Questions tagged [lie-algebras]
For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.
6,595
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On the Killing splitting of some algebras
It is well-known that the algebra $ so(4,1) $ admits the Killing splitting as vector spaces as follows: $ so(4,1)=so(3,1) \oplus \mathbb{R}^{3,1} $. In the same context, is there a Killing splitting ...
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Applying theorems from Fulton and Harris to Lie groups/algebras with non-$\mathbb{C}$ coefficients
I recently asked this question about $SL_4(\mathbb{Q})$ representations. Commenters warned me that Fulton and Harris is about representations of complex Lie groups/algebras so I should be careful ...
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Book on $SL(2,C)$
Is there a book, which treats $SL(2,C)$ in detail as a group, Lie group, its Lie algebra, geometry of its subgroups etc.? It is often seen as an example in Lie Algebra/Group books but it always ...
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Moving between $Sp_{2n}(\mathbb{C})$ reps and $SL_{n}(\mathbb{C})$ reps
Say I have some irreducible $Sp_{2n}(\mathbb{C})$ representation, such as $\Gamma_{0,1,0,1}$.
Consider the subgroup of $Sp_{2n}(\mathbb{C})$ isomorphic to $SL_n(\mathbb{C})$, consisting of matrices of ...
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Jacobi identity as a cocycle condition
The Jacobi identity in a Lie algebra $\mathfrak{g}$ looks like this (for $x,y,z\in \mathfrak{g}$):
$$[[x,y],z]-[x,[y,z]]+[y,[x,z]]=0.$$ This just says that the map $y\mapsto [x,y]$ is a derivation ...
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Does a special orthogonal group $SO(4) \supset SU(2)$ contain a special unitary group? [closed]
Note that in the context of Lie group,
The spin group $Spin(4) = SU(2) \times SU(2)$ is a product of two special unitary groups.
Can you answer:
Does $SO(4)=Spin(4)/{\mathbf{Z}/2}=\frac{SU(2) \times ...
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What is the unitary subset of Lie algebra $su(2)$?
Lie algebra $su(2)$ consists of the $2\times 2$ skew-hermitian complex matrices with addition and multiplication by real numbers as vector space operations and commutator as Lie bracket. The i-...
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Short exact sequence of Lie Algebra representations
We have the following SES of a complex Lie Algebra representation,
$$0\to \mathbb{C} \to W \to \mathbb{C} \to 0$$ Now the easy claim would be: $(\rho,W)$ is a two dimension representation and $\rho(x)$...
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Producing a nontrivial proper ideal in the (Kac-Moody) Lie algebra $\mathfrak{g}(A)$
Proposition 1.7 in Kac's book Infinite dimensional Lie algebras states that if $A=(a_{ij})$ is any complex matrix (that is, not necessarily a generalized Cartan matrix) then the Lie algebra $\mathfrak{...
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Elementary consequences of the root system axioms
On Root_system: Elementary_consequences_of_the_root_system_axioms (wikipedia), from the relation $\langle \alpha, \beta \rangle = (2\cos(\theta))^2 \in \mathbb Z$, the value $\cos(\theta)$ can only be ...
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Representations of a simply-connected non-compact Lie group induced from Lie algebra representations
It is a known fact that for simply connected Lie groups, each representation of the Lie algebra comes from a representation of the Lie group [See this]. Consequently, we study representations of the ...
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Is it true that for a semisimple Lie algebra $\mathfrak{g}$, $[\mathfrak{g},\mathfrak{g}^e]\subset[e,\mathfrak{g}]$ for $e$ a regular nilpotent?
Let $\mathfrak{g}$ be semisimple Lie algebra, let $e \in \mathfrak{g}$ be a regular nilpotent element ie. an element whose adjoint endomorphism is nilpotent and of maximal rank. Denote by $\mathfrak{g}...
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Does the structural theory of irreducible reprensentation of $\mathfrak{sl}(2,\mathbb C)$ also apply to $\mathfrak{sl}(2,\mathbb R)$?
By Theorem 1.66 (page 62) of Knapp's Lie groups: beyond an introduction, 2nd ed:
For each positive integer $n$, if $\pi$ is an $(n+1)$-dimensional irreducible representation of $\mathfrak{sl}(2,\...
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dual representation map as a linear map [closed]
For $\rho\colon G\to GL(V)$, it dual representation is given by $\rho^*(g) = \rho(g^{-1})^T$.
How to understand this transposition in the definition $\rho(g^{-1})^T$ when we consider as a linear map $...
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How can I show this property about lie algebras and lie subgroups?
I have given the following problem:
Let $G$ be a closed matrix group with lie algebra $\mathfrak{g}$. Let $\mathfrak{h}$ be a commutative lie subalgebra of $\mathfrak{g}$. Let $H$ be the corresponding ...
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Show explicitly the rank-24 Leech lattice that is also symmetric unimodular matrix with integer entries?
A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix:
$$\begin{pmatrix}
2 &−1 &0 &0& 0& 0& 0& 0\\
−1& 2 &−1& 0& 0& 0& 0& 0\\
0& −1&...
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Is there a way to convert an element of $\mathfrak{so}(3)/\mathfrak{so}(2)$ into a "geometric" tangent vector on $S^2$?
From my understanding, the tangent space at the identity of the homogeneous space $\rm SO(3) / \rm SO(2)$ is just the quotient space $\mathfrak{so}(3) / \mathfrak{so}(2)$. An element in this quotient ...
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Rank-24 Leech matrix cannot have simultaneous integer entries with unit determinant or integer determinant?
A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix:
...
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Uniqueness of Lie algebra structure of quotient under canonical projection mapping.
Let $\mathfrak{g}$ be a Lie algebra, and $\mathfrak{h} \subset \mathfrak{g}$ a Linear subspace which is an ideal. That is, for every $X \in \mathfrak{h}$ and every $Y \in \mathfrak{g}$, one has $[X,Y] ...
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Extensions of simple Lie groups with a topological splitting
This is a follow-up to my question here.*
I'm interested in knowing about (finite-dimensional) Lie group extensions, where $G$ is simple and $H$ is connected (and hopefully abelian): $$1\to H\to G'\to ...
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algoritm and program for modelling a Free Nilpotent Lie algeabra [closed]
The question is solved below by Eero Hakavuori
thanks to him!
I need to compute in a Free Nilpotent Lie Algebra L given by a finite list of generators.
For example, put the generators {A, B}. So, ...
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The order in which diffeomorphisms should be applied
I’m reading ‘Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras’ by Kac and Raina. And there is a following paragraph in the first chapter.
The Lie algebra $\...
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Two alternative descriptions of $\mathrm{so}_{2n}(\mathbb{C})$
It seems that there are two ways to define the Lie algebra $\mathrm{so}_{2n}(\mathbb{C})$. The first one is
$\mathrm{so}_{2n}(\mathbb{C})_{(1)}:=\{M \in \mathrm{gl}_{2n}(\mathbb{C}) \ | \ M + M^t = 0\}...
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Commutation relationship of the generator of SO(N)
The group $S O(N)$ is defined as the set of $N \times N$ matrices $R_{i j}$ that leave the $N$-dimensional Euclidean metric invariant, i.e.,
$$
\delta_{i j}=R_{i k} R_{j l} \delta_{k l},$$
which also ...
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Non-trivial $\frac{1}{2}$-derivations Lie algebras
I came across the following theorem:
Let $\mathcal{L}$ be a Lie algebra without non-trivial
$\frac{1}{2}$-derivations. Then every transposed Poisson
structure defined on $\mathcal{L}$ is trivial.
If ...
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lie algebra cohomology
Suppose that I have a semi-simple lie algebra $\mathfrak{g}$ and I have a linear map from a vector space $V$ to $\mathfrak{g}$. This map is surjective. Can I show that the corresponding map from $V$ ...
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Expression for the symmetric BCH-formula to linear order in $Y$
Let's say I have an operator $Y$ which can be expanded in powers of small $\varepsilon$ such that $Y$ to lowest order is already of order $\varepsilon$. I now want to find the expression for $\log(e^X ...
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Lorentz algebra
Suppose we have a lie algebra with a $\mathbb{Z}_3$-decomposition: $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\mathfrak{g}_2$. Remember the relation $[\mathfrak{g}_i,\mathfrak{g}_j]\subset \...
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Lie algebras are to Lie groups what Lie modules are to …?
I am trying to understand the relationship between Lie algebras, Lie groups, and Lie modules. I know that a Lie algebra is a vector space with a bilinear, antisymmetric, and Jacobi-satisfying bracket ...
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Adjoint representation as matrices of Lie Algebra - difference between physics and math literature [closed]
In math literature (e.g. Sattinger and Weaver or wikipedia) the structure constants are defined as
$[e^a,e^b]=\Sigma_c C^{abc}e^c$ and the adjoint representation as $[Ad(e^a)]^{de}=C^{aed}$.
In ...
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Simple modules of $\mathcal{g}\otimes\mathbb{C}[x]$ for a lie algebra $\mathcal{g}$
Let $\mathcal{g}$ be a simple, finite-dimensional complex Lie algebra, and let $\mathcal{M}$ be a representative system of finite-dimensional simple $\mathcal{g}$-modules (up to isomorphism). ...
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Isomorphic connected Lie subgroup
This is a question (5. 13) from Brian C.Hall's textbook (GTM 222).
Let $G$ be a matrix Lie group with Lie algebra $\frak{g}$, let $\frak{h}$ be a subalgebra of $\frak{g}$, and let $H$ be the unique ...
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Vectors belonging to each component of decomposition of semi-simple Lie group representation
Say that $W$ is a $SL_4(\mathbb{Q})$ representation that decomposes into irreducibles (using the notation of Fulton-Harris) $$W = \Gamma_{a,b,c} \oplus \Gamma_{d,e,f}$$
Now say that I have a vector $v ...
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I am stuck with this one problem related to nilpotent. Please guide me to write the proof.
I know how to go about when it is given that $L/Z(L)$ is nilpotent and we need to prove that $L$ is nilpotent. But how should I proceed if it is given that $L$ is nilpotent and we need to prove that $...
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Equivalence definition to semidirect product in the operated context
An operated semigroup (or a semigroup with an operator) is a semigroup $U$ together with an operator $\alpha : U \to U$ that is called the distinguished operator on $U$. Is there any definition ...
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When is the simply connected Lie group of a Lie algebra flat?
I would like to know if given an $n$-dimensional Lie algebra $\mathfrak{g}$, there is a way to know whether the simply-connected Lie group integrating it is topologically $\mathbb{R}^n$.
At the moment,...
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Semisimple elements in compact Lie groups
Suppose that $\mathfrak{g}$ is a Lie algebra. We call $\mathfrak{g}$ a compact Lie algebra if there exists an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak{g}$ such that $$\langle[Z,X],Y\...
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Isomorphic representation for lie algebra $[x,y]=x$
I am currently working on task 28 from the following sheet:
https://agag-lassueur.math.rptu.de/~lassueur/en/teaching/LIESS16/LASS16/Blatt8.pdf
I have already completed parts 1 and 3. For part 2, I ...
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Number of type $A_{n-1}$ sub-systems of root system $A_n$
If I am not mistaken, it should be true that a type $A_n$ root system contains $n+1$ different sub-systems of type $A_{n-1}$. In my geometric application these do appear quite naturally, but is there ...
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$G$-equivariant map between Lie group representations
Say $f: A \rightarrow B$ is a $G$-equivariant map between finite dimensional $G$-representations, for some semi-simple Lie group $G$. If I know:
$B$ decomposes into irreducibles as $B = B_1 \oplus ......
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Complexification of $G-$representation where $\mathrm{End}_G(V)\cong\mathbb{C}.$
Let $G$ be a compact Lie group and $V$ a finite dimensional real $G$ representation. Suppose that $\mathrm{End}_G(V)\cong \mathbb{C}.$ I want to show that the complexification $V\otimes_{\mathbb{R}}\...
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Find an element $x\in\mathfrak{sl}(2,\mathbb{C})$ sucht that the exponential is not a local diffeomorphism at $x$.
Recently I have been studying Lie algebra, and I have been solving some problems to get familiarized with the concepts.
A problem that I recently solved was to prove that in a global sense, the ...
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Is the exponential map a global diffeomorphism when g is nilpotent?
I have a question about the exponential aplication: if $\mathfrak{g}$ is a complex semisimple and nilpotent Lie algebra, is true that the exponential map is a global diffeomorphism?
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Definition of Universal enveloping algebra of a lie algebra
I am a physics student, and I am trying to understand the Casimir operator from a formal perspective; therefore, I come to learn what's Universal enveloping algebra.
Two definitions of Universal ...
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Explicit expression for simple roots of the root systems $A_2$, $B_2$ and $G_2$ in 2D
I often find explicit expression for rank-2 root systems as $A_2$, $B_2$ and $G_2$ in a 3D Euclidean space. Does anybody have an explicit expression for the simple roots in terms of $e_1,e_2$ in the ...
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Multiplication map for centrally extended Lie group
First of all apologies for my "hands on" language with coordinates and indices, I am aware that this is not how Lie algebras are normally discussed, but it is the language I'm familiar with ...
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Process for decomposing tensor products of Lie algebra representations into irreducibles?
I'm trying to decompose tensor products of semisimple Lie group/algebra representations into direct sums of irreducible representations. I know this question has been asked many times before, with ...
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The notation for a non-abelian two dimensional Lie algebra
Most of the introductions into Lie algebras start with the notion of a two-dimensional non-abelian Lie algebra $\mathfrak{g} = \langle x,y\rangle$ such that $[x,y]=x$. Is there a common notation for ...
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Other direction of Weyl's theorem on semisimple Lie-algebras
Weyl's theorem states
Let $\mathfrak{g}$ be a finite dimensional semisimple Lie-algebra over an algebraically closed field with characteristic 0. Then all finite dimensional representations are ...
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Question about solvable finite dimensional Lie algebras
Let $\mathfrak{g}$ be a finite dimensional solvable Lie algebra. Given that we can take subspaces $\mathfrak{a}_{i}$ of $\mathfrak{g}$ such that $\mathrm{dim}(\mathfrak{a}_{i}/\mathfrak{a}_{i+1})=1$ ...
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