**3**

votes

**1**answer

92 views

### Are all the Lie bialgebra structure on $sl_n$ coboundary?

In the case of $sl_2$, there are three Lie bialgebra structures. We have three cobrackets $\delta: sl_2 \to \Lambda^2 sl_2$. Each $\delta$ can be written as $\delta=d r$ for some matrix $r$. Therefore ...

**0**

votes

**0**answers

75 views

### Picard scheme analogy with Lie algebra

I'm far from an expert, but its classic to study for a smooth, proper scheme over a field Pic(X) its Picard scheme and to note from deformation theory that the tangent space at the identity of Pic(X) ...

**1**

vote

**0**answers

55 views

### Low-dimensional classical r-matrices

Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties:
(1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$.
(2) $[r_{12}, ...

**2**

votes

**0**answers

131 views

### Lie Algebra of Aut(GL(n,R))

What is the Lie Algebra of $Aut(Gl(n,F))$ when $F$ is either $\mathbb{R}$ or $\mathbb{C}$?
Is it enough to consider the injection via Hochschild:
$Aut(GL(n)) \to Aut(\mathfrak{gl}(n))$?
Edit: The ...

**2**

votes

**2**answers

165 views

### Recover Poisson bracket on $C[G]$ using the Lie cobracket $\delta: g \to \Lambda^2 g$

By a theorem of Drinfeld, there is a one to one correspondence between Lie bialgebras and Poisson Lie groups. Therefore given a Lie cobracket $\delta: g \to \Lambda^2 g$, there is a Poisson bracket on ...

**2**

votes

**1**answer

146 views

### Classifications of Lie bialgebras

What is the current status of the classifications of Lie bialgebras? In particular, has the following problem been solved? Let $gl_n$ be the general linear Lie algebra. Classify all Lie cobrackets ...

**1**

vote

**0**answers

167 views

### Solvable Lie algebra whose nilradical is not characteristic

It is well known that the nilradical of a finite-dimensional Lie algebra over a field of characteristic p > 0 need not be characteristic (that is, invariant under all derivations of the algebra), but ...

**2**

votes

**0**answers

146 views

### Exponential analogue of formal connections

Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation:
$$
g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).
$$
Here, ...

**2**

votes

**0**answers

139 views

### Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n ...

**2**

votes

**0**answers

97 views

### Systematic treatment of folding and valued graphs

I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed ...

**3**

votes

**0**answers

85 views

### Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations

I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations:
$[U_i, U_j] = 0$ for $|i-j|>1$
...

**6**

votes

**0**answers

81 views

### Subquotients of Jantzen Filtration for Kac-Moody algebras

Let $\mathfrak{g}$ be a complex symmetrizable Kac-Moody algebra, with triangular decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$. Let $\lambda \in \mathfrak{h}^*$, and ...

**0**

votes

**0**answers

45 views

### Generalized Gaussian Decomposition

Let $G$ be a connected complex semisimple Lie group. Let $H$ be a maximal torus of $G$, let $W$ be the Weyl group of $G$, and let $N_\pm$ be a pair of opposite maximal unipotent subgroups. For each ...

**8**

votes

**1**answer

319 views

### Intuition for the Cartan connection and “rolling without slipping” in Cartan geometry

Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$.
The Cartan connection is supposed to formalize what it means to "roll ...

**1**

vote

**0**answers

43 views

### Extra-Lorentzian Kac-Moody algebras

My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with ...

**4**

votes

**1**answer

144 views

### Lie functor preserves “surjections” in synthetic differential geometry?

In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism.
As pointed out ...

**3**

votes

**1**answer

209 views

### Affine analog of the theory of sheets

In the study of adjoint orbits in a complex semi-simple lie algebra, there is a well known object known as a "sheet". These are the irreducible components of the union of orbits of the same dimension. ...

**4**

votes

**1**answer

199 views

### The parity of the full automorphism group order of finite non-abelian groups of prime exponent

Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?

**0**

votes

**1**answer

122 views

### Tensor products of simple modules over algebras [closed]

Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules.
We can regard $M\otimes_{\mathbb C}N$ as $A\otimes_{\mathbb C} B$-modules in natural ...

**6**

votes

**0**answers

135 views

### del Pezzo surfaces and exceptional algebras

It is well-known that $H^2(dP_n, Z)$ for a del Pezzo surface of degree $9-n$ which is $\mathbb{P}^2$ blown up at $n$ generic points, is encoded by the exceptional Lie algebra $E_n$. However, the Mori ...

**1**

vote

**1**answer

231 views

### Soliton equation and non-killing potential vector field

I am searching for a non-Killing vector field $\zeta \in\frak X\rm (M)$ where $(M,g)$ is a Riemannian manifold such that
$$\frac12 \frak L_\zeta \rm g+Ric=\lambda g$$
$$ \frak L_\zeta \rm Ric=\lambda ...

**5**

votes

**1**answer

124 views

### The action of $GL_{\infty}$ on the infinite wedge space

This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya.
Consider the following objects:
the ...

**16**

votes

**5**answers

547 views

### Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?

Let $G$ be a simple algebraic group group over $\mathbb C$.
Let $V$ be a self-dual representation of $G$.
Let $\lambda$ be the highest weight of $V$.
Write $\lambda$ as a sum of fundamental weights: ...

**2**

votes

**0**answers

82 views

### Are singular critical points isolated for control systems on compact semisimple Lie groups

Given a control system on $\mathrm{SU}(n)$ (or any other compact, semi-simple Lie group I suspect) of the form:
$\frac{d U_t}{dt} = (A + w(t)B)U_t$
where $A,B \in \mathfrak{su}(n)$ generate the ...

**5**

votes

**1**answer

96 views

### Ideals of $U(\mathrm{gl}(n,\mathbb{C}))$ and their intersection with center

Does every non-null two-sided ideal of $U(\mathrm{gl}(n,\mathbb{C}))$ have nonzero intersection with the center of $U$?

**14**

votes

**2**answers

475 views

### factorization of the regular representation of the symmetric group

Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation.
Question: Does there exist a representation $V$ (of dimension ...

**0**

votes

**1**answer

51 views

### Authomorphisms of factoralgebra A_5 over its center in characteristic 2

Let L be classical Lie algebra of type A_5 over field of characteristic 2, M - factoralgebra L/Z(L) where Z(L) - center of L.
What about group of automorphisms of M?
Does anybody know answer or at ...

**3**

votes

**2**answers

152 views

### Commutator 2-forms on Lie groups

Let $G$ be a compact Lie group and $\mathfrak g$ its Lie algebra.
For any $f$ in the dual space $\mathfrak g^*$, we can define a skew-symmetric bi-linear form on $\mathfrak g$ by $(A,B)\mapsto ...

**0**

votes

**1**answer

82 views

### A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to $L_{n}$

We define $$L_{n}=\{A=(a_{ij})\in M_{n}(\mathbb{R})\mid \sum_{i=1}^{n} a_{ij}=0 \;\;\;\text{for every fixed j}\}$$
This is a Lie subalgebra of $M_{n}(\mathbb{R})$.
A dynamic-geometric proof for ...

**1**

vote

**0**answers

91 views

### A question about the associative classes of parabolic subgroups

Let $\mathbb{A}$ denote the adele group of $\mathbb{Q}$. Suppose that $P_1$ and $P_2$ are parabolic subgroups of a reductive algebraic group $G$, and consider their Langlands decomposition
$$
...

**3**

votes

**0**answers

325 views

### Mystery behind ADE Dynkin diagram [duplicate]

ADE Dynkin diagram classifies:
finite type quiver (Gabriel's theorem)
du Val singularity (minimal resolution of $\mathbb{C}^2/G$, $G$ is a finite subgroup of $SL_2(\mathbb{C})$)
simply laced simple ...

**2**

votes

**0**answers

101 views

### The existence of zero-divisors in the universal enveloping algebra of an infinite-dimensional Lie algebra

The intuition for this problem comes from $\S$17 Exercise 1 Humphreys' Introduction to Lie Algebras and Representation Theory which essentially asks us to use PBW in order to prove that if a Lie ...

**2**

votes

**0**answers

123 views

### Differential of the adjoint quotient map

My question is regarding a paper by R.W Richardson titled "Derivatives of invariant polynomials on a semisimple Lie Algebra" ** . In this paper, he reports on computations of the rank of the ...

**4**

votes

**2**answers

163 views

### References about Hasse diagrams of root systems

This is to ask about references of Hasse diagrams of irreducible root systems. I found here and there nice pictures of root systems of type $E$. I would like to ask for Hasse diagrams of classical ...

**4**

votes

**0**answers

158 views

### Hochschild cohomology of SU(2)

I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth.
Let's ...

**6**

votes

**1**answer

78 views

### Constructing symmetric invariants of the exceptional simple Lie algebra as restrictions

Let $\mathfrak{g}$ be a complex simple Lie algebra with maximal torus $\mathfrak{h}$, Weyl group $W$. The adjoint representation $\operatorname{ad} : \mathfrak{g} \rightarrow \mathfrak{gl(g)}$ extends ...

**1**

vote

**0**answers

33 views

### Iwasawa decomposition and Non-Abelian Centraliser of A

I'm studying Knapp's book "Representation Theory of Semisimple Groups" and am trying to understand the structure theory of non-compact groups. Namely, let $G=KAN$ be the Iwasawa decomposition, ...

**5**

votes

**0**answers

222 views

### Recurrence Formula for Zernike polynomials

I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...

**3**

votes

**2**answers

109 views

### Covering derivations of a quotient algebra

Let $(\mathcal{A},+,·)$ an algebra and $\mathcal{I}$ an ideal of $\mathcal{A}$.
Is easy to check that if $D\in Der(\mathcal{A})$ with $D(\mathcal{I})\subseteq I$ then $D$ induces a derivation $D_I$ ...

**2**

votes

**1**answer

78 views

### Seeking for abelian subalgebra of fixed dimension in finite Lie algebra

The problem is: I want to know if there is abelian subalgebra of dimension $k$ in Lie algebra of dimension $n$. My Lie algebra is given by its structure constant table. There are some algorithms ...

**11**

votes

**2**answers

555 views

### What happened to the fourth paper in the series “On the classification of primitive ideals for complex classical Lie algebras” by Garfinkle?

In a series of papers in Compositio Math. entitled On the classification of primitive ideals for complex classical Lie algebras I, II and III, Garfinkle describes an algorithm that allows one to ...

**5**

votes

**2**answers

142 views

### Relationship between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?

Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the Lie algebra of vector fields on $M$; that is ...

**6**

votes

**2**answers

418 views

### Why is the trace of the Casimir on the irrep of a semisimple algebra nonzero?

A crucial step in the "purely algebraic" proof of Weyl's semisimplicity theorem is that the Casimir element $C\in U\mathfrak{g}$ acts by nonzero scalars on a nontrivial irrep $V$. However, at least ...

**3**

votes

**1**answer

120 views

### Singular curves of affine distributions on a Lie group

Are there any results about the rigidity of singular curves of rank 1 affine distributions on a connected compact Lie group?
Specifically the case of a right invariant affine distribution: $D_{U} = ...

**2**

votes

**1**answer

150 views

### Length of Weyl group element mapping highest root to a simple root

Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the ...

**2**

votes

**0**answers

85 views

### Conjugacy classes of involutions in Kac-Moody groups

Let $A=(a_{s,s'})_{s,s'\in S}$ be a generalized Cartan matrix.
Let $G=G(A)$ be the corresponding simply connected complex Kac-Moody group with Cartan subgroup $H$ and Weyl group $W$ acting on $H$.
...

**3**

votes

**1**answer

132 views

### A converse to Whitehead's Second Lemma (and more)

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{h}$ be a finite dimensional $k$-Lie algebra. I'm interested in knowing which (Lie algebraic) properties $\mathfrak{h}$ ...

**7**

votes

**1**answer

287 views

### A technical question about root systems

I'm studying root systems and coming up with an observation:
Let $\Phi$ be an irreducible root system and $\Phi^+$ be a positive root system. Denote by $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_n\}$ ...

**2**

votes

**0**answers

93 views

### Prove that a Verma module is projective only if its highest weight is dominant?

Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over $\mathbb{C}$ with a fixed Cartan subalgebra $\mathfrak{h}$ and a fixed system of simple roots. It is stated in Exercise 3.11 of ...

**3**

votes

**1**answer

117 views

### Maximal possible dimension of abelian Lie subalgebra of Heisenberg Lie algebra of dimension $2n+1$? [closed]

Fix $n \in \mathbb{N}$, and let $\mathfrak{h}_n$ denote the Heisenberg Lie algebra of dimension $2n+1$ (over any given field $k$). Namely, $\mathfrak{h}_n$ is the Lie algebra with basis $x_1, \dots, ...