**1**

vote

**0**answers

35 views

### Applications of finite presentation of lie algebra

If you know a finite presentation of say a certain Poisson algebra (finite presentation as a Lie algebra), in what ways might this be useful? Does this allow you to extract information that would ...

**0**

votes

**1**answer

82 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
...

**3**

votes

**1**answer

134 views

### The Jordan Plane and Enveloping Algebras

Let $k$ denote a field of characteristic $0$ (assume algebraically closed for convenience). Define $J=k\langle x,y|[x,y]=y^{2}\rangle$. This noncommutative algebra (which can be viewed as a derivation ...

**2**

votes

**1**answer

315 views

### Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...

**1**

vote

**0**answers

33 views

### How to define the determinant of a morphism between graded Lie algebras?

I have the following question. Suppose $g_1$ and $g_2$ are two finite dimensional, nilpotent, stratified Lie algebras and $A:g_1\to g_2$ is a morphism of the graded Lie algebra. I wonder whether there ...

**-2**

votes

**0**answers

84 views

### Lie algebra and Lie groups [on hold]

A complex Lie algebra $L$ has a representation on $Der(L)$ by just putting $x.D=-ad_{Dx}$. For semisimple Lie algebras, by Weyl's theorem, $Der(L)$ decomposes into irreducible subspaces $D_i$. Then my ...

**5**

votes

**0**answers

69 views

### How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...

**-1**

votes

**1**answer

157 views

### Reductive space & Reductive Lie algebra

If $M=G/H$ is a reductive space and $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ be the canonical decomposition, then are $\mathfrak{g}$ or $\mathfrak{h}$ or both reductive lie algebras? (in this case, ...

**2**

votes

**0**answers

122 views

### How to write BRST-BV for dg-Lie?

The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc.
Where is there written a corresponding formula incorporating the differential of
a dg Lie algebra and module?

**1**

vote

**1**answer

113 views

### Explicit generators of $Z(U(\mathfrak{g}))$

Let $\mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field. By Harish-Chandra, the center of its universal enveloping algebra $Z(U(\mathfrak{g}))$ is a polynomial ring and the ...

**2**

votes

**0**answers

104 views

### Free-field representations: how to study highest-weight submodules of the Fock module?

Suppose we have a representation of some affine Lie algebra $\mathfrak{g}=\mathfrak{n}_- \oplus \mathfrak{h} \oplus \mathfrak{n}_+$ on a Fock space $V$. The module $V$ will contain a lot of ...

**1**

vote

**1**answer

145 views

### Is $M=E_{7(7)}/SU(7)\times\mathbb{R}^{+}$ a Kahler-Hodge manifold? (possible open problem)

I have been told that the following is an open problem in mathematics, but I am pretty sure that experts in the topic surely know the answer. The problem is:
Is the manifold
...

**0**

votes

**0**answers

18 views

### How to find the character of $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$? [migrated]

Let $\mathfrak g$ be a Kac-Moody algebra.
Then
$$
\mathfrak{n}_{-}=\oplus_{\alpha\in\varPhi_{+}}\mathfrak{g}_{-\alpha}
$$
and for $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$ the ...

**2**

votes

**0**answers

47 views

### Homology of derivations of Differential Graded Lie algebra

Let $(L,d)$ be a Differential Graded Lie Algebra ($L=\bigoplus L_i$ and $d:L_i \to L_{i-1}$ satisfying the graded Leibniz rule).
On the algebra $\mathrm{Der}L$ of derivations of $L$ define a grading ...

**2**

votes

**0**answers

94 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 ...

**1**

vote

**0**answers

131 views

### rational representation of semisimple algebraic group

Let $G$ be a connected semisimple algebraic group defined over $\mathbb Q$. Could some expert give me a complete classification of finite dimensional $\mathbb Q$-irreducible representations of $G$?
...

**0**

votes

**0**answers

85 views

### List of irreducible representations whose weights are in a single Weyl group orbit

Let $\mathfrak g$ be a (finite dimensional) simple (nonabelian) Lie algebra
over $\mathbb C$. I need a complete list of irreducible (nontrivial) representations $V$
of $\mathfrak g$ such that the Weyl ...

**0**

votes

**0**answers

21 views

### problem with the notation for derivation of contact lie algebra

For contact Lie algebras $K$ over field of characteristic $p>2$ we know that contact lie algebra is the derived lie algebra of the image of contact operator which is defined as $D_K:A(n,m) \mapsto ...

**3**

votes

**3**answers

368 views

### classifying space and cohomology of integer general linear group

I have obtained that the classifying space
$$
BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty)
$$
is the Grassmannian.
I have also obtained that the mod 2 cohomology is the polynomial ...

**5**

votes

**0**answers

114 views

### LS paths construction

Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on ...

**8**

votes

**0**answers

106 views

### Comparing a Chevalley basis with the canonical basis of the adjoint module?

First some background: Given a simple Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, fix a Cartan decomposition $\mathfrak{g} = \mathfrak{h} ...

**3**

votes

**0**answers

165 views

### Bruhat decomposition of $G/Q$

Let $G$ be a semisimple algebraic group over $\mathbb C$, $T$ be a maximal torus and $B$ be a Borel subgroup of $G$ containing $T$. Let $R^+$ be the set of positive roots with respect to $B$. Let $Q$ ...

**0**

votes

**0**answers

41 views

### format of grading Witt Lie Algebra

Let $W(n,m)$ be generalized Jacobson-Witt algebra over a field of characteristic p>3, according to the grading of $W(n,m)$ , we know that it inherit the grading from $A(n,m)$ as follows: ...

**0**

votes

**0**answers

62 views

### An embedding of modules by tensor product over a Noetherian domain

I have a problem on Ring theory. I would like to prove or disprove the following statement:
Let $R$ be a Noetherian domain. Then by the Goldie theorem $R$ have $Q$ as a full ring of quotients and $Q$ ...

**1**

vote

**0**answers

186 views

### A Lie algebra associated with a one dimensional foliation

A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras ...

**3**

votes

**1**answer

92 views

### Reference for exercises with solutions for affine Lie algebras

I am doing self-study of affine Lie algebras, and while I have all recommended reference material, I find it hard to prepare for the exam.
It was quite easy to study finite-dimensional simple Lie ...

**3**

votes

**0**answers

101 views

### Invariant Laurent polynomials under cyclic group action

Start with the cyclic group $G:=\mathbb{Z}/p$ of prime order $p$ and and an integer lattice $P:=\mathbb{Z}^p$. Let $G$ act on $P$ by cyclic permutation of coordinates. There is an induced action on ...

**-1**

votes

**1**answer

115 views

### cartan killing metric [closed]

I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant ...

**4**

votes

**2**answers

203 views

### An algorithm to compare two representations of a simple Lie algebra?

I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis.
the first one is the adjoint ...

**2**

votes

**0**answers

45 views

### Computing maximal ideals of a Lie algebra

Would you know an algorithm (or an automatic method) that computes all maximal ideals $J$ of a given Lie algebra?
Or an algorithm that computes all maximal ideals $J$ containing a given minimal ideal ...

**2**

votes

**0**answers

187 views

### The geometry of the holomorph of a Lie group

Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)
Is Hol$(G)$ always a Lie group?
If the answer is yes our main questions:
1.For a left ...

**2**

votes

**3**answers

294 views

### What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...

**1**

vote

**1**answer

98 views

### Commutators of Schur polynomials of Lie algebra elements

Question:
Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...

**1**

vote

**0**answers

81 views

### representation of $SO(p)\times SO(q)$ with $p,q$ odd

Assume $p,q$ odd. We denote by $\sigma_p$ the standard representation of $SO(p)$, that is the representation of $SO(p)$ acting on $\mathbf{R}^p$ as matrix. So is $\sigma_q$.
Take $K=SO(p)\times ...

**5**

votes

**2**answers

234 views

### Convention about “long” roots for simple Lie algebras of types ADE?

The classification of simple Lie algebras (over $\mathbb{C}$ or other sufficiently large field of characteristic 0) correlates these Lie algebras with the irreducible reduced root systems (in ...

**2**

votes

**1**answer

191 views

### highest weight representations inside tensor product

Let $G$ be a semisimple simply connected group over an algebraically closed field $k$ of characteristic zero, $B$ a Borel and $T$ a maximal torus.
Let $\lambda,\mu,\nu$ be dominant characters of $T$.
...

**4**

votes

**0**answers

273 views

### A vector space associated with a vector field on a symplectic manifold

Let $(M,\omega)$ be a $2n$ dimensional symplectic manifold and $X$ is a smooth vector field on $M$. Consider the following subvector space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in ...

**5**

votes

**1**answer

183 views

### Closure order on nilpotent orbits in exceptional Lie algebras

Let $G$ be a simple algebraic group over the algebraically closed field $k$ of positive characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent ...

**4**

votes

**0**answers

115 views

### Finite-codimension subalgebras of generalized Kac-Moody lie algebras

Do generalized Kac-Moody lie algebras of infinite dimension contain subalgebras of finite codimension? If so, is there a classification?

**0**

votes

**0**answers

37 views

### References about reality of minimal affinizations of quantum affine algebras

Let $U_q(\widehat{\mathfrak{g}})$ be the quantum affine algebra associated to a complex simple Lie algebra $\mathfrak{g}$. A simple module $M$ of $U_q(\widehat{\mathfrak{g}})$ is called real if $M ...

**0**

votes

**0**answers

99 views

### Connection between Lie algebras and fusion rings

Example: Take the irreps of $SU(2)$: $0,1/2,1,...$ (Spin notation.) The quantum dimensions are $1,q+1/q,q^2+q+1+1/q+1/q^2,...$. At $q=(-1)^{1/5}$ this evaluates to $1,\phi,0,...$ and you get the ...

**7**

votes

**3**answers

242 views

### Polarizations generate the ring of invariants?

The symmetric group $S_n$ acts on $\mathbb R^n$ by permuting the coordinates and the ring of polynomial invariants is generated by the elementary symmetric polynomials. If we restrict the action to ...

**3**

votes

**1**answer

219 views

### Weyl groups of $E_6$ and $E_7$

The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...

**4**

votes

**1**answer

143 views

### Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky

The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, ...

**1**

vote

**0**answers

79 views

### Can we find structure constants of Lie Algebra for Lie Symmetry of ODE without solving determining equations?

Let's consider (for example) one scalar ODE.
We are searching for Lie Symmetries of it.
There is well-known result, that we can find size of Symmetry Group without solving determining equations.
...

**3**

votes

**1**answer

90 views

### Extension of an involutive automorphism

Suppose that $g$ is a complex semi-simple Lie algebra and $g'$ its reductive subalgebra.
If $\tau$ is an involutive automorphism of $g'$, can $\tau$ be extended to an involutive automorphism of $g$ ...

**2**

votes

**1**answer

105 views

### Link between Virasoro algebra and Heisenberg algebra

I'm reading these notes on infinite-dimensional Lie algebras.
On page 5, author defines Heisenberg algebra and shows that certain infinite sums of elements of Heisenberg algebra (I'm being a little ...

**0**

votes

**1**answer

125 views

### Inequality for the index of a Lie algebra using its Levi decomposition

The answers to this question indicate that there is a fairly vast literature on the index of Lie algebras. Unfortunately I was not able to find in this literature (or maybe to extract from it) an ...

**2**

votes

**2**answers

237 views

### Closure relations between Bruhat cells on the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...

**2**

votes

**1**answer

77 views

### Conjugacy class of semisimple elements

Let $G$ be a complex simple Lie group with Lie algebra $\mathfrak g$ and $G_1$ be a complex simple subgroup of $G$ with Lie algebra $\mathfrak g_1$. We may assume $G$ and $G_1$ to be of adjoint type. ...