Nilpotent Varieties Research Articles

On the computation of the nilpotent pieces in bad characteristic for algebraic groups of type G2, F4, and E6

Let G be a connected reductive algebraic group over an algebraically closed field k, and let Lie(G) be its associated Lie algebra. In his series of papers on unipotent elements in small characteristic, Lusztig defined a partition of the unipotent variety of G. This partition is very useful when working with representations of G. Equivalently, one can consider certain subsets of the nilpotent variety of g called pieces. This approach appears in Lusztig's article from 2011. The pieces for the exceptional groups of type G2,F4,E6,E7, and E8 in bad characteristic have not yet been determined. This article presents a solution, relying on computational techniques, to this problem for groups of type G2, F4, and E6.

Asymptotic semigroups and two-sided weak orders

Various partial orders related to the structures of dual canonical monoids are investigated. It is shown that the nilpotent variety of a dual canonical monoid is equidimensional; its dimension is found. It is shown in type A that certain intervals of the Putcha poset of a dual canonical monoid are isomorphic to the Renner monoids of matrices. The notion of a two-sided weak order on a normal reductive monoid is introduced. A criterion, in terms of type maps, for the covering relations in a two-sided weak order to have degree 2 is found. It is shown that, for the unique equivariant divisor of a dual canonical monoid (the asymptotic semigroup), the covering relations of the two-sided weak order are always of degree 1. These computations provide new insights for the two-sided weak orders on Coxeter groups. In type A, some enumerative results for the covering relations are presented.

Nilpotent varieties in symmetric spaces and twisted affine Schubert varieties

We relate the geometry of Schubert varieties in twisted affine Grassmannian and the nilpotent varieties in symmetric spaces. This extends some results of Achar–Henderson in the twisted setting. We also get some applications to the geometry of the order 2 nilpotent varieties in certain classical symmetric spaces.

Microlocal characterization of Lusztig sheaves for affine quivers and 𝑔-loops quivers

We prove that for extended Dynkin quivers, simple perverse sheaves in Lusztig category are characterized by the nilpotency of their singular support. This proves a conjecture of Lusztig in the case of affine quivers. For cyclic quivers, we prove a similar result for a larger nilpotent variety and a larger class of perverse sheaves. We formulate conjectures concerning similar results for quivers with loops, for which we have to use the appropriate notion of nilpotent variety, due to Bozec, Schiffmann and Vasserot. We prove our conjecture for g g -loops quivers ( g ≥ 2 g\geq 2 ).

Nilpotent varieties and metabelian varieties

We deal with varieties of nonassociative algebras having polynomial growth of codimensions. We describe some results obtained in recent years in the class of left nilpotent algebras of index two. Recently the authors established a correspondence between the growth rates for left nilpotent algebras of index two and the growth rates for commutative or anticommutative metabelian algebras that allows to transfer the results concerning varieties of left nilpotent algebras of index two to varieties of commutative or anticommutative metabelian algebras.

PBW parametrizations and generalized preprojective algebras

Geiß-Leclerc-Schröer (2017) [24] has introduced a notion of generalized preprojective algebras associated with generalized Cartan matrices and their symmetrizers. These algebras realize crystal structures on the set of maximal dimensional irreducible components of the nilpotent varieties (Geiss et al. 2018, [26]). For general finite types, we give stratifications of these components via partial orders of torsion classes in module categories of generalized preprojective algebras in terms of Weyl groups. In addition, we realize Mirković-Vilonen polytopes from generic modules of these components, and give an identification as crystals between the set of Mirković-Vilonen polytopes and the set of maximal dimensional irreducible components. This generalizes results of Baumann-Kamnitzer (2012) [8] and Baumann-Kamnitzer-Tingley (2014) [10].

Nilpotency of Alternative and Jordan Algebras

We study the polynomial identities of alternative and Jordan algebras which imply the nilpotency of the algebras. In the case of a field of characteristic 0 we describe all these systems of identities as well as all almost nilpotent varieties of alternative and Jordan algebras. In particular, we establish a connection between the Engel identity for the Lie algebras and the standard identity for the Jordan algebras.

Correspondence between some metabelian varieties and left nilpotent varieties

In the class of left nilpotent algebras of index two it was proved that there are no varieties of fractional polynomial growth ≈ n α with 1 < α < 2 and 2 < α < 3 instead it was established the existence of a variety of fractional polynomial growth with α = 7 2 . In this paper we investigate similar problems for varieties of commutative or anticommutative metabelian algebras. We construct a correspondence between left nilpotent algebras of index two and commutative metabelian algebras or anticommutative metabelian algebras and we prove that the codimensions sequences of the corresponding algebras coincide up to a constant. This allows us to transfer the above results concerning varieties of left nilpotent algebras of index two to varieties of commutative or anticommutative metabelian algebras.

IA-automorphisms of relatively free nilpotent torsion-free groups and Lie algebras

For a free metabelian and nilpotent group G of finite rank n and class c, with and we show that for all and we extend this result to several torsion-free nilpotent varieties of groups. By our methods, which are based on a connection of automorphisms of groups with automorphisms of Lie algebras, we also show the analogous result for several relatively free nilpotent Lie algebras.

On the number of points of nilpotent quiver varieties over finite fields

We give a closed expression for the number of points over finite fields (or the motive) of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has 1-loops or oriented cycles, there are several possible variants of the Lusztig nilpotent variety, and we provide formulas for the point count of each. This involves nilpotent versions of the Kac A-polynomial, which we introduce and for which we give a closed formula similar to Hua's formula for the usual Kac A-polynomial. Finally we compute the number of points over a finite field of the various stratas of the Lusztig nilpotent variety involved in the geometric realization of the crystal graph.

Dn Dynkin quiver moduli spaces

We study 3d quiver gauge theories with gauge nodes forming a Dn Dynkin diagram and their relation to nilpotent varieties in . The class of good Dn Dynkin quivers is completely characterised and the moduli space singularity structure fully determined for all such theories. The class of good Dn Dynkin quivers is denoted where is an integer, and are integer partitions and denotes membership of one of two broad subclasses. Small subclasses of these quivers are known to realise some nilpotent varieties with their moduli space branches. We fully determine which nilpotent varieties are realisable as Dn Dynkin quiver moduli spaces and which are not. Quiver addition is introduced and is used to give large subclasses of Dn Dynkin quivers poset structure. The partial ordering is determined by inclusion relations for the moduli space branches. The resulting Hasse diagrams are used to both classify Dn Dynkin quivers and determine the moduli space singularity structure for an arbitrary good theory. The poset constructions and local moduli space analyses are complemented throughout by explicit checks utilising moduli space dimension matching.

The nullcone of the Lie algebra of [formula omitted

This paper investigates the nilpotent conjugacy classes of the Lie algebra of the simple algebraic group of type G2. These classes are determined by first finding the stratification, and then finding the classes within the strata. Except for characteristic 3, the classes coincide with the strata. In characteristic 3, one stratum splits into two orbits. If the characteristic differs from 2 and 3, the classes are determined by the singularities of the nilpotent variety. In characteristic 3, the matter is undecided yet. In characteristic 2, different classes have the same singularities.

The nilpotent variety of W(1;n)p is irreducible

In the late 1980s, Premet conjectured that the nilpotent variety of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic [Formula: see text] is irreducible. This conjecture remains open, but it is known to hold for a large class of simple restricted Lie algebras, e.g. for Lie algebras of connected algebraic groups, and for Cartan series [Formula: see text] and [Formula: see text]. In this paper, with the assumption that [Formula: see text], we confirm this conjecture for the minimal [Formula: see text]-envelope [Formula: see text] of the Zassenhaus algebra [Formula: see text] for all [Formula: see text].

Moduli space singularities for 3d mathcal{N}=4 circular quiver gauge theories

The singularity structure of the Coulomb and Higgs branches of good 3d mathcal{N}=4 circular quiver gauge theories (CQGTs) with unitary gauge groups is studied. The central method employed is the Kraft-Procesi transition. CQGTs are described as a generalisation of a class of linear quivers. This class degenerates into the familiar class Tρσ(SU(N)) in the linear case, however the circular case does not have the degeneracy and so the class of CQGTs contains many more theories and much more structure. We describe a collection of good, unitary, CQGTs from which the entire class can be found using Kraft-Procesi transitions. The singularity structure of a general member of this collection is fully determined, encompassing the singularity structure of a generic CQGT. Higher-level Hasse diagrams are introduced in order to write the results compactly. In higher-level Hasse diagrams, single nodes represent lattices of nilpotent orbit Hasse diagrams and edges represent traversing structure between lattices. The results generalise the case of linear quiver moduli spaces which are known to be nilpotent varieties of mathfrak{s}{mathfrak{l}}_n .

On the Varieties of Commutative Metabelian Algebras

The paper presents new results on varieties of commutative metabelian algebras over a field of zero characteristic. We study the structure of the multilinear part of the variety of all commutative metabelian algebras as a module of the symmetric group. Two almost nilpotent varieties are introduced and studied in this class of algebras. We prove the nonexistence of other almost nilpotent commutative metabelian varieties of subexponential growth.

Equations for Some Nilpotent Varieties

AbstractLet ${\mathcal{O}}$ be a Richardson nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ of rank $n$ over $\mathbb C$, induced from a Levi subalgebra whose $s$ simple roots are orthogonal, short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $\overline{\mathcal{O}}$ in $S \mathfrak{g}^{\ast }$. In such cases, the ideal is generated by bases of either one or two copies of the representation whose highest weight is the dominant short root, along with $n-s$ fundamental invariants of $S \mathfrak{g}^{\ast }$. This extends Broer’s result for the subregular nilpotent orbit, which is the case of $s=1$. Along the way we give another proof of Broer’s result that $\overline{\mathcal{O}}$ is normal. We also prove a result relating a property of the invariants of $S \mathfrak{g}^{\ast }$ to the following question: when does a copy of the adjoint representation in $S \mathfrak{g}^{\ast }$ belong to the ideal in $S \mathfrak{g}^{\ast }$ generated by another copy of the adjoint representation together with the invariants of $S \mathfrak{g}^{\ast }$?

Quivers with relations for symmetrizable Cartan matrices IV: crystal graphs and semicanonical functions

We generalize Lusztig's nilpotent varieties, and Kashiwara and Saito's geometric construction of crystal graphs from the symmetric to the symmetrizable case. We also construct semicanonical functions in the convolution algebras of generalized preprojective algebras. Conjecturally these functions yield semicanonical bases of the enveloping algebras of the positive part of symmetrizable Kac-Moody algebras.

НОВЫЕ СВОЙСТВА ПОЧТИ НИЛЬПОТЕНТНЫХ МНОГООБРАЗИЙ С ЦЕЛЫМИ ЭКСПОНЕНТАМИ

Almost nilpotent varieties of nonassociative algebras over a field of zero characteristic in the class of all algebras satisfying identical relation x(yz) ≡ 0 are studied. Earlier in this class of algebras for each natural number m ≥ 2 the algebra Am generating the almost nilpotent variety var(A m ) of exponential growth with exponent of m was defined. In the paper numerical characteristics of varieties var(A m ) are studied. To this end in the relatively free algebras of the varieties var(A m ) the spaces of multilinear elements corresponding to left normed polynomials with fixed variable on the first position are considered. Each space is considered as completely reducible module of the symmetric group and multiplicities in the decomposition of the corresponding cocharacter into sum of irreducible characters are calculated. The multiplicities corresponding to the multilinear parts of relatively free algebras of the variety var(A m ) are defined by the calculated values. Colengths of the varieties var(A m ), m≥ 2 are obtained using this method. For each m ≥ 2 the set of identical relations that defines the variety var(A m ) is obtained.

Sturmian words and uncountable set of almost nilpotent varieties of quadratic growth

An almost nilpotent variety of quadratic growth is constructed and studied in the paper for any infinite Sturmian word.

An uncountable family of almost nilpotent varieties of polynomial growth

A non-nilpotent variety of algebras is almost nilpotent if any proper subvariety is nilpotent. Let the base field be of characteristic zero. It has been shown that for associative or Lie algebras only one such variety exists. Here we present infinite families of such varieties. More precisely we shall prove the existence of1) a countable family of almost nilpotent varieties of at most linear growth and2) an uncountable family of almost nilpotent varieties of at most quadratic growth.