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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.

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Lie bialgebras and the Jacoci identity

Let $\mathfrak{g}$ be an $n$-dimensional real Lie algebra and let $\xi : \mathfrak{g}\to\wedge^2\mathfrak{g}$ be a linear map which is a $1$-cocycle, i.e. $\xi([x,y])=\mathrm{ad}_x\xi(y)-\mathrm{ad}_y\...
user56980's user avatar
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Mapping between vectors in irreducible Sp-representation

Let $V$ be the standard Sp-representation with symplectic basis $\{ a_i, b_i \}$. I believe the vector $(b_1 \wedge b_2) \otimes (b_1 \wedge b_2 \wedge a_3 \wedge a_4)$ lies inside the irreducible ...
Chase's user avatar
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1 answer
39 views

Does the formal character determine the representation?

Suppose $V,W$ are two finite-dimensional representations of a Lie algebra $\mathfrak{g}$. Is it true that if their formal characters coincide, $$\mathrm{ch}_V=\mathrm{ch}_W ,$$ then the ...
Minkowski's user avatar
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1 vote
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"Linear independency" of Lie Brackets

I was watching this eigenchris video. At 21:49, he says: $$[g_i, g_j]=\sum_k {f_{ij}}^{k}g_k$$ for $\mathfrak{so}(3)$. Does this mean $[g_i, g_j]$ and $g_i, g_j$ can be linear independent? What about ...
Cro's user avatar
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Relation between the enveloping algebra $\mathcal{U}(\mathfrak{g})$ and the group von Neumann algebra $W^*(G)$

Let $G$ be a Lie group. Is it true that the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ of the associated Lie algebra $\mathfrak{g}$ generates the group von Neumann algebra $W^*(G) := \...
szantag's user avatar
  • 101
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Complexification of complex Lie algebras like $\mathfrak{su}(2)$

I'm reading Brian Hall's book on Lie theory. He defines the complexification $V_{\mathbb{C}}$ of a real vector space $V$ as the linear combinations $v_1+iv_2$, with $v_1,v_2\in V$. Next, he proceeds ...
Gabriela Martins's user avatar
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Velocity vector vs Matrix Differential

I am having trouble understanding the equivalence of taking the derivative of the matrix and taking the velocity vector. I came across a proof of the Lie Algebra of $O(n)$ as follows: Let $\gamma(t)$ ...
user1335336's user avatar
1 vote
1 answer
39 views

uniqueness of generators of Lie groups

Are the generators of any particular Lie group always unique? Let's take $SU(3)$ group as an example. It does have 8 generators which are explicitly written as eight $3\times3$ matrices in the ...
physics_2015's user avatar
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Lemma in proof of Cartan's Criterion (Humphreys' 4.3)

In Humphreys proof of Cartan's criterion there is a step I do not understand: $\text{ad y}=r(\text{ad s})$. I tried to check this for $\{e_{ij}\}$: $$ r(\text{ad s})(e_{ij}) \stackrel{r(0)=0}{=} r(\...
Yotam Ohad's user avatar
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Antisymmetric Structure Constants, $f_{ijk}$ of su(N) for generalised Gell-man/Pauli Matrices, $k$ is unique for a given $i,j$

I want to prove that for a fixed $(i,j)$ there exists only a single $k$ such that $f_{ijk} \neq 0$. I did this by considering the Generalized Pauli matrices: $$ \hat{T}_{\alpha_{nm}}=\frac{\hbar}{2}(|...
LieAlgebraGuy1999's user avatar
2 votes
1 answer
53 views

Real Commutant Algebra of a Set of Matrices

Suppose I have a collection of $N\times N$ real, symmetric matrices $R_1, R_2, \dots$ and I want to find their orthogonal commutant---that is, the group of real, orthogonal matrices that commute with ...
Matt Mitchell's user avatar
1 vote
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Simple Lie subalgebra of a semisimple Lie algebra

For a semisimple Lie algebra with a nondegenrate trace form, it's well-known that it can be decomposed as the direct sum of simple Lie ideals, and hence has a simple Lie ideal. But in general, how to ...
Youness EL KHARRAF's user avatar
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Invariant Solutions of PDEs-Linear fokker-planck equation with an odd drift

I am currently writing my master thesis on Lie group analysis and recently I came across this infinitesimal generator: I am trying to obtain group invariant solutions in their implicit form and so ...
George Beliyiannis's user avatar
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2 answers
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The action of $SL(n,\Bbb{R})$ on the tangent space of $SL(n,\Bbb{R})/SO(n)$.

SETUP. It is a standard result that $\text{GL}(n,\Bbb{R})/O(n)$ is isomorphic to the set $P'$ of positive definite $n\times n$ matrices, as manifolds: the basic idea is that $\text{GL}(n,\Bbb{R})$ ...
SomeCallMeTim's user avatar
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product of structure constants [closed]

I'm not that familiar with Lie Algebra, however for a basis $F_{i} \in \mathfrak{su}(N)$ such that $$[F_{i}, F_{j} ] = \sum_{k}f_{ijk}F_{k}$$ I have seen it written that $f_{ikl}f_{jkl} = \delta_{ij}$....
LieAlgebraGuy1999's user avatar
3 votes
1 answer
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Grothendieck ring of $Rep(\mathfrak{sl}_2)$

The Grothendieck ring of the abelian category $Rep(\mathfrak{sl}_2)$ of finite-dimensional representations of $\mathfrak{sl}_2$ is, according to Bakalov-Kirillov's Lecture notes on tensor categories ...
Minkowski's user avatar
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Can't "finite" be removed from the definition of a root system? [duplicate]

In every definition of finite root systems I could find (reduced or not-necessarily-reduced), the root system is always assumed to be finite in the definition. It's a very minor point, but isn't this ...
Terence C's user avatar
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Nilpotent Lie-Algebra $g$: $g^{i+1} ⊆ g^i$ ideal in $g$?

Assume $g$ to be a nilpotent Lie-Algebra. Nilpotency means that we can find an index $n$ such that: $g^n = \{0\}$ for the series defined as: $g^0 = g$ $g^{i+1} = \operatorname{span}\{[g,g^i]\}$ Why is ...
melmo99's user avatar
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Finite cover of Weyl group act on lie algebra

The following text is from "A guide to quantum groups". The Weyl group $W$ of a complex simple Lie algebra $\mathfrak{g}$ acts as a group of reflections on the Cartan subalgebra $\mathfrak{...
Peter Wu's user avatar
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Bounding norms of symplectic matrix factorisations and non-seperable Hamiltonian flows

Problem setup: Let $e^{hJM}$ be the time-$h$ flow corresponding to the ODE $\dot{x} = JMx$, with $M = \left(\begin{array}{cc} A & C\\ C^T & B\\ \end{array}\right)$ symmetric positive ...
Ben94's user avatar
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How to prove that $\mathfrak{su}(n)$ is a simple Lie algebra? Relationship with the complexification to $\mathfrak{sl}(n,\mathbb{C})$?

I know that there's counterexample which a simple lie algebra whose complexification is not simple, see here. However, I'm not sure if a simple complex lie algebra's real form is still simple or not. ...
Kenny S's user avatar
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Calculate differential $dF_j$ of $F:j\mapsto j^2+I$ on $O(n)$

Given the map $F:j\mapsto j^2+I$ on the orthogonal group $O(n)$, what is the differential $dF_j$ ? How do I calculate this? I am trying to understand Example 7 in Lecture Notes on Symmetric Spaces by ...
Andrius Kulikauskas's user avatar
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26 views

Nilpotent Lie algebra decomposed into direct sum of root spaces for a torus

In this article Solvable complete lie algebras. I written by Meng Dao Ji and Zhu Lin Sheng. They use a decomposition using root systems, but when I searched about the Root systems I found that it is ...
Mary Maths's user avatar
2 votes
0 answers
57 views

When does a Lie algebra's outer automorphism group 'inherit' a representation?

In the following I am considering finite dimensional representations of semi-simple Lie algebras over fields of characteristic $0$. Examples should illuminate what I am getting at. Consider $\mathfrak{...
Craig's user avatar
  • 821
1 vote
1 answer
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Finding inverse to inner automorphism (Humphreys' Lie Algebra book)

This is basically an algebra question. In Humphreys' book on Lie algebras he states that one can find the inverse to $\exp \delta$, where $\delta$ is a nilpotent derivation - say $\delta^k=0$, by ...
raynea's user avatar
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Why $\omega|_S\in\Omega^1(S,\mathfrak{g})$ iff $\omega(\nabla,e)\in\Omega^1(U,\mathfrak{g})$?

Let $M$ be a smooth manifold, let $\mathcal{S}$ be a $G$-strucutre on $M$ and let $\nabla$ be a connection on $TM$. Let $\omega\in\Omega^1(\text{Fr}(TM),\mathfrak{gl}_r)$ be the connection 1-form ...
Armando Patrizio's user avatar
2 votes
2 answers
127 views

Does isometry on PSD matrices preserve eigenvalues?

Let $S$ be the set of symmetric matrices and $T: S\rightarrow S$ be a linear isometry. Moreover, let $T$ be a bijection from the space of PSD matrices to the set of PSD matrices. Must $T$ preserve ...
curiousperson's user avatar
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1 answer
28 views

Lie algebra question about the highest weight of a root system

I'm reading the book James E. Humpreys's book Introduction to Lie algebras and Representation Theory . In this book the section 13.4 said in case $\Phi$ is irreducible, there is a unique highest root (...
Zoël Li's user avatar
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Bruhat decomposition of $SL_3$

I would like to know the Bruhat decomposition of $SL_3$. In Chevalley Group Theory, we have the following theorem: Let $G$ be the Chevalley group, and $G=\langle\mathfrak{X}_\alpha | \alpha \;\text{...
Tommk's user avatar
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Why do we have $b_n\cdot 1=0$ in Heisenberg vertex algebra for the annihilation operator $b_n$?

I am reading "Vertex Algebras and Algebraic Curves Second Edition" by Edward Frenkel and David Ben-Zvi. In section 2.1.2, they define the Heisenberg Lie algebra $\mathcal{H}$ as the central ...
user117521's user avatar
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How to prove that submodules of $L = b(n,F)$ indecomposable

Let $F$ be a field,$L = b(n, F)$ and $V = F^n$. Let $e_1, ..., e_n$ be the standard basis of $F^n$. Define $W_r = Span\{e_1,...,e_r\}$, with $1 \leq r \leq n$. It can be proved that $W_r$ a submodule ...
Chen's user avatar
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1 vote
1 answer
43 views

Prove that $XY-YX\in \mathfrak{g}$ if $X,Y\in \mathfrak{g}$, where $\mathfrak{g}$ is the Lie algebra of a matrix Lie group $G$.

Definition $3.18$. Let $G$ be a matrix Lie group (a closed subgroup of $GL_n(\mathbb{C})$). The Lie algebra of $G$, denoted $\mathfrak{g}$, is the set of all matrices $X$ such that $e^{tX}$ is in $G$ ...
Bowei Tang's user avatar
  • 1,545
4 votes
0 answers
56 views

Does $SU(2,2)$ act on $2\times 2$ skew-Hermitian matrices?

We can put a Hermitian form of type $(2,2)$ on a four dimensional complex vector space and define the group $SU(2,2)$ be the group of matrices that preserve the Hermitian form and have determinant ...
Zhaoting Wei's user avatar
  • 1,094
2 votes
0 answers
47 views

Proof of Invariance lemma

I am reading Introduction to Lie Algebra by Kirdman. I had a doubt. Assume that $F$ has characteristic zero. Let $L$ be a Lie subalgebra of $\text{gl}(V )$ and let $A$ be an ideal of $L$. Let $\...
Raheel's user avatar
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0 votes
1 answer
37 views

Relation between linear representation and "induced adjoint representation" of Lie algebra?

Consider a representation $\rho \colon \mathfrak{g} \mapsto \mathrm{End}(V)$ of a Lie algebra $\mathfrak{g}$ on a vector space $V$. What can we say about the induced representation on the space of ...
Another User's user avatar
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65 views

Vector field of rotation around an axis in spherical coordinates

This is a qualifying exam question in Differential Geometry. I'm new to the subject and am reading up from John Lee's Introduction to Smooth Manifolds. Let $M=S^2\subset\mathbb{R}^3$ be the unit ...
giraffe's user avatar
1 vote
1 answer
63 views

Binomial theorem for ideals

I was proving the statement that if $I$ and $J$ solvable ideals of Lie algebra $L$, then $I + J$ is a solvable ideal of $L$. The proof is we know $$(I+J)/J\cong I/I\cap J.$$ Since $I,J$ are solvable ...
Raheel's user avatar
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1 vote
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Is my proof valid for Exercise 7.3 of book "Introduction to Lie Algebras" by Karin Erdmann and Mark J. Wildon

Exercise 7.3 of book "Introduction to Lie Algebras" by Karin and Mark: Show that $V$ is irreducible if and only if for any non-zero $v \in V$ the submodule generated by $v$ contains all ...
Chen's user avatar
  • 45
1 vote
0 answers
32 views

Possible issues with the Identification between $Lie(G)$ and $T_e G$ for lie group $G$

I was reading Lee's Introduction to Smooth Manifolds and had a question regarding Theorem 8.37. Theorem 8.37: Let $G$ be a (finite-dimension) Lie Group. The evaluation map $\varepsilon : Lie(G) \...
Keshav Balwant Deoskar's user avatar
2 votes
1 answer
47 views

Question on the Proof of Proposition 6.6 in the Book Introduction to Lie Algebras by Karin Erdmann and Mark J. Wildon

I have yet another question around Engel's theorem, this time on the Proof of Proposition 6.6 near the end of page 50 of Book "Introduction to Lie Algebras" by Karin and Mark. The ...
Chen's user avatar
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2 votes
0 answers
58 views

Real forms of a solvable Lie algebra

Consider the Lie algebra $\mathfrak{r}_2=\mathfrak{aff}(\mathbb{C})$ of the group of affine maps of $\mathbb{C}$, and let $\mathfrak{g}=\mathfrak{r}_2 \oplus \mathfrak{r}_2$. I am interested in ...
JRojo's user avatar
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0 answers
42 views

Complexification of the nilradical of a real Lie algebra

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra and let $\mathfrak{g}^{\mathbb{C}}$ denote its complexification. Is it true that the nilradical of $\mathfrak{g}^{\mathbb{C}}$ is the ...
Agustina C.'s user avatar
4 votes
1 answer
71 views

Geometric interpretation and upper bounds on Euclidean distance in Lie algebra of SO(3)

$\newcommand{\Log}{\operatorname{Log}} \newcommand{\Exp}{\operatorname{Exp}}$ Let $\Exp : \mathbb{R}^3 \mapsto SO(3)$ denote the composition of the standard $SO(3)$ exponential map with the natural ...
japreiss's user avatar
  • 433
1 vote
1 answer
54 views

Question on the proof of Engel's Theorem step 1 in book Introduction to Lie Algebras

I am reading chapter 6.1 Engle's Theorem of book "Introduction to Lie Algebras" by Karin Erdmann and Mark J. Wildon, and have a question on Step 1 of the Proof for Proposition 6.2. It says ...
Chen's user avatar
  • 45
0 votes
0 answers
35 views

Locally Compact Lie Groups and Matrix Lie Groups

Let $G$ be any Lie group with a Lie algebra $\mathfrak{g}(n, \mathbb{R})$; but assume that I only deal with a locally compact, connected (topologically) component around its $e \in G$ element, say $K \...
iliTheFallen's user avatar
4 votes
1 answer
76 views

Irreducible complex representations of some abelian Lie groups

I wanted to classify all irreducible complex representations of the following basic abelian Lie groups: $\mathbb{S}^1$ the circle in the complex plane, $\mathbb{R}_{>0}$ the positive real numbers, $...
Don Abbondio's user avatar
2 votes
1 answer
52 views

The Lie algebra of an algebraic group

I have a question concerning different visions of the Lie algebra of an algebraic group. To be more specific, let $G = \operatorname{Spec}(H)$ be a smooth (finitely generated) algebraic group over a ...
Tomas Fernandez's user avatar
2 votes
2 answers
47 views

What is the semidirect product we use in Levi Decomposition

So using the Levi Decomposition for any lie algebra $\mathfrak{g}$, there exists a semisimple subalgebra $\mathfrak{s}$ such that: $Rad(\mathfrak{g})$$\ltimes$$\mathfrak{s}$=$\mathfrak{g}$ However in ...
Albi's user avatar
  • 69
2 votes
1 answer
73 views

Representation/factorising of symplectic groups elements

According to Hall Chap. 3 Corollary 3.47: for a connected matrix Lie group $G$, every element $A\in G$ can be written in the form $A=e^{X_1}e^{X_2}...e^{X_k}$ for some $X_i\in g$, where $g$ is the Lie ...
Ben94's user avatar
  • 108
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0 answers
36 views

exponential map of lie groups

In "A micro Lie theory for state estimation in robotics" by Joan Sola`, there is a formular on page 6: $$\exp((t+s)τ^∧) = exp(tτ^∧)exp(sτ^∧) \quad (17)$$ And another one on page 10: $$Exp(...
Charles Ju's user avatar

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