The group algebra of the infinite symmetric group

The rational group algebra of the infinite symmetric group is studied using Young diagrams. Maximal and prime ideals are characterized and the maximal condition on ideals is proved. Research supported by the National Research Council of Canada.     

On quasifree representations of infinite dimensional symplectic group

We consider an infinite dimensional generalization of metaplectic representations (Weil representations) for the (double covering of) symplectic group. Given quasifree states of an infinite dimensional CCR algebra, projective unitary representations of the infinite dimensional symplectic group are constructed via unitary implementors of Bogoliubov automorphisms. Complete classification of these representations up to quasi-equivalence is obtained.     

Solvable Jordan algebras of compact operators

It is proved that a Jordan algebra of compact operators which is closed is either an Engel Jordan algebra, or contains a nonzero finite rank operator; Moreover, it is showed that any solvable Jordan algebra of compact operators on an infinite dimensional Banach space is triangularizable.     

Endomorphisms of Free Metabelian Lie Algebras Which Preserve Orbits

For the free rank 2 metabelian Lie algebra over an infinite field we prove that an endomorphism of the algebra which preserves the automorphic orbit of a nonzero element is an automorphism. We construct some counterexamples over finite fields.     

Modular group rings of the finitary symmetric group

The paper discusses approaches to constructing two-sided ideals of the modular group algebra of finitary symmetric group. In memory of Professor S. A. Amitsur     

The Hecke group algebra of a Coxeter group and its representation theory

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, …).In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. It turns out to be closely related to the monoid algebras of respectively nondecreasing functions and nondecreasing parking functions, the representation theory of which we describe as well.This defines three towers of algebras, and we give explicitly the Grothendieck algebras and coalgebras given respectively by their induction products and their restriction coproducts. This yields some new interpretations of the classical bases of quasi-symmetric and noncommutative symmetric functions as well as some new bases.     

Group algebras with symmetric units satisfying a group identity

We study group algebras FG for which the symmetric units under the natural involution: g^*=g⁻¹ satisfy a group identity. For infinite fields F of characteristic ≠2, a classification of torsion groups G whose symmetric units U⁺(FG) satisfy a group identity was given in [3] by Giambruno-Sehgal-Valenti. We extend this work to non torsion groups. Research supported by NSERC of Canada and MIUR of Italy.     

Factorisation properties of group scheme actions

Let H be an algebraic group scheme over a field k acting on a commutative k-algebra A which is a unique factorisation domain. We show that, under certain mild assumptions, the monoid of nonzero H-stable principal ideals in A is free commutative. From this we deduce, in certain special cases, results about the monoid of nonzero semi-invariants and the algebra of invariants. We use an infinitesimal method which allows us to work over an arbitrary base field.     

On the Dirac cohomology of complex lie group representations

This note studies the problem of classifying all the irreducible unitary representations with nonzero Dirac cohomology for a complex Lie group G. We reduce it to the classification of spherical ones with nonzero Dirac cohomology on the Levi level. Then in the spherical unitary dual, by computing spin norm and utilizing Vogan pencil, we show how to further reduce the classification to fairly few candidate representations.     

Cyclic group actions on polynomial rings

We consider a cyclic group of order pⁿ, acting on a module incharacteristic p, and show how to reduce the calculation ofthe symmetric algebra to that of the exterior algebra.     

Nilpotent elements in group rings

The main theorem gives necessary and sufficient conditions for the rational group algebra QG to be without (nonzero) nilpotent elements if G is a nilpotent or F·C group. For finite groups G, a characterisation of group rings RG over a commutative ring with the same property is given. As an application those nilpotent or F·C groups are characterised which have the group of units U(KG) solvable for certain fields K.This work has been supported by N.R.C. Grant No. A-5300.     

Boolean algebras admitting a countable minimally acting group

The aim of this paper is to show that every infinite Boolean algebra which admits a countable minimally acting group contains a dense projective subalgebra.     

The automorphism group of certain higher degree forms

We consider symmetric indecomposable d-linear (d>2) spaces of dimension n over an algebraically closed field k of characteristic 0, whose center (the analog of the space of symmetric matrices of a bilinear form) is cyclic, as introduced by Reichstein [B. Reichstein, On Waring’s problem for cubic forms, Linear Algebra Appl. 160 (1992) 1-61]. The automorphism group of these spaces is determined through the action on the center and through the determination of the Lie algebra. Furthermore, we relate the Lie algebra to the Witt algebra.     

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

Lattice congruences, fans and Hopf algebras

We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto–Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern avoidance. Applying these results, we build the Malvenuto–Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations.     

The socle of a nondegenerate Lie algebra

We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element.     

Weyl group action and semicanonical bases

Let U be the enveloping algebra of a symmetric Kac–Moody algebra. The Weyl group acts on U, up to a sign. In addition, the positive subalgebra U⁺ contains a so-called semicanonical basis, with remarkable properties. The aim of this paper is to show that these two structures are as compatible as possible.     

The unitary group preserving maps (the infinite dimensional case)

Marcus's theorem on the unitary group preserving map is extended to the infinite dimensional case. The method of proof uses the Jordan automorphisms of the algebra of hermitian operators. A connection with the group of (linear bijective) isometries is established.     

The unit group of FS 4

A complete characterization is given for the unit group U(FS ₄) of the group algebra FS ₄ of the symmetric group S ₄ of degree 4 over a finite field F.      

Degenerate alternative algebras

We prove that each degenerate alternative algebra of characteristic ≠ 2 contains a nonzero ideal with an additive basis consisting of the absolute zero divisors of an arbitrary large order. As a corollary we establish the existence of infinitely many nonisomorphic commutative prime alternative algebras and existence of infinite series of strict and nonstrict exceptional alternative algebras with different sets of proper identities.