Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie ...

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2
votes
0answers
105 views

Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves: $\frac{d U_s}{ds} = (a + w(s)b)U_s$ for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...
-2
votes
0answers
21 views

Compactness and noncompactness of Lie groups [on hold]

I have read that if one complexifies a Lie group, (i.e. a group which has elements written as $G=e^{i\lambda_a T^a}$, where $T^a$ are the Lie algebra elements which generate the group, and $\lambda_a$ ...
30
votes
14answers
4k views

What does the generating function $x/(1 - e^{-x})$ count?

Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series $$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty ...
1
vote
1answer
110 views

Is the endpoint map smooth

Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE: $\frac{d U_t}{dt} = (a + w(t)b)U_t$ consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...
2
votes
0answers
84 views

Lie algebra of holomorphic vector fields

It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly. For example, take $X=\mathbb{P}^n$, ...
1
vote
1answer
113 views

Curves on $SU(4)$ whose adjoint action on $\mathfrak{su}(4)$ integrates to $0$

Given $\xi \in \mathfrak{su}(4)$ and positive $T \in \mathbb{R}$, is it possible to find all smooth curves $U_s \in SU(4)$ with $U_0 = I$ such that $$\int_0^T U_s \xi U_s^{\dagger} ds =0\; ?$$
3
votes
3answers
353 views

Form of elements of a Lie algebra

How can we write the elements of a free Lie algebra? Is there any reference concerning this subject?
0
votes
1answer
208 views

generalization of highest weight theorem for semisimple lie algebras

Let $\mathfrak g$ be a real semisimple Lie algebra (without compact factors) with Iwasawa decomposition $\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak u$. Let $\mathfrak p$ be a ...
0
votes
2answers
104 views

Does an analytic tensorial Lie structure on $S^2$ gives a fiberwise Abelian Lie algebra structure?

Motivated by the answer to this question we ask: Is it true to say that for every real analytic tensorial Lie algebra structure $\alpha$ on $\chi^{\infty}(S^2)$, all fibers are necessarily ...
1
vote
1answer
114 views

Manifold_Lie algebra compatibility

In this question we try to improve some parts of this post as follows: What is an example of a manifold $M$ and a Lie algebra $L$ (with the same dimension) such that $M$ does not admit ...
6
votes
1answer
295 views

Some questions about the Malcev completion

Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n ...
3
votes
1answer
231 views

A question about $O(3,1)$

Recall that $O(3,1)$ is the collection of matrices $A\in M_4(\mathbb R)$ such that $$A\begin{pmatrix}1 ...
3
votes
1answer
119 views

Faithful linear representation of a nilpotent Lie algebra

Let \begin{align} \mathfrak{g} = Span_{\mathbb{C}}\{ e_1, e_2, e_3, e_4, e_5: \text{ non-zero brackets are } [e_1, e_i]=e_{i+1}, i=2,3,4, [e_2, e_3]=e_5 \} \end{align} be a $5$-dimensional Lie ...
1
vote
1answer
117 views

A proof of the Ibragimov et al. commutation relation

Let $u_a(x),\,a=1,2,\ldots n$ be a $n$-component field in Minkowski spacetime $x^\mu,\,\mu=0,1,2,3$ and let $u_{a,\,\mu}=\frac{du_a}{dx^\mu}$. Let us introduce two operators (we use Einstein ...
0
votes
1answer
162 views

Meaning of $[A,B]$ when $A$, $B$ are self-adjoint

This is just a question about notation, but it got no useful answers on math.stackexchange. Let $L$ be the Lie algebra of $n\times n$ Hermitian matrices, with Lie bracket $(A,B)\mapsto i(AB-BA)$. ...
0
votes
0answers
52 views

Outer automorphisms of semi-simple Lie algebras [migrated]

It is known that outer automorphisms of semi-simple Lie algebras are automorphisms of their corresponding Dynkin diagrams. But would it be correct to say that for a semi-simple Lie algebra all outer ...
1
vote
1answer
73 views

Normalized invariant form on a Kac-Moody Algebra

For a symmetrizable Kac-Moody Algebra, we can define a normalized invariant form that performs the same role as the Killing form in the finite dimensional case. My question is, do these forms ...
11
votes
5answers
2k views

Understanding moment maps and Lie brackets

I'm trying to learn about moment maps in symplectic topology (suppose our Lie group is $G$ with Lie algebra $\mathfrak g$, acting on the symplectic manifold $(M,\omega)$ by symplectomorphisms). I'm ...
34
votes
7answers
6k views

What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic ...
16
votes
1answer
331 views

What is the homomorphism between the third exterior and third symmetric power of the adjoint representation of a simple Lie algebra?

Let $\mathfrak{g}$ be the adjoint representation of a simple Lie algebra (which is not of type $A$). Then the space of intertwiners between the third exterior power of $\mathfrak{g}$ and the third ...
0
votes
1answer
244 views

A problem about a matrix norm on $\mathfrak{su}(4)$

Given a fixed $B \in \mathfrak{su}(4)$ is it possible to solve for $F$: $\sigma^{\text{max}}\left(\frac{A}{F(A)} + B \right) = 1$, $\forall A \in \mathfrak{su}(4)$. A theorem in ...
4
votes
1answer
140 views

Matrix from a homomorphism of simply connected groups

Let $H$ be a simple algebraic group of type $\mathbf{G}_2$ over $\mathbb{C}$. Let $\rho$ be the standard 7-dimensional complex representation $$ \rho\colon H=\mathbf{G}_2\to \mathrm{SO}_7.$$ We ...
2
votes
1answer
72 views

Do the values of the differential of a function on a Lie group with a single maximum span the Lie algebra?

Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a ...
1
vote
1answer
68 views

Classification of finite dimensional Lie subalgebras of $\mathbb R[q^1,\dots,q^n,p_1,\dots,p_n]$

Do there exist results towards answering the following question? Consider the Poisson algebra of regular functions $A=\mathbb R[V]$ on the symplectic vector space $V:=T^* \mathbb R^n$. Using ...
4
votes
1answer
172 views

Check symplectomorphism property on infinitesimal generators

I stumbled over the following question: First, let me give the basic definition of a symplectic group action: Let $(M, \omega)$ be a symplectic manifold and $G$ a Lie group. A smooth action $\Phi:G ...
3
votes
0answers
122 views

What does “control of a deformation problem” mean?

Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...
4
votes
1answer
92 views

Characterizations of Jacobson-Morozov parabolics associated to a nilpotent

Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a Jacobson-Morozov parabolic ...
6
votes
1answer
326 views

Torsion in profinite groups

Is there a finitely generated profinite group $G$ with a closed subgroup of infinite index $K \leq G$ such that for every $g \in G$ there exists some $n \in \mathbb{N}$ for which $g^n \in K$ ? Can ...
4
votes
2answers
177 views

What are the “tensor-closed” object of the BGG category $\mathcal{O}$ of a semisimple Lie algebra $\mathfrak{g}$?

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra and we can consider its BGG category $\mathcal{O}$. It is well-known that $\mathcal{O}$ is not closed under tensor product, ...
2
votes
3answers
585 views

A question on the root systems of simple Lie algebras in the 90 degree case

I've been taking a look at simple Lie algebras for particle physics and I've found myself wondering about the following question. It can be shown that the adjacent roots in a root diagram ...
0
votes
0answers
79 views

Poincare inequality for connected Lie groups

Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is: symmetric, adapted (in the sense that there is no proper subgroup $H$ such ...
5
votes
2answers
125 views

Lie Algebras over DVRs and basechange to the completion

Let $R$ be a discrete valuation ring containing an algebraically closed field $K$ of characteristic zero and let $L$ be a Lie algebra over $R$ whose underlying $R$-module is finitely generated and ...
5
votes
2answers
167 views

Permutable (Lie) subgroups

Let's recall that, a group $G$ being given, two subgroups $A,B\subset G$ are called permutable iff $AB=BA$ for the Minkowski law. It is straightforward to see that $(A,B)$ are permutable iff $AB$ ...
0
votes
1answer
70 views

Solving $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$ [closed]

Given an $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$? Cross posted from ...
8
votes
5answers
725 views

Applications of Chevalley Restriction Theorem

Let $G$ be a simple linear algebraic group (over $\mathbb{C}$, say) and $\mathfrak{g}$ be its Lie algebra, $\mathfrak{t}\subset \mathfrak{g}$ the Lie algebra of a maximal torus in $G$ and $W$ the ...
3
votes
3answers
243 views

Jacobson-Morozov theorem

Jacobson-Morozov theorem for a semisimple algebraic group $G$ (presumably I am working over algebraically closed field) states that: given a unipotent u, there exists a homomorphism $\phi$ from $SL_2$ ...
0
votes
1answer
99 views

Gradient on $SU(n)$

I'm trying to calculate the gradient (wrt to the bi-invariant metric) of the following functions $F_1, F_2 : SU(n) \rightarrow \mathbb{R}$ defined by $F_1(U) = | Tr (G^{\dagger} U) |^2$, $F_2(U) = \Re ...
3
votes
0answers
61 views

Homological dimension of Joseph quotients

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ not isomorphic to $sl(n)$. Let $\mathcal O$ be the minimal nilpotent orbit in $\mathfrak g^*$. Joseph proved that there exists unique ...
0
votes
1answer
74 views

Non-graded representations over Lie superalgebra $\mathfrak{gl}(m,n)$

I have the following questions: Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra ...
4
votes
1answer
131 views

Infinite-dimensional admissible representations of SL(2,C)

I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper "Infinite Irreducible ...
5
votes
1answer
168 views

What is Jantzen's formula for the determinant of the Shapovalov form in the case of generalized Verma modules?

The best reference I found is [Kac, Kazhdan '79] which extends the results of Shapovalov and Jantzen to the case of infinite dimensional Lie algebras. Theorem 1 of this paper gives the Shapovalov ...
4
votes
1answer
112 views

Restricted Burnside Problem: Lower bound nilpotency class

Let $p$ be a prime and let $F$ be a free group of rank $d\geq 1$. Kostrikin [1] proved that the $d$-generated Burnside group $B=B(d,p)=F/F^p$ of exponent $p$ has a maximal finite quotient ...
2
votes
0answers
88 views

Is there an E8 symmetry in the zero-field Ising model?

In the paper On classification of modular tensor categories by Rowell, Stong and Wang, they list the Ising modular category $I$ as having 3 objects $1$, $\sigma$ and $\psi$, with fusion rules ...
3
votes
1answer
167 views

Is every weight of an integrable highest weight module in the Tits cone?

Let $\mathfrak{g}$ be a Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, Weyl group $W$, and simple roots and coroots $\alpha_i, \check{\alpha_i}, i \in I$, respectively. Let $L$ be an ...
0
votes
0answers
45 views

Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action

I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
2
votes
0answers
354 views

Differential and pre-differential of Jacobi identity

Let M be a manifold. To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied? That is a Lie algebra structure for which $[X,fY]=f[X,Y]$. (For ...
3
votes
0answers
165 views

Can the product of a simple and a non-simple indecomposable representation be semisimple?

Consider two (possibly infinite-dimensional) representations $\rho$, $\pi$ of a semisimple Lie algebra $\mathfrak{g}$, with $\rho$ irreducible and $\pi$ indecomposable but not irreducible (i.e., not ...
2
votes
2answers
246 views

What is twisted in a twisted Poisson structure?

The usual definition of a twisted Poisson algebra involving a 3-form does not seem to refer to a Poisson algebra that is then twisted, i.e. twisting either the commutative product or the Lie bracket. ...
8
votes
3answers
891 views

Hopf algebra structure on the universal enveloping algebra of a Leibniz algebra?

A Leibniz algebra L may be thought of as a noncommutative generalisation of a Lie algebra. One drops the requirement that the bracket be alternating and substitutes the Jacobi identity for the ...
2
votes
1answer
172 views

Understanding the Weyl Character Formula

Let $G$ be a compact (connected) Lie group with a maximal torus $T$. For each (analytically) integral weight $\lambda$ the Weyl character formula $$\Theta_{\lambda}(H)=\frac{\sum_{w\in ...