Commutative quotients of finite W-algebras

In this paper we reduce the problem of 1-dimensional representations for the finite W-algebras and Humphreys' conjecture on small representations of reduced enveloping algebras to the case of rigid nilpotent elements in exceptional Lie algebras. We use Katsylo's results on sections of sheets to determine the Krull dimension of the largest commutative quotient of the finite W-algebra U(g,e).     

Critère d’existence d’idempotent basé sur les algèbres de Rétrocroisement

We study the relationship of backcrossing algebras with mutation algebras and algebras satisfying ω-polynomial identities: we show that in a backcrossing algebra every element of weight 1 generates a mutation algebra and that for any polynomial identity f there is a backcrossing algebra satisfying f. We give a criterion for the existence of idempotent in the case of baric algebras satisfying a nonhomogeneous polynomial identity and containing a backcrossing subalgebra. We give numerous genetic interpretations of the algebraic results.     

Two results on centralisers of nilpotent elements

Let g be a semisimple Lie algebra. We provide a short proof of McNinch’s result on centralisers of the sum of commuting nilpotent elements (J. Pure Appl. Algebra 206 (2006), 123-140) and characterise nilpotent elements e∈g having the property that the orbit G⋅e is the largest nilpotent orbit meeting the centraliser of e.     

Classification of solvable Lie algebras with a given nilradical by means of solvable extensions of its subalgebras

We construct all solvable Lie algebras with a specific n-dimensional nilradical n_n,3 which contains the previously studied filiform (n-2)-dimensional nilpotent algebra n_n-2,1 as a subalgebra but not as an ideal. Rather surprisingly it turns out that the classification of such solvable algebras can be deduced from the classification of solvable algebras with the nilradical n_n-2,1. Also the sets of invariants of coadjoint representation of n_n,3 and its solvable extensions are deduced from this reduction. In several cases they have polynomial bases, i.e. the invariants of the respective solvable algebra can be chosen to be Casimir invariants in its enveloping algebra.     

Hochschild cohomology and stratifying ideals

Suppose B is an algebra with a stratifying ideal BeB generated by an idempotent e. We will establish long exact sequences relating the Hochschild cohomology groups of the three algebras B, B/BeB and eBe. This provides a common generalization of various known results, all of which extend Happel’s long exact sequence for one-point extensions. Applying one of these sequences to Hochschild cohomology algebras modulo the ideal generated by homogeneous nilpotent elements shows, in some cases, that these algebras are finitely generated.     

On the communication complexity oft-intersection problems in generalized Boolean algebras

We consider the following game: Two players independently choose a chain in a partially ordered set. How many bits of information have to be communicated until at least one of the players knows whether the chains have exactlyt elements in common? This model generalizes thet-intersection problem for subsets of a finite set. We establish the deterministic communication complexity in general. For the special cases of generalized Boolean algebras, we present improved nondeterministic and probabilistic protocols that are of optimal order of complexity for classes with fixed widthq.     

Étude des ω-PI Algèbres Commutatives de Degré 4: II. Algèbres Non Barycentriques Invariantes par Gamétisation

In this article we study the algebras satisfying the ω-polynomial identity x ² x ² ? x ⁴ = δ(x ² ? x) with δ ≠ 0 but do not satisfy any monomial identities of degree ≤4. We show that there exist such algebras for all δ ≠ 0 and they have a unique baric function. We give conditions for the existence of idempotents of weight 0 or 1, and we construct the three Peirce decompositions associated to these idempotent elements.     

On reachable elements and the boundary of nilpotent orbits in simple Lie algebras

Let g be a simple Lie algebra. An element x∈g is said to be reachable, if it is contained in the commutant of its centraliser. Any reachable element is necessarily nilpotent. We study various properties of reachable elements, and a relationship between the property of being reachable and the codimension of the boundary of the corresponding orbit. Some general estimates for the boundary of an arbitrary nilpotent orbit is given.     

Nilmanifolds with a calibrated G2-structure

We introduce obstructions to the existence of a calibrated G₂-structure on a Lie algebra g of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from g to a six-dimensional Lie algebra h with kernel contained in the center of g, then h has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G₂-structure.     

Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity

An absolute valued algebra is a non-zero real algebra that is equipped with a multiplicative norm. We classify all finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity, up to algebra isomorphism. This completes earlier results of Ramírez Álvarez and Rochdi which, in our self-contained presentation, are recovered from the wider context of composition k-algebras with an LR-bijective idempotent.     

The Idempotent Manifold in a Clifford Algebra

The idempotent sets in sufficiently sophisticated algebras form manifolds and Hausdorff spaces. In this paper it is shown how the idempotents in a real Clifford algebra Cl_p,q can be calculated by nilpotents and reflections. Minimal sets of nilpotents are given and generating relations are defined. It is shown that the manifold thus constructed is complete. Every idempotent in the manifold can be calculated in the way proposed here, namely by a nilpotent multinomial form.     

Étude des ω-PI Algèbres Commutatives de Degré 4: I. Algèbres Non Barycentriques Non Invariantes par Gamétisation

We define and study the properties of baric algebras defined by ω-polynomial identities, called ω-PI algebras. We show that every finite dimensional baric algebra is ω-PI. Next we introduce the study of ω-PI algebras of degree 4 with one indeterminate. By gametization we reduce their study to four types. We study the first type corresponding to algebras that are neither barycentric nor invariant by gametization. We show that the variety of these algebras is partitioned into only two subvarieties admiting an unique ponderation, an idempotent, and verifying ω-monomial identities of degrees > 4.     

A direct product decomposition of QMV algebras

We study the direct product decomposition of quantum many-valued algebras (QMV algebras) which generalizes the decomposition theorem of ortholattices (orthomodular lattices).In detail,for an idempo- tent element of a given QMV algebra,if it commutes with every element of the QMV algebra,it can induce a direct product decomposition of the QMV algebra.At the same time,we introduce the commutant C(S) of a set S in a QMV algebra,and prove that when S consists of idempotent elements,C(S) is a subalgebra of the QMV algebra.This also generalizes the cases of orthomodular lattices.     

Semiconic idempotent residuated structures

An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form avariety SCIL, which is not locally finite, but it is proved that SCIL has the finite embeddability property (FEP). More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains.     

1-Dimensional representations and parabolic induction for W-algebras

A W-algebra is an associative algebra constructed from a reductive Lie algebra and its nilpotent element. This paper concentrates on the study of 1-dimensional representations of W-algebras. Under some conditions on a nilpotent element (satisfied by all rigid elements) we obtain a criterium for a finite dimensional module to have dimension 1. It is stated in terms of the Brundan–Goodwin–Kleshchev highest weight theory. This criterium allows to compute highest weights for certain completely prime primitive ideals in universal enveloping algebras. We make an explicit computation in a special case in type E₈. Our second principal result is a version of a parabolic induction for W-algebras. In this case, the parabolic induction is an exact functor between the categories of finite dimensional modules for two different W-algebras. The most important feature of the functor is that it preserves dimensions. In particular, it preserves one-dimensional representations. A closely related result was obtained previously by Premet. We also establish some other properties of the parabolic induction functor.     

Lie Algebras with Nilpotent Length Greater than that of each of their Subalgebras

The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call minimal non- \({\mathcal N}\). To facilitate this we investigate solvable Lie algebras of nilpotent length k, and of nilpotent length ≤k, and extreme Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-\({\mathcal N}\) Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index ≤3.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Invariance of dimension of p-subspaces in bernstein algebras *

The purpose of this paper is to prove that some vector subspaces, called p-subspaces, obtained from the Peirce decomposition of a Bernstein algebra A relative to an idempotent have dimensions which are independent of the idempotent used to decompose A. In particular, for Bernstein-Jordan algebras, this fact is true for every such subspace and this implies that all p-subspaces of a Bernstein algebra, contained in V, for A = Ke + U + V, have invariant dimension. Finally we classify all p-subspaces of degree ≥ 3, contained in U, in a Bernstein algebra A, relative to the invariance (or not) of dimension.