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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3
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2answers
154 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 $SL_2$ ...
4
votes
1answer
99 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
64 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 ...
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votes
0answers
9 views

equivalence of Lie group and Lie algebra intertwiner [migrated]

I encountered this problem while working on my research. Let $G$ be a Lie group, and consider an intertwiner of the complex representations (possibly infinite-dimensional) $$ \pi:G\rightarrow ...
2
votes
1answer
141 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 ...
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votes
0answers
36 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 ...
3
votes
0answers
161 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
1answer
147 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 ...
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vote
0answers
21 views

Symplectic group action [migrated]

Let $(M,\omega)$ be a symplectic manifold. We say that a group action $\phi: G \times M \rightarrow M$ is symplectic if each $\phi(g,.)$ is a symplectomorphism. Now, I am going through some lecture ...
0
votes
2answers
217 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
263 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 ...
2
votes
0answers
50 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 ...
4
votes
1answer
116 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 ...
0
votes
0answers
47 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) ...
0
votes
0answers
44 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 ...
3
votes
0answers
127 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 ...
0
votes
1answer
200 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 ...
9
votes
2answers
172 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: ...
9
votes
3answers
284 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 ...
4
votes
0answers
103 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
78 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
69 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 ...
1
vote
0answers
38 views

General quantum highest-weights dimension formulas

The formulas hold modulo typos :-) It is well known (tl;dr fun fact: not well enough for me, I forgot where I saw it so I guess-computed it from the data in the Hayashi paper; promptly I found it in ...
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vote
0answers
28 views

artinian quotients of U(g)

Suppose that $G$ is a connected Lie Group with Lie algebra $\mathfrak{g}$ and $\Gamma$ is a cocompact lattice, and you choose a left-invariant metric on $G/\Gamma$ and let $\Delta$ be the Laplacian. ...
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votes
2answers
159 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 $F_n=e^{\pi \cdot i \cdot g_n / 2}$. ...
9
votes
1answer
225 views

One identity in Lie algebras

Let $L$ be a (non-restricted) Lie algebra over a field of prime characteristic $p,$ $UL$ be its universal enveloping algebra and $a_1,\dots, a_p \in L$ (the number of elements is equal to the ...
4
votes
3answers
329 views

$A \wedge A \wedge A$ in Chern-Simons

I am confused with the wedging operations of Lie algebra valued differential forms. Especially, for instance, I have some problems with the Chern-Simons 3-form $$A \wedge dA + \frac{2}{3}A \wedge A ...
2
votes
1answer
251 views

Earliest source for a Lie algebra construction

I am looking for the earliest reference to the fact that any associative algebra becomes a Lie algebra with bracket $AXB-BXA$, where $X$ is a fixed element of the algebra. This is observed in the ...
4
votes
1answer
177 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 ...
3
votes
0answers
102 views

n-homology of a Harish-Chandra module

Let $G$ be a connected real reductive Lie group and let $K$ be its maximal compact subgroup. Let $P=MAN$ a parabolic subgroup. Let $K_M^0=M^0\cap K$ be connected component of the maximal compact ...
0
votes
1answer
119 views

Generalization of the Lie group exponential map and its derivative

Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$, and $exp:\mathfrak{g}\to G$ be its exponential map. The group $G$ could be finite or infinite dimensional. Let $G$ have the property that ...
3
votes
2answers
128 views

About supersolvable Lie algebras

A colleague of mine asked me the question below, and since I could not answer it, I thought I might have more luck on MO. In Encyclopedia of Mathematics, a finite dimensional Lie algebra $L$ over a ...
1
vote
1answer
89 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 ...
1
vote
0answers
65 views

Symmetric and antisymmetric powers of SU(2) representations [closed]

Recently, I took a course in representation theory at Imperial College, and on the first homework the questions were about certain sneaky relationships when it came to representations of SU(2). ...
0
votes
1answer
49 views

Centralizer of the derived algebra in a non-perfect Lie algebra

Is there a non-perfect Lie algebra for which the centralizer of the derived algebra is trivial?
0
votes
0answers
124 views

Reference request: the formula $\langle x, [f, g] \rangle = \langle \delta(x), f \otimes g \rangle$

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}^*$ the dual vector space of $\mathfrak{g}$ which is also a Lie algebra with natural brackets. Let $\delta: U(\mathfrak{g}) \to U(\mathfrak{g}) ...
3
votes
0answers
42 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
38 views

Motivation for a representation of the Witt algebra

I'm reading through these notes and I'm confused by the definition of a representation of the Witt algebra on page 6. Precisely, the sentence This is the representation that one would discover by ...
1
vote
1answer
78 views

Dimension of Span of Adjoint orbit in $\mathfrak{su}(n)$

Given two elements $A,B \in \mathfrak{su}(n)$ what is the dimension of the span of the following adjoint orbit: $\{Ad_{e^{sA}}(B) \ | \ s \in [0,t]\}$ for different values of $t$. Does it ever change ...
0
votes
0answers
36 views

Maximal nilpotent substructures

In finite group theory of solvable groups maximal nilpotent subgroups are important. These are Carter subgroups, nilpotent injectors, nilpotent projectors and Fischer-subgroups. In Lie-theory there ...
8
votes
5answers
524 views

Representation Theory of Lie Groups: Reference Request

I am looking for a reference that describes the correspondence between the (finite-dimensional) representations of (real) Lie groups and the representations of their Lie algebras. More precisely, ...
1
vote
0answers
58 views

Largest dimensional Lie subgroup of $SU(N)$ [duplicate]

What is the largest (Lie) subgroup of $SU(n)$ in the sense of its dimension. I am aware of this potential duplicate subgroup of SU(N) with maximal manifold dimension , however, the title of this ...
0
votes
1answer
77 views

About Kahler curvature operator

I have problems on how to consider the Kahler curvature operator. I know that one can consider the Riemannian curvature operator $R$ as a linear transformation from $\mathfrak{so}(n,\mathbb{R})$ to ...
3
votes
0answers
94 views

scalar multiple of Young symmetrizer

The following is a lemma from Fulton and Harris' book -Representation theory,a first course (page 53): Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar ...
2
votes
0answers
80 views

Contraction of the maximal submodule in a Verma module

Suppose $\mathfrak{g}$ is a real semisimple Lie algebra, $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ is a Cartan decomposition, and $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{k}$. ...
5
votes
0answers
81 views

What is the importance of the number $k+h^{∨}$ (level+dual Coxeter number)?

The number $k+h^{∨}$ appears at many places in the representation theory of affine Lie algebras (and probably elsewhere). Here $h^{∨}$ is the dual Coxeter number of the root system, and $k$ is the ...
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votes
0answers
49 views

Centralizer of a non-regular Lie algebra element

It is well understood that the centralizer of a regular element $A$ of a Lie algebra of complex (square, diagonalizable) matrices consists of polynomials $p(A)$ in that element of degree less than $n$ ...
2
votes
1answer
111 views

Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$

Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which: $K \left(A, Ad_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the ...
7
votes
0answers
123 views

How to show the compatibility between Duflo isomorphism and Harish-Chandra isomorphism for semi-simple Lie algebras?

I was told that the Duflo isomorphism is compatible with the Harish-Chandra isomorphism when the Lie algebra $\mathfrak{g}$ is semi-simple. However I cannot see why this is true. All I can show is ...
0
votes
1answer
136 views

Existence of a Lie algebra element orthogonal to the adjoint orbit of another element

Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Denote the Killing form by $K$. Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) ...