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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11
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359 views

Source of a formula for tensor product multiplicities?

This is a follow-up to a recent question by Allen Knutson here, involving a special type of tensor product multiplicity for a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (or other ...
10
votes
0answers
309 views

“Special” meanders

One of the open problems in combinatorics is enumeration of meanders. Here on MO I only could find them under the heading not-especially-famous-long-open-problems-which-anyone-can-understand. Since ...
10
votes
0answers
486 views

Combinatorial identity involving the Coxeter numbers of root systems

The setup is: $R$ = irreducible (reduced) root system; $D$ = connected Dynkin diagram of $R$, with nodes numbered $1,2,...,r$; $\hat D$ = extended Dynkin diagram, nodes numbered $0,1,2,...,r$; ...
9
votes
0answers
198 views

Cohomological Proof of Serre Relations for a Symmetrizable Kac-Moody Algebra

In 'Infinite Dimensional Lie Algebras, 3rd edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'. Does anyone know how ...
9
votes
0answers
197 views

differentiating positive energy LG reps

Background:Let $G$ be a cscsc¹ Lie group, and let $\widetilde{LG}$ be the universal central extension (center = $S^1$) of $LG:=Map_{C^\infty}(S^1,G)$, with the topology inherited from the $C^\infty$ ...
8
votes
0answers
254 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 ...
8
votes
0answers
278 views

Explicit change of basis for the Schur-Weyl basis

The Schur-Weyl duality states that $\bigotimes_{m=1}^n \mathbb{C}^k$ can be decomposed as a direct sum over the tensor product of irreductible representations of $SU(k)$ and of the symmetric group ...
7
votes
0answers
394 views

Dual versions of “folding” symmetric ADE Dynkin diagrams?

Start with the Dynkin diagram of an irreducible root system, typically associated with a simple Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE diagrams ...
6
votes
0answers
110 views

Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids

When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie ...
6
votes
0answers
245 views

Injectivity of Lie group exponential function

If $G$ is a (finite-dimensional) Lie group, then the exponential function $\exp\colon\mathfrak{g}\to G$ is injective on some identity neighbourhood. If, moreover, $\mathfrak{g}$ is semi-simple and ...
6
votes
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159 views

The meaning of a “subcomplex” of the Cartan-Eilenberg of a Lie algebra

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^* $ be the dual vector space. We have the standard Cartan-Eilenberg complex $(\wedge^{\cdot} \mathfrak{g}^* ...
6
votes
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236 views

Idempotent non-associative algebras

This is inspired by that question by Andreas Thom. Let $L$ be a finitely generated Lie ring (or Lie algebra over a field) such that $L=[L,L]$, that is the Abelianization of $L$ is 0. Is it true ...
6
votes
0answers
265 views

Reference for the Thick Affine Grassmanian

Let $G$ be a reductive group and $LG$ be the algebraic loop group of $G$; i.e. $LG(k) = G( k((t)) )$. There is a fair amount of literature on the affine Grassmanian $LG(k)/G(k[[t]])$ and its Picard ...
6
votes
0answers
291 views

Kazhdan-Lusztig graph for the Springer fiber of the minimal special unipotent class?

This graph was determined in the case of simply-laced root systems by Igor Dolgachev and Norman Goldstein: "On the Springer resolution of the minimal unipotent conjugacy class" (J. Pure Appl. Algebra ...
6
votes
0answers
433 views

Small sum of group elements acting as rank 1 matrix.

I am interested in constructing small (as possible) group $G$ with large dimensional irreducible representation $\rho,V$ such that exist three elements of $g_1,g_2,g_3\in G$ such that for some ...
6
votes
0answers
550 views

Baker-Campbell-Hausdorff formula: prime divisors of denominators

Consider the Baker-Campbell-Hausdorff formula (Wikipedia page): $$Z(X,Y) := X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + \dots$$ Many sources, including the Wikipedia ...
6
votes
0answers
315 views

Approximating the radical of a Lie algebra by Killing radicals

The following question might be filed under "idle curiosity", but I'm hoping some expert on Lie algebras can answer it. Background One of the fundamental structural results about Lie algebras is the ...
5
votes
0answers
113 views

Generators for invariant tensors of lie algebras

EDITED FOR (hopeful) CLARITY: For a simple Lie algebra $\mathfrak{g}$ (over $\mathbb{C}$) and its adjoint group G, the $G$-invariant polynomials on $\mathfrak{g}$ are linear combinations of products ...
5
votes
0answers
138 views

A question on the resolution of parabolic Verma module $M_I(\lambda)$ in BGG category O

I am reading Humphrey's book "Representations of semisimple Lie algebras in the BGG Category O" on Page 189, Proposition 9.6, where he remarked that "Note that if we had developed the full BGG ...
5
votes
0answers
139 views

On an interesting subalgebra of the functions on the cotangent bundle of the flag variety

Setup Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $p$ where $p = 0$ or $p > 0$ is a good prime for $G$. Fix a Borel subgroup $B \subseteq G$ corresponding to the ...
5
votes
0answers
117 views

Does the normal ordered product on differential operators lift to $U\left(\mathfrak{gl}_n\right)$ ?

Let $n\in\mathbb N$. Let $k$ be a commutative ring. Let $\Omega$ denote the $k$-algebra of polynomial differential operators on $n$ variables $x_1$, $x_2$, ..., $x_n$ over $k$. (The multiplication in ...
5
votes
0answers
569 views

Hodge theory for Lie algebra (co)homology

Let $G$ be a simple Lie group and $P$ its parabolic subgroup such that on the level of Lie algebras we have $\mathfrak{p} = \mathfrak{g}_0 \oplus \mathfrak{n}$. The dual $\mathfrak{n}^{\*}$ is ...
5
votes
0answers
314 views

Lie locally nilpotent associative algebras

Let $A$ be an associative algebra over a field. Then $A$ can be regarded as a Lie algebra via the Lie bracket defined by $[a,b]=ab-ba$ for every $a,b\in A$. The algebra $A$ is called Lie locally ...
5
votes
0answers
304 views

The Hochschild-Serre spectral sequence relative to an ideal containing the derived subalgebra

Is the Hochschild-Serre spectral sequence $$H_\bullet(\mathfrak g/\mathfrak h,H_\bullet(\mathfrak h,k))\Rightarrow H_\bullet(\mathfrak g,k)$$ for an extension of Lie algebras $$0\to\mathfrak ...
5
votes
0answers
447 views

Applications (and source) of Bourbaki exercise on root systems with two root lengths?

In Chapters 4-6 of Bourbaki's Groupes et algebres de Lie, Exercise 20 for Section VI.1 concerns irreducible (reduced) root systems with roots of two lengths: in other words, systems of types $B_\ell, ...
5
votes
0answers
578 views

Category O of Kac-Moody algebra

Category $\mathcal{O}$ of semisimple Lie algebra has been understood very well. One can decompose the category into different blocks by central characters, and evey block is Noetherian and Aritian, ...
4
votes
0answers
90 views

When does finite presentability of the associated graded Lie algebra of a group imply the group is finitely presented?

Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite ...
4
votes
0answers
80 views

Adjoint orbit of two vectors

Let $G$ be a simple compact real Lie group and let $\mathfrak g$ be its Lie algebra. Let $u,v\in \mathfrak g$ be two distinct unit vectors and $H\subset \mathfrak g$ be a hyperplane with normal vector ...
4
votes
0answers
166 views

How to find the unitary matrices in this exponential matrix representation

In the following post Representing a product of matrix exponentials as the exponential of a sum there is a statement regarding the result of the multiplication of two matrix exponentials: if $A$ and ...
4
votes
0answers
157 views

Bracket of lyndon words?

Here is a simple question regarding the standard Lyndon basis for the free Lie Algebra. Suppose I take two lyndon words $m$ and $n$ and their standard bracketings $B(m)$ and $B(n)$ as elements in the ...
4
votes
0answers
138 views

The Killing form on quantized enveloping algebras and reduction to the classical case

Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...
4
votes
0answers
175 views

Is the “Toeplitz algebra” the representation ring of a Hopf algebra related to SU(2)?

More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional ...
4
votes
0answers
216 views

Lie-infinity structure in Lagrangian Floer theory ?

Is there (besides the A-infinity structure) also a L-infinity structure in Lagrangian Floer theory (forming together a G-infinity structure) - like in Hochschild cohomology ?
4
votes
0answers
205 views

$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$

While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...
4
votes
0answers
149 views

Exotic Chains for Group Cohomology of a Complex Lie Group

Related Question: Exotic Chains for Group Homology of a Complex Lie Group Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural ...
4
votes
0answers
195 views

How to decide if two surfaces occurring in Springer theory are isomorphic?

In the study of a simple algebraic group (say over $\mathbb{C}$) and related geometry of its flag variety associated with the Springer correspondence, one encounters pairs of surfaces which have some ...
4
votes
0answers
248 views

What is the dual bialgebra structure in this special case?

Hi, I would like to study a special case of Lie bialgebras. Let $(\mathcal{G},<,>)$ a Lie algebra endowed with a scalar product $<,>$ such that $$\mathcal{G}=S\oplus D(\mathcal{G}),$$ ...
4
votes
0answers
304 views

about ℓ-adic and Perverse Stuff and ℓ-adic cohomology with compact support

this question is trivial. We know from this paper link text, Springer constructed rep of the Weyl group $W$ on the cohomology of the Springer fibre. Also, Deligne-Lusztig constructed the linear rep ...
4
votes
0answers
224 views

Norm in the fundamental representations of Lie algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra, $\omega_0$ a dominant weight, $\rho$ a fundamental representation with highest weight $\Lambda$. Fix some weight $w$ in this representation. Let ...
4
votes
0answers
278 views

Working with quadratic Lie algebras

A quadratic Lie algebra is a Lie algebra with an invariant inner product and the main examples are semisimple Lie algebras. This definition then makes sense in any linear symmetric monoidal category. ...
3
votes
0answers
51 views

Explicit description of graded (counital) cofree cocommutative coalgebras

Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$. Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...
3
votes
0answers
302 views

Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation

Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$? Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...
3
votes
0answers
155 views

What is a pure algebraic interpretation for this dynamical property?

According to comments of Yves Cornulier to the previous version of this question I revise the question as follows: To what extent the following types of Lie algebras $A$ are classified? And what is ...
3
votes
0answers
118 views

Examples of divisible Lie algebras

We say that a nonzero Lie algebra $L$ is divisible, if for all elements $a$ and $b$ with $a\neq 0$, there exists $x\in L$ such that $[a, x]=b$. What are examples of divisible Lie algebras?
3
votes
0answers
138 views

Homotopes of simple Lie algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra with bracket $[x,y]$. For which $z\in \mathfrak{g}$ defines $$ \mu(x,y)=ad (z)([x,y])=[z,[x,y]] $$ another Lie bracket on the same vector space ? For ...
3
votes
0answers
218 views

Reductive Lie Groups and Complexification

Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, ...
3
votes
0answers
248 views

The normalizer a maximal compact subgroup of a semi-simple Lie group

Let $G$ be a semi-simple real Lie group such that $|\pi_0(G)|<\infty$ and let $K$ be a maximal compact subgroup of $G$. Q1: How does one prove that $N_G(K)=K$? So I know a nice (and low-tech) ...
3
votes
0answers
113 views

Number of submodules in $\wedge^2 V$ and $S^2V$ isomorphic to $\mathfrak{g}$

Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $\mathfrak{g}\subset\mathfrak{so}(V)$ be an orthogonal irreducible representation. It can be shown that the number of ...
3
votes
0answers
147 views

Generators and relations for the enveloping algebra of a unipotent radical

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0 and let $B \subseteq G$ be a Borel subgroup with unipotent radical $U$. Let $P \supseteq B$ be a ...
3
votes
0answers
462 views

Commutator Baker-Campbell-Hausdorff formula

Consider the Baker-Campbell-Hausdorff formula $\Phi(X,Y)\in\mathbb{Q}\langle\langle X,Y\rangle\rangle$ in non-commutative variables. Define $X*Y:=\Phi(X,Y)$ and $[X,Y]=(-X)*(-Y)*X*Y$, and then as ...