**3**

votes

**1**answer

61 views

### Restricted Lie algebras with a $p$-nilpotent basis

Let $L$ be a finite-dimensional restricted Lie algebra over a field of characteristic $p>0$. An element $x$ of $L$ is called $p$-nilpotent if $x^{[p]^k}=0$ for some positive integer $k$. If $L$ is ...

**5**

votes

**0**answers

67 views

### Lie subalgebras of $\chi^{\infty}(M)$ of codimension $n=dim M$

For a connected $n$ manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a pointe $p\in M$ we define $L_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\} $....

**0**

votes

**0**answers

4 views

### For which category (if any) are Lie algebras the algebras of a monad? [migrated]

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...

**1**

vote

**0**answers

50 views

### About Composition diamond lemma

Composition Diamond lemma for Lie algebra over a field $F$ has already been investigated in several papers :
L.Bokut and Y.Q.Chen Groebner-Shirshov bases for Lie algebras
and
A.I Shirshov, ...

**6**

votes

**1**answer

207 views

### The minimum codimension of Lie subalgebra of $\chi^{\infty}(M)$

Assume that $M$ is an arbitrary manifold.
Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?
If not, what is a counter ...

**7**

votes

**1**answer

292 views

### Is the assignment of a root system to a complex semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...

**6**

votes

**1**answer

111 views

### Chevalley restriction theorem for non-split Cartan

Let $G$ be a reductive group over a field $k$ with maximal torus $H$. Let $\mathfrak{g}$ and $\mathfrak{h}$ denote the corresponding Lie algebra. If $k$ is algebraically closed, we have a theorem of ...

**18**

votes

**0**answers

294 views

### Follow-up to Steinberg's problem (12) in his 1966 ICM talk?

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (with ...

**3**

votes

**1**answer

210 views

### Lie algebra of invariant polynomials or invariant smooth functions

Is there a symplectic structure on $M_{2n}(\mathbb{R})$, not necessarily with constant coefficients, such that the space of smooth invariant functions, those smooth functions $f:M_{2n}(\mathbb{...

**-2**

votes

**0**answers

36 views

### Irreducible modules of dimension $\leq d$ is finite [migrated]

Let $L$ be a finite dimensional semisimple Lie algebra and $V(\lambda)$
denote the unique irreducible (upto isomorphism of $L$ modules) standard cyclic module of highest weight $\lambda$.For each $p\...

**2**

votes

**1**answer

107 views

### “Nice” basis for highest-weight irreducible module of a simple Lie algebra

Let $\mathfrak{g}$ be a simple complex Lie algebra, $\mathfrak{h}$ a Cartan subalgebra, $\Phi \subset \mathfrak{h}^*$ the associated root system, $\Sigma = \{\sigma_i : i\in I\}$ a basis of simple ...

**10**

votes

**1**answer

276 views

### Uncle of Witt algebra

A Witt algebra W is an infinite-dimensional Lie-algebra defined by the generator relations:
W: $[L_{j},L_{k}]:=(j-k)\cdot L_{j+k}$
And my first thought was: What about the analogous algebra defined by
...

**3**

votes

**1**answer

77 views

### What is known about the morphism $H^*_{Lie}(L,L)\to H^*_{Lie}(L,UL)$ induced by $L\hookrightarrow UL$

Let $L$ be a (differential) graded Lie algebra over a field $k$ of characteristic 0, and let $UL$ be the universal enveloping algebra of $L$.
The inclusion $L\hookrightarrow UL$ induces a morphism of ...

**3**

votes

**1**answer

66 views

### Explicit generators of the Lie algebra $spin(9)$

It is well known that the Lie group $Spin(9)$ acts on the vector space $\mathbb{R}^{16}$ (see e.g. Harvey's book "Spinors and calibrations".) It is convenient to identify this vector space with the ...

**1**

vote

**1**answer

97 views

### Bounded operators $T: B(H)\to H$ whose Kernel is a Lie algebra

Assume that $H$ is an infinite dimensional Hilbert space.The space of all bounded operators on $H$ is denoted by $B(H)$.We consider the Lie algebra structure $[T,S]=TS-ST$ on $B(H)$.
Is there a ...

**1**

vote

**0**answers

104 views

### Generating $\mathfrak{so}(7)$

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is actually a subalgebra isomorphic to $\...

**1**

vote

**0**answers

56 views

### Comodules of the $B,C$ and $D$ series quantum groups

In Section 11.5 of Klimyk and Schmudgen's book on quantum groups, explicit presentations of the isomorphism classes of comodules of ${\cal O}(GL_q(N))$ are given in terms of its "quantum minors". In ...

**0**

votes

**1**answer

82 views

### Hamiltonian potential invariant under lie group action?

Let $(M,g)$ be a riemannian manifold and $G$ a Lie group acting on $M$ by isometries.
Taking a $G$-invariant function $f\colon M \to \mathbb{R}$, we have the riemannian gradient $\operatorname{grad} ...

**2**

votes

**0**answers

33 views

### $TSU(n)$ completely integrable with 3 $SU(2)$ invariant functions?

Consider the Lie group $SU(n)$ endowed with the standard bi-invariant metric. Then $SU(n)$ can be viewed as a symmetric space of $K_{n,n} := SU(n) \times SU(n)$.
Define $M := K_{n,n} /SU(n)$. Using ...

**0**

votes

**1**answer

76 views

### Exterior derivative on principal bundle [closed]

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...

**6**

votes

**1**answer

97 views

### Involutions and Little Adjoint Representations of Simple Algebras

In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$.
Is easy to see that ...

**4**

votes

**2**answers

210 views

### Invariant theory for parabolics

Let $G$ be a connected reductive group over $\mathbb{C}$ of (reductive) rank $\ell$. Let $P$ be a parabolic of $G$ and let $P=LN$ denote the Levi decomposition. Let $\mathfrak{g}, \mathfrak{p}, \...

**1**

vote

**0**answers

58 views

### Symplectic gradients whose span doesn't intersect Lie group orbits

I asked a similar question some hours ago. But thinking about my problem, I found a loophole in my arguments, so that my question wasn't the right one. This one is what I wanted to ask:
Let $G$ be a ...

**4**

votes

**1**answer

154 views

### Sum of Young symmetrisers of a given shape

Preliminaries and notation:
Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it ...

**2**

votes

**0**answers

64 views

### Center of affine W-algebras

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $k$ a complex number. Denote by $\hat{\mathfrak{g}}$ the corresponding affine Lie algebra ($\hat{\mathfrak{g}}=\mathfrak{g}((t)...

**1**

vote

**1**answer

81 views

### slice theorem for proper actions

I'm trying to understand the slice-theorem for proper Lie-group actions.
Having a smooth manifold $M$ and a Liegroup $G$ acting on $M$ in a proper way, we have the slice theorem, saying that at each ...

**3**

votes

**1**answer

217 views

### Irreducible representations containing simple actions of $\mathrm{SL}(2,\mathbb{C})$

Let $G$ be a complex semisimple Lie group and let $\rho: G \longrightarrow \mathrm{SL}(n,\mathbb{C})$ be a faithful irreducible representation of $G$ with $n \geq 3$. Suppose that $G$ contains a copy ...

**0**

votes

**0**answers

73 views

### Sandwich rule for Lie algebras

On an infinite dimensional vector space an operator can be onto but not one-to-one (and vice versa). This arises the following question. Let $L_1$ and $L_2$ be Lie algebras (infinite dimensional, over ...

**6**

votes

**0**answers

113 views

### Software for explicit computations in representations of classical Lie algebras

I'm pretty sure many a mathematician has longed for such a tool but I wasn't able to find such a question here, so here we go.
Is there, by any chance, an existing package or program that allows one ...

**3**

votes

**0**answers

137 views

### Invariant functions on the dual Lie algebra

Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$.
...

**2**

votes

**3**answers

108 views

### How can one show G/T is a coadjoint orbit for G a compact lie group and T it's maximal torus

Let $G$ be a compact lie group and $g$ it's lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) is a coadjoint orbit. However it is not at all clear to me ...

**-1**

votes

**1**answer

127 views

### Crystal basis for quantum groups and Lie algebras

Lie $g$ be a finite dimensional complex simple Lie algebra and $U_q(g)$ the corresponding quantum group, where $q$ is not a root of unity. Every simple finite dimensional $g$-module is of the form $V(\...

**1**

vote

**0**answers

126 views

### What inner autmorphism really do in abstract algebra? [closed]

The definition of inner automorphism is easy to understand but I wonder what these inner automorphism really do in abstract Lie algebra, especially when we are talking about Lie algebra ?
In Lie ...

**5**

votes

**2**answers

136 views

### Multiplication in universal enveloping algebra in terms of PBW isomorphism

Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt ...

**6**

votes

**0**answers

168 views

### Is there an explicit construction of Tate-Beilinson residue?

Background: Tate's construction of abstract residues is generalized in Beilinson's constructon as follows:
Let $E$ be a unital, cubically decomposed $k$-algebra, i.e., (a) there are two sided ideals $...

**2**

votes

**2**answers

137 views

### Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra

Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...

**5**

votes

**1**answer

227 views

### Motivating the existence of Canonical Bases for Representations

In Representation Theory, the theme of the existence of a canonical basis has been explored quite a lot. I will limit myself in this question to the kind of canonical bases that arise from the ...

**10**

votes

**0**answers

461 views

### Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem.
Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ ...

**4**

votes

**0**answers

187 views

### A soft question on Gauge Equivalence in Integrable Systems

I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations (...

**3**

votes

**1**answer

134 views

### Isomorphisms between extension group and $\mathfrak{u}$-cohomology

Let $\mathcal{O}^{\mathfrak{p}}$ be a parabolic subcategory (see, Humphrey's BGG's Ch.9) of the BGG category $\mathcal{O}$ (w.r.t the Cartan $\mathfrak{h}$ and Borel $\mathfrak{b}$) over a finite-...

**2**

votes

**1**answer

236 views

### The Quasimirs and Casimirs of a Lie algebra

Casimirs are not completely trivial to compute, and somewhat ill defined (if $C_4$ is a quartic casimir and $C_2$ a quadratic, $C_4'=C_4+p*C_2^2+q*C_2$ is another quartic casimir). With generalized ...

**6**

votes

**2**answers

424 views

### Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra

Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$.
Tilting module theory play an important role in the ...

**4**

votes

**1**answer

123 views

### Vanishing of power of nilpotent operator $\mathrm{ad} \, \;e$ in different characteritics

This question needs some background:
(1) In his influential 1959 paper here, Kostant studied the adjoint representation of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (which can be ...

**10**

votes

**2**answers

434 views

### From Weyl groups to Weyl groupoids?

I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed.
Let $\mathfrak{g}$ be a semisimple lie algebra.
...

**5**

votes

**1**answer

350 views

### A criterion for orbits of complex reductive group to be closed

I am having some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following:
Let $G=K_{\Bbb C}$ be a ...

**1**

vote

**0**answers

50 views

### Integrable modules and comodules

Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? ...

**4**

votes

**2**answers

226 views

### A finiteness property for semi-simple algebraic groups

Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any semi-...

**2**

votes

**0**answers

51 views

### Are there nilpotent Manin Triples?

Let $\mathfrak{g}$ be a Lie bialgebra and denote by $\mathfrak{d}$ the double of $\mathfrak{g}$, i.e. $\mathfrak{d}$ is a Manin triple. Are there known examples or conditions on $\mathfrak{g}$ for ...

**0**

votes

**0**answers

61 views

### Try to understand the relation between Poisson Lie groups and Lie bialgebras

I am trying to understand the relation between Poisson Lie groups and Lie bialgebras. In the book Lectures on quantum groups, Page 20, Theorem 22 says that given a Poisson Lie group, we can construct ...

**6**

votes

**2**answers

199 views

### “Diagonalizing” Littlewood-Richardson coefficients

Let's consider the Littlewood-Richardson coefficients $c^{\lambda}_{\mu \nu}$ so that
\begin{equation}
V_\mu \otimes V_\nu = \bigoplus_\lambda V_\lambda^{\oplus c^{\lambda}_{\mu \nu}}
\end{equation}
...