Questions tagged [lie-algebras]
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 groups and differentiable manifolds.
758
questions with no upvoted or accepted answers
8
votes
0
answers
244
views
Induction from the Borel subalgebra to BGG category $\mathcal{O}$
Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with a choice of Cartan subalgebra $\mathfrak{h}$, Borel $\mathfrak{b}$, and nilpotent radical $\mathfrak{n}$. Let $\mathcal{O}...
- 2,450
8
votes
0
answers
260
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 ...
- 141
8
votes
0
answers
223
views
Root-theoretic formulation of characteristic polynomial
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra of rank $n$ over $\mathbb{C}$. Let $G$ denote the corresponding simple simply connected algebraic group. By Chevalley's Theorem, $\mathbb{...
- 2,681
8
votes
0
answers
193
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?
- 66.8k
8
votes
0
answers
562
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 ...
- 301
8
votes
0
answers
1k
views
Computational complexity of multiplication in a nilpotent group?
What is the computational complexity of multiplication in a Carnot group ?
Background: A Carnot group is a real nilpotent Lie group $N$ whose Lie algebra $Lie(N)$ admits a direct sum decomposition
...
- 765
7
votes
0
answers
193
views
Chevalley-Solomon formula and Weyl character formula
Let $\Phi\subset V$ be a root system of rank $r$ with Weyl group $W$, a choice of positive roots $\Phi_+$ and exponents $d_1, \ldots, d_r$ (i.e. the invariant algebra $(\operatorname{Sym}^\bullet V)^W$...
- 3,024
7
votes
0
answers
725
views
What's the point of geometric representation theory?
Please forgive the provocative title, what I mean is the following:
One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
- 157
7
votes
0
answers
167
views
Constructive theory of Lie algebras
I'm looking for references on constructive Lie algebra theory, e.g. the sort of theory you could develop in Martin-Löf type theory or internal to some topos with a NNO. Obviously excluded middle is ...
- 5,933
7
votes
0
answers
681
views
Is this construction related to the geometric Langlands program perhaps?
Given a complex Lie algebra $\mathfrak{g}$, a choice of Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ and a dominant integral weight $\lambda$ of $\mathfrak{g}$, there is a natural construction ...
- 5,011
7
votes
0
answers
181
views
Combinatorial bases of simple Lie algebras
The Lie algebra $\mathfrak{so}_3$ has a basis $x_1,x_2,x_3$ with the multiplication table $[x_1,x_2]=x_3$, $[x_2,x_3]=x_1$, $[x_3,x_1]=x_2$. Moreover there is an isomorphism $\mathfrak{so}_3(\mathbb C)...
- 17.5k
7
votes
0
answers
133
views
Learning roadmap for admissible representations of $\widehat{\mathfrak{g}}$ (affine Lie algebras)
Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over $\mathbf{C}$. A priori one might expect the representation theory of the affine Lie algebra $\widehat{\mathfrak{g}}$ (the Lie ...
- 5,565
7
votes
0
answers
291
views
Associativity of the Campbell-Baker-Hausdorff operation on a Banach-Lie algebra
Let $(\mathfrak{g}, [\cdot,\cdot]_\mathfrak{g}, \Vert \cdot \Vert_\mathfrak{g})$ be an infinite-dimensional Banach-Lie algebra, and let us define for any $a,b \in \mathfrak{g}$ the series
$$~ Z^\...
- 171
7
votes
0
answers
215
views
Strange formula for the dimension of a certain space of noncommutative polynomials
Consider a vector space $V_r(n)$ spanned by (noncommutative) monomials in variables $x_1,\ldots,x_r$
$$
x_{1}^{n_1}x_{2}^{n_2}\ldots x_{r}^{n_r}
$$
of total degree $n.$ Inside this space consider a ...
- 4,296
7
votes
0
answers
197
views
On the center of Koszul Lie algebras
The short question is the following: If a positively graded Lie algebra $\mathfrak g$ over a field $F$ is Koszul, is the center of $\mathfrak g$ concentrated in degree $1$?
Let us be more precise. A ...
7
votes
0
answers
120
views
Softwares to determine semi-simple types of Lie algebras generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of matrices
I wish to determine the type of a Lie algebra generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of square matrices with irrational elements. For example,
\begin{align}
n^+ =
\begin{pmatrix}
...
- 535
7
votes
0
answers
216
views
Demazure modules and dimension of weight spaces
Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra, $w \in W$ an element of the Weyl group, and $\lambda$ an integral dominant weight with $V(\lambda)$ the associated irreducible highest weight ...
- 495
7
votes
0
answers
334
views
Why are fundamental weights denoted by omega?
In my field (and many others, I believe) the absolutely standard notation for the fundamental weights of a root system is lowercase omega: $\omega$. Recently I was taken aback to receive a copyedited ...
- 3,493
7
votes
0
answers
136
views
When is an algebra derived indecomposable?
Call a finite dimensional (acyclic) quiver $K$-algebra A derived indecomposable in case $A$ is not derived equivalent to an algebra of the form $B \otimes_K C$.
For example when the number of simples ...
- 26.1k
7
votes
0
answers
174
views
Regarding $F_4$ and $G_2$ Lie algebras, do there exist $F_n$ or $G_n$ families of Lie algebras?
For example, $E_6$ exceptional Lie algebra is part of the $E_n$ series of Lie algebras (Kac-Moody algebras). Are $F_4$ or $G_2$ maybe also parts of some $F_n$ or $G_n$ series of Lie algebras or are ...
7
votes
0
answers
236
views
$X$ with $H^*(X)=$affine Verma module?
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbf{C}$, and $\widehat{\mathfrak{g}}_\kappa$ the associated affine Lie algebra. It is the central extension of the loop algebra $...
- 5,565
7
votes
0
answers
754
views
Serre presentations over $\mathbb{Z}$
Given a Cartan matrix $A=(a_{ij})_{i,j\in I}$, a classical result of J.-P. Serre asserts that the complex semisimple Lie algebra $\mathfrak g=\mathfrak g(A)$ corresponding to $A$ admits a presentation ...
- 563
7
votes
0
answers
218
views
Universal Enveloping algebra of a L$_\infty$ algebra
In their paper Strongly homotopy Lie algebras, Lada and Markl first show, that there is a symmetrization functor $(-)_L:\mathcal{A}(m)\rightarrow \mathcal{L}(m)$ from the category of $A(m)$-algebras ...
- 183
7
votes
0
answers
132
views
Formal Group Laws in a lined topos
I am aware of the following: in the context of synthetic differential geometry (SDG) one obtains a Lie algebra by exponentiating a microlinear group by a standard infinitesimal object and taking the ...
7
votes
0
answers
711
views
Hochschild cohomology of a universal enveloping algebra of a Lie algebra
I was told that the following equation is true:
Given a finitely generated Lie algebra $\mathfrak g$, there is a Gerstenhaber algebra isomorphism
$$ HH(U\mathfrak g) \cong HH(\wedge^* \mathfrak g^\vee,...
- 323
7
votes
0
answers
170
views
Relationship between R-matrix and Casimir element?
Given a simple Lie algebra $\mathfrak{g}$, is there any relation between its Casimir element and the $R$-matrix of the related Yangian $Y(\mathfrak{g})$?
- 671
7
votes
0
answers
484
views
Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space
Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...
- 1,890
7
votes
0
answers
165
views
"Non standard" formulas for eigenspaces in $V_\rho$
In the context of the Simple Lie Algebras Representations, let $\rho$ be half-the-sum of the positive roots and let $V_\rho$ be the irreducible representation of highest weight $\rho$.
Let$\mu$ be a ...
- 233
7
votes
0
answers
194
views
Structure constants of Lyndon-Shirshov basis of the free Lie ring
Let $X$ be an alphabet, ${\sf Lyn}$ be the set of Lyndon words on $X$ and $L$ be the free Lie ring on $X.$ For $w\in {\sf Lyn}$ we denote by $[w]$ the corresponding element of the Lyndon-Shirshov ...
- 1,312
7
votes
0
answers
132
views
Symmetric pairs of holomorphic type
Let $G$ be a real simple Lie group of Hermitian type; that is, $G/K$ carries a structure of a Hermitian symmetric space where $K$ is a maximal compact subgroup of $G$. Equivalently, the center $Z(\...
- 831
7
votes
0
answers
420
views
Reconstructing a nilpotent Lie algebra from its cohomology with $A_{\infty}$-structure
Let $L$ be a nilpotent Lie algebra (over a field of char 0) and $CE^{\bullet}(L)$ be its Chevalley-Eilenberg dg-algebra. By homotopy transfer, there exists a structure of an $A_{\infty}$-algebra on ...
- 1,233
7
votes
0
answers
551
views
Is there an E8 symmetry in the zero-field Ising model?
In the paper On classification of modular tensor categories by Rowell, Stong and Wang, they list the Ising modular category $I$ as having 3 objects $1$, $\sigma$ and $\psi$, with fusion rules $\sigma^...
- 1,821
7
votes
0
answers
163
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 ...
- 8,707
7
votes
0
answers
904
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 $\...
- 1,040
7
votes
0
answers
310
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 ...
- 71
7
votes
0
answers
181
views
Deformation of Noether's first theorem
Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...
- 3,850
7
votes
0
answers
394
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 ...
- 3,938
7
votes
0
answers
509
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 $c_1,...
- 2,149
7
votes
0
answers
1k
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, ...
- 1,447
6
votes
0
answers
236
views
Torsion in the Lie algebra cohomology of gl(n,Z)
What is known about the Lie algebra cohomology $H^*(\mathfrak{gl}_n(\mathbb{Z}),\mathbb{Z})$? After passing to $\mathbb{Q}$-coefficients, the question is classical: $H^*(\mathfrak{gl}_n(\mathbb{Q}),\...
- 4,467
6
votes
0
answers
251
views
Tits construction of algebraic groups of type D₆ and E₇ via C₃
As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input
an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
- 1,419
6
votes
0
answers
337
views
Asymptotically nilpotent Lie sets of matrices
A matrix $A\in\textbf{Mat}_n(\mathbb{R})$ is called asymptotically nilpotent if for each vector $v$, ${\lim}_{k\to\infty}A^k(v) = 0$.
Question 1. Assume that $\mathcal{A}$ is the subset of $\textbf{...
- 291
6
votes
0
answers
148
views
Sections of $\mathcal{L}_{\lambda}$ on intersections of open cover on a flag variety
Let $G$ be a reductive complex algebraic group, $P$ a parabolic subgroup, $\mathbb{C}_{\lambda}$ a one-dimensional representation of $P$ and $\mathcal{L}_{\lambda}$ the corresponding line bundle on $G/...
- 1,025
6
votes
0
answers
97
views
Explicit formula for star product on the symmetric algebra of a Lie algebra via standard ordering
There is a well known vector space isomorphism $\phi:\mathcal{S}(\frak{g})\rightarrow U(\frak{g})$ given by the symmetrization (or Weyl ordering), i.e.
$$ \phi(t_{i_1}\dots t_{i_k})=\frac{1}{k!}\sum_{\...
- 175
6
votes
0
answers
106
views
$S_n$-invariant polynomials on the dual of reflection representation
Let $W=S_n$ (the symmetric group) acting on $V=k^n$ via permutation of the indices, where $k$ is an algebraically closed field. Closely related to this are the reflection (sub-)representation $V_0=\{ (...
- 1,184
6
votes
0
answers
156
views
$\mathfrak{sl}_2(\mathbb{Z})$'s properties as a Lie algebra over a ring
I was wondering what was known about $\mathfrak{sl}_2(\mathbb{Z})$ as a Lie algebra; in particular, what is known about its representation theory? I know of some texts which treat Lie algebras ...
- 61
6
votes
0
answers
370
views
Group-like elements of universal enveloping algebra
Suppose $\mathfrak{g}$ is a finite-dimensional Lie algebra over $\mathbb C$. Take $A=U(\mathfrak g[[t]])$, a universal enveloping algebra of $\mathfrak g[[t]]$ over $\mathbb C[[t]]$.
Then we may ...
- 471
6
votes
0
answers
185
views
Word/Loop in $L(\Lambda)$
Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, with Chevalley generators $e_i,f_i$ ($i=1,...,n$).
Let $L(\Lambda)$ denote the irreducible module with highest weight $\Lambda$. Let $v$ denote ...
- 688
6
votes
0
answers
170
views
Equivalence between $\mathcal{D}_\lambda$ modules and $\mathcal{D}_{0}$ modules
Fix $G$ a finite dimensional reductive group and $\lambda$ a weight. Apparently the category of $\mathcal{D}_\lambda$ modules on $G/B$ is equivalent to the category of $\mathcal{D}_0$ modules on $G'/B'...
- 5,565
6
votes
0
answers
185
views
Eigenvalues of spherical function on $\mathrm{SL}(2,\mathbb{R})$
Lie algebraically, the eigenvalue of the spherical function
\begin{align*}
\phi_{\lambda}(g)=\int_{K} e^{(i \lambda+\rho)(A(k g))} \mathrm{d} k \quad (g \in G,\,\lambda\in\mathfrak{a}^*)
\end{align*}
...
- 83