We give upper bounds for the number of irreducible representations of dimension at most n for a compact semisimple Lie group. In particular, we prove that there are at most n irreducible representations of dimension at most n for a simple compact Lie group. We use this prove that the number of conjugacy classes of maximal subgroups of a compact simple Lie group is O(r) where r is the Lie rank. We also give a short proof that the dimensions of the weight spaces of a maximal torus are small relative to the dimension of an irreducible module.
The paper studies the minimum polynomial degrees of $p$-elements in cross-characteristic representations of simple groups of exceptional Lie type whose BN-pair rank is at most 2. Specifically, we prove that the degree in question equals the order of the element.
We characterize finite-dimensional thick representations over ${\Bbb C}$ of connected complex semi-simple Lie groups by irreducible representations which are weight multiplicity-free and whose weight posets are totally ordered sets. Moreover, using this characterization, we give the classification of thick representations over ${\Bbb C}$ of connected complex simple Lie groups.
We use the language of Lie pseudoalgebras to gain information about the representation theory of the simple infinite-dimensional linearly compact Lie superalgebra of exceptional type $E(5,10)$. This technology allows us to prove that the degree of singular vectors in minimal Verma modules is $\leq 14$. A few technical adjustments allow us to refine the bound, proving that the degree must always be $\leq 12$ and it is actually, except for a finite number of cases, $\leq 10$.
For any prime $p$ and $S$ a $p$-group isomorphic to a Sylow $p$-subgroup of a rank $2$ simple group of Lie type in characteristic $p$, we determine all saturated fusion systems supported on $S$ up to isomorphism.
For an arbitrary representation $ρ$ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan-Kronecker invariants of $ρ$. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of $ρ$. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.
We prove that the tensor product of a simple and a finite dimensional $\mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $\mathfrak{q}(n)$-supermodules to that of simple $\mathfrak{sl}_n$-modules. Rough structure of simple $\mathfrak{q}(n)$-supermodules, considered as $\mathfrak{sl}_n$-modules, is described in terms of the combinatorics of category $\mathcal{O}$.
Let $(\mathfrak{g},[p])$ be a restricted Lie algebra over an algebraically closed field $k$ of characteristic $p\!\ge \!3$. Motivated by the behavior of geometric invariants of the so-called $(\mathfrak{g},[p])$-modules of constant $j$-rank ($j \in \{1,\ldots,p\!-\!1\}$), we study the projective variety $\mathbb{E}(2,\mathfrak{g})$ of two-dimensional elementary abelian subalgebras. If $p\!\ge\!5$, then the topological space $\mathbb{E}(2,\mathfrak{g}/C(\mathfrak{g}))$, associated to the factor algebra of $\mathfrak{g}$ by its center $C(\mathfrak{g})$, is shown to be connected. We give applicat…
We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}^-$. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an $\mathbb{N}$-filtration on any finite dimensional simple $\mathfrak{g}$-module. We prove for type $\tt{A}_n$, $\tt{C}_n…
The deformed current Lie algebra was introduced by the author to study the representation theory of cyclotomic q-Schur algebras at q=1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.
In this paper, the correspondence between the finite dimensional representations of a simple Lie algebra and their characteristic polynomials is established, and a monoid structure on these characteristic polynomials is constructed. Furthermore, the characteristic polynomials of sl(2, C) on some classical simple Lie algebras through adjoint representations are studied, and we present some results of Borel subalgebras and parabolic subalgebras of simple Lie algebras through characteristic polynomials.
In this paper, we describe restricted one-dimensional central extensions of all finite dimensional simple restricted Lie algebras defined over fields of characteristic $p\ge 5$.
In a previous paper by the authors, we obtain the first example of a finitely freely generated simple $\mathbb Z$-graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying $\mathbb Z$-graded simple Lie conformal algebras ${\cal G}=\oplus_{i=-1}^\infty{\cal G}_i$ satisfying the following, (1) ${\cal G}_0\cong{\rm Vir}$, the Virasoro conformal algebra; (2) Each ${\cal G}_i$ for $i\ge-1$ is a ${\rm Vir}$-module of rank one. These algebras includ…
We classify all simple bounded highest weight modules of a basic classical Lie superalgebra $\mathfrak g$. In particular, our classification leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of $\mathfrak g$.
In this paper, we study a class of infinite simple Lie conformal algebras associated to a class of generalized Block type Lie algebras. The central extensions, conformal derivations and free intermediate series modules of this class of Lie conformal algebras are determined. Moreover, we also show that these Lie conformal algebras do not have any non-trivial finite conformal modules. Consequently, these Lie conformal algebras cannot be embedded into $gc_N$ for any positive integer $N$.
Let $W_n^+$ be the Lie algebra of the Lie algebra of vector fields on $\C^n$. In this paper, we classify all simple bounded weight $W_n^+$ modules. Any such module is isomorphic to the simple quotient of a tensor module $F(P,M)=P\otimes M$ for a simple weight module $P$ over the Weyl algebra $K_n^+$ and a finite dimensional simple $\gl_n$ module $M$.
In the recent paper [A14], a geometric realization to minimal representations of simple real Lie groups of non Hermitian type is given, based on the geometric setting introduced in [A11]. We give in this paper a geometric realization to minimal representations of simple real Lie groups of Hermitian type.
The standard way to compute the structure constants of semi-simple Lie algebras involves the additive structure of the roots. In earlier work, I described how ideas of Jacques Tits could be applied to do this by using the structure of the Weyl group. In this work, I explain how to combine some of the earlier ideas with some suggestions of Kottwitz to present a very simple algorithm which comes close to constructing a canonical basis of the Lie algebra. Unfortunately, unlike previous methods, it does not work for arbitrary Kac-Moody algebra.
Based on an idea in [Gan--Savin, Represent. Theory (2005)], we give a classification of minimal representations of connected simple real Lie groups not of type $A$. Actually, we prove that there exist no new minimal representations up to infinitesimal equivalence.
We generalize the results in [2] giving a reduction algorithm allowing to compute the index of seaweed subalgebras of classical simple Lie algebras. We thus are able to obtain the index of some interesting families of seaweed subalgebras and to give new examples of large classes of Frobenius Lie algebras among them.
In this article, we are interested in some subspaces of the exterior algebra of a simple Lie algebra g. In particular, we prove that some graded subspaces of degree d generate the g-module d (g) for some integers d.
Any graded restricted simple Lie algebra of Cartan type contains a subalgebra isomorphic to the Witt algebra over a field of prime characteristic. As some analogue of study on branching rules for restricted non-classical Lie algebras, it is shown that each graded restricted simple Lie algebra of Cartan type can be decomposed into a direct sum of restricted baby Verma modules and simple modules as an adjoint module over the Witt algebra. In particular, the composition factors are precisely determined.
For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional $Q$-graded simple Lie algebras over any algebraically closed field of characteristic $0$ based on the recent classification of gradings on finite dimensional simple Lie algebras. We also reduce classification of simple graded modules over any $Q$-graded Lie algebra (not necessari…
The purpose of this paper is to study representations of simple multiplicative Hom-Lie algebras. First, we provide a new proof using Killing form for characterization theorem of simple Hom-Lie algebras given by Chen and Han, then discuss the representations structure of simple multiplicative Hom-Lie algebras. Moreover, we study weight modules and root space decompositions of simple multiplicative Hom-Lie algebras, characterize weight modules and provide examples of representations of $\mathfrak{sl}_2$-type Hom-Lie algebras.
Given a sequence $\vec d=(d_1,\dots,d_k)$ of natural numbers, we consider the Lie subalgebra $\mathfrak{h}$ of $\mathfrak{gl}(d,\mathbb{F})$, where $d=d_1+\cdots +d_k$ and $\mathbb{F}$ is a field of characteristic 0, generated by two block upper triangular matrices $D$ and $E$ partitioned according to $\vec d$, and study the problem of computing the nilpotency degree $m$ of the nilradical $\mathfrak{n}$ of $\mathfrak{h}$. We obtain a complete answer when $D$ and $E$ belong to a certain family of matrices that arises naturally when attempting to classify the indecomposable modules of certain so…
The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of…
