: ''$\text{Ad}_G(h)|_{\mathfrak{h}} = \text{Ad}_H(h)$ due to functoriality in the Lie ...
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1answer
26 views

Every irreducible representation of $G_2$ appears in some tensor power of the standard representation

In the Book "Representation Theory" by Fulton and Harris, this fact ist stated on page 353 after looking at the weight diagrams of the complex Lie-Algebra $G_2$. The authors deduce that with $V=\...
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50 views

Irreducible Representation of $sl(3,C)$

We know that the the roots of $\mathbb{g} = sl(3,C)$ under the adjoint action are given by $L_i - L_j$ where $L_i (diag(a_1, a_2, a_3))=a_i$ for $i = 1,2,3$. If $V$ is any irreducible representation ...
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1answer
18 views

Isotropy algebra for $U(n)$? [duplicate]

Let $G = U(n)$ be the Liegroup of $n \times n$ unitary matrices and $\mathfrak{g}$ the corresponding Lie algebra. Now $G$ can act on $\mathfrak{g}$ by the Adjoint-action. Since $G$ is a subgroup of $\...
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1answer
16 views

Show that we have a smooth path in $T_1(G)$, the tangent space of a matrix group

Consider the path $D_s(T)=A(s)B(t)A(s)^{-1}B(t)^{-1}$ in $G$ for some fixed value of s. Then the Lie bracket $[X,Y]$ can be related to the commutator of $A(s)B(t)A(s)^{-1}B(t)^{-1}$ of smooth paths $A(...
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1answer
53 views

Infinite Lie algebra

It is well known fact that the finite dimensional Lie algebra will always be closed under commutation relation. But I have doubt about infinite Lie algebra. In some of the case it is closed under ...
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0answers
62 views

The cohomology of $\mathrm{GL}_n$ over an algebraically closed field

How does one go about computing the cohomology groups $H^*(\mathrm{GL}_{\kern{0.1em}{m}}(\overline{\mathbb{F}}_p),M)$? I am particularly interested in the case when $M$ is an algebraic representation. ...
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33 views

Non-Lie character of Leibniz algebra

Let $J$ be the largest ideal of Leibniz algebra $L$ which denotes the non-Lie character of $L$. Is it possible to write $L=L_{Lie}\cap J$? We know that $L_{Lie}= L/J$. I am going to give the following ...
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0answers
34 views

The product of dg Lie algebras

I am trying to understand what are products and coproducts in the category of dg Lie algebras. I am okay with coproducts. For products, however, this Wikipedia article says that given $\mathfrak{g},\...
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2answers
38 views

A semisimple Lie group has no character; Am I right?

Let $G$ be a compact connected Lie group with semisimple Lie algebra ${\frak g}$. With the following reasoning, I show that there is no non-trivial Lie group homomorphism $$\chi:G\to S^1.$$ Is that ...
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0answers
20 views

Exponentiating an ``affine subalgebra''

Consider the Lie algebra $u(N)$. If I exponentiate an $x \in u(N)$, I will obtain an element of the $U(N)$ group. My understanding is that $exp(u(N))= \{exp(x)| x \in u(N)\}$ is, in fact, the group $U(...
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1answer
29 views

Algebraic groups and restricted Lie algebras

If $G$ is an algebraic group with coordinate algebra $A=\mathcal O(G)$, say over a field $k$ of characteristic $p$, then its Lie algebra $\mathfrak g$ can be endowed with the structure of a restricted ...
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29 views

Topologies of partially exponentiated lie algebras, especially in regard to $SU(2)$

Consider the fundamental respresentation of $\mathfrak{su}(2)$ given in terms of the Pauli matrices as $\mathfrak{su}(2) = \langle \frac{i\sigma_1}{2},\frac{i\sigma_2}{2},\frac{i\sigma_2}{2}\rangle_{\...
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36 views

Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
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1answer
21 views

What's its use of the nonsingular 2-step nilpotent Lie algebras

What's its use of the nonsingular 2-step nilpotent Lie algebras which form an a class of 2-step nilpotent Lie algebras ? Recall: A $2$-step nilpotent Lie algebra $N$ is non-singular if $ad X : N \to ...
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21 views

how to show that the representation of $SL(2, \mathbb{C})$ is holomorphic

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to GL(...