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 ...

learn more… | top users | synonyms

2
votes
1answer
64 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 ...
0
votes
1answer
192 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 ...
1
vote
1answer
63 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
164 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
84 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
84 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 ...
0
votes
0answers
16 views

Tangent vectors to coadjoint orbits [migrated]

Let $O_x:=\{Ad^*_g (x); g \in G\}$ be the orbit of $x \in \mathfrak{g}^*$ and $Ad$ the adjoint map. Now take $\xi \in \mathfrak{g}$ then $g(t):=Ad^*_{e^{t \xi}}(x)$ defines a map $g: I \rightarrow ...
6
votes
1answer
323 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
169 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, ...
0
votes
0answers
44 views

Stabilizer subgroup in adjoint action [migrated]

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...
2
votes
3answers
572 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
76 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 ...
4
votes
2answers
111 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
165 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
68 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 ...
1
vote
1answer
78 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 ...
8
votes
5answers
712 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
235 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
97 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
57 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
73 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 ...
0
votes
1answer
217 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
130 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
167 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
109 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
83 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
149 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
43 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
325 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
164 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
244 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
889 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
161 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 ...
9
votes
3answers
289 views

Real and Quaternionic Representations according to Weights

According to this question, it is easy to know whether a representation is self dual or not: just check if the weight distribution in space is symmetric about the origin. Now, for self dual ...
0
votes
2answers
229 views

Representation Theory of $U(N)$

(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...
2
votes
0answers
54 views

Vanishing of finite difference operators by composition under a cyclic condition

Consider $n$ finite difference operators $D_1, \ldots, D_n$ acting on real-valued functions $f_1 (y), \ldots , f_n (y)$ of a variable $y$, with the following properties: (i) $D_i f_i (y) = 0$ for ...
5
votes
1answer
124 views

How to embed $U(1)$ into a bigger group, using Dynkin diagrams

I am trying to find the embedding and the branching rules for some group decompositions. For example, I consider $E_7$ and its maximally compact subgroup $SU(8)$ and I want to "see" how the Dynkin ...
4
votes
1answer
183 views

A different Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$

In this question $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$ is the space of all smooth vector fields on the plane or sphere. A limit cycle for a vector field $X$ is an isolated closed ...
0
votes
0answers
51 views

Equivalent definitions of positive root system

$\bullet$ I begin with a definition of positive root systems of a root system over Euclidean space. A subset $\Delta$ of root system $\Phi$ is called a simple root system (or base) in $\Phi$ if (1) ...
9
votes
2answers
178 views

What's the most simple proof of Kostant's version of Borel-Weil-Bott for Lie Algebra cohomology?

Besides Kostant's original proof (in http://www.math.tamu.edu/~jml/kostant61.pdf) of the above mentioned theorem (using the Lie Algebra Laplacian), there are a few other approaches: ...
0
votes
1answer
203 views

Lie Algebra, counterexample [closed]

I am trying to find an example of an algebra over a field of characteristic p (prime) which satisfies anti-symmetry and Jacobi identity but is not a lie algebra. i.e., [x,x] is not zero. Can one ...
3
votes
0answers
132 views

The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$

It is commonly known that we have a chain of embeddings $$SU(4)\subset Spin(7)\subset SO(8)$$ (there is more than one possible $Spin(7)$, just take one). Which is the explicit analog for the Lie ...
8
votes
2answers
353 views

Bracket of lyndon words?

Here is a simple question regarding the standard Lyndon basis for the free Lie Algebra. Suppose I take two lyndon words $m$ and $n$ and their standard bracketings $B(m)$ and $B(n)$ as elements in the ...
24
votes
1answer
1k views

Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...
10
votes
3answers
425 views

Which nilpotent Lie algebras appear as nilradicals of parabolic subalgabras?

I am interested to identify (ideally classify) nilpotent Lie algebras that occur as nilradicals of parabolic subalgebras in (say) reductive Lie algebras. For example, all Heisenberg Lie algebras ...
15
votes
3answers
773 views

Is a retract of a free object free?

I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?
5
votes
0answers
105 views

Root-theoretic formulation of characteristic polynomial

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra of rank $n$ over $\mathbb{C}$. Let $G$ denote the corresponding simple simply connected algebraic group. By Chevalley's Theorem, ...
4
votes
0answers
81 views

Polynomials invariant with respect to a nilpotent Lie algebra

Let $\mathfrak{u}$ be a nilpotent Lie algebra and let $\mathbb{C}[\mathfrak{u}]$ be the space of polynomials with the natural coadjoint action of $\mathfrak{u}$. Can one describe ...
2
votes
0answers
72 views

When the Lie algebra of matrices with zero last rows is Frobenius?

Let $\mathcal{A}_{n,k}$ be the Lie algebra of $n \times n$ matrices over $\mathbb{C}$ for which the last $k$ rows are equal to zero. Suppose that $k$ does not divide $n$. How to prove that ...
0
votes
2answers
160 views

A question on an set of 8 matrices related to the SU(3) generators

SU(2) and SU(3) differ quite a bit. The Lie algebra of SU(2) formed by the three generators $g_n$ is the same as the algebra formed by the SU(2) matrices/elements $F_n=e^{\pi \cdot i \cdot g_n / 2}$. ...