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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-2
votes
0answers
82 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 ...
6
votes
5answers
552 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 ...
-1
votes
1answer
156 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, ...
5
votes
0answers
68 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 ...
2
votes
0answers
121 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?
17
votes
1answer
1k views

Isometry group of a homogeneous space

Background Let $(M,g)$ be a finite-dimensional riemannian (or more generally pseudoriemannian) manifold. Suppose that I know that a certain Lie group $G$ acts transitively and isometrically on $M$ ...
1
vote
1answer
112 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 ...
1
vote
1answer
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 ...
2
votes
0answers
103 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 ...
0
votes
0answers
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
0answers
44 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
0answers
93 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
0answers
130 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
0answers
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
1answer
160 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 ...
0
votes
0answers
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 ...
2
votes
1answer
143 views

Maurer-Cartan elements of the extension of an $L_{\infty}$-algebra

Let $g$ be a nilpotent $L_{\infty}$-algebra. For every commutative differential graded algebra $A$, one can form the extension $g\otimes A$ and endow it with a nilpotent $L_{\infty}$-algebra ...
3
votes
3answers
364 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 ...
12
votes
0answers
373 views

Lie algebras vs. graph complexes

A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...
5
votes
0answers
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 ...
1
vote
0answers
183 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 ...
8
votes
0answers
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
1answer
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 ...
2
votes
0answers
186 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 ...
3
votes
0answers
164 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
0answers
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
0answers
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 $Q$ ...
16
votes
2answers
1k views

Invariants for the exceptional complex simple Lie algebra $F_4$

This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise. Let $\mathfrak{g}$ denote a complex simple Lie algebra of ...
57
votes
11answers
13k views

why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
3
votes
0answers
100 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 ...
4
votes
2answers
202 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 ...
-1
votes
1answer
114 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 ...
5
votes
2answers
232 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
0answers
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 ...
1
vote
3answers
290 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 ...
4
votes
0answers
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 ...
1
vote
1answer
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
0answers
79 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 ...
2
votes
1answer
190 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 $T$. ...
5
votes
1answer
181 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 ...
6
votes
5answers
1k views

Weight lattice and the fundamental group

Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively. Then, $\frak{t}$ can be considered as a Cartan subalgebra of ...
6
votes
4answers
749 views

Lie algebra $\mathfrak{so}(9)$ as a subalgebra of $\mathfrak{f}_4$

What are the explicit Lie algebra mono-morphisms from $\mathfrak{so}(9)$ into the exceptional Lie algebra $\mathfrak{f}_4$ and from $\mathfrak{f}_4$ to $\mathfrak{e}_6$? Are these explicitly described ...
4
votes
0answers
113 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
0answers
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
0answers
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
3answers
240 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
1answer
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
1answer
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, ...
8
votes
2answers
817 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 ...
3
votes
1answer
87 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 $g$ ...