343 research outputs found
Idempotent generated algebras and Boolean powers of commutative rings
A Boolean power S of a commutative ring R has the structure of a commutative
R-algebra, and with respect to this structure, each element of S can be written
uniquely as an R-linear combination of orthogonal idempotents so that the sum
of the idempotents is 1 and their coefficients are distinct. In order to
formalize this decomposition property, we introduce the concept of a Specker
R-algebra, and we prove that the Boolean powers of R are up to isomorphism
precisely the Specker R-algebras. We also show that these algebras are
characterized in terms of a functorial construction having roots in the work of
Bergman and Rota. When R is indecomposable, we prove that S is a Specker
R-algebra iff S is a projective R-module, thus strengthening a theorem of
Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is
a torsion-free R-module. For an indecomposable R, we prove that the category of
Specker R-algebras is equivalent to the category of Boolean algebras, and hence
is dually equivalent to the category of Stone spaces. In addition, when R is a
domain, we show that the category of Baer Specker R-algebras is equivalent to
the category of complete Boolean algebras, and hence is dually equivalent to
the category of extremally disconnected compact Hausdorff spaces. For a totally
ordered R, we prove that there is a unique partial order on a Specker R-algebra
S for which it is an f-algebra over R, and show that S is equivalent to the
R-algebra of piecewise constant continuous functions from a Stone space X to R
equipped with the interval topology.Comment: 18 page
Vertex rings and their Pierce bundles
In part I we introduce vertex rings, which bear the same relation to vertex
algebras (or VOAs) as commutative, associative rings do to commutative,
associative algebras over the complex numbers. We show that vertex rings are
characterized by Goddard axioms. These include a generalization of the
translation-covariance axiom of VOA theory that involves a canonical
Hasse-Schmidt derivation naturally associated to any vertex ring. We give
several illustrative applications of these axioms, including the construction
of vertex rings associated with the Virasoro algebra. We consider some
categories of vertex rings, and the role played by the center of a vertex ring.
In part II we extend the theory of Pierce bundles associated to a commutative
ring to the setting of vertex rings. This amounts to the construction of
certain reduced etale bundles of vertex rings functorially associated to a
vertex ring. We introduce von Neumann regular vertex rings as a generalization
of von Neumann regular commutative rings; we obtain a characterization of this
class of vertex rings as those whose Pierce bundles are bundles of simple
vertex rings
Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras
We study the representation theory of three towers of algebras which are
related to the symmetric groups and their Hecke algebras. The first one is
constructed as the algebras generated simultaneously by the elementary
transpositions and the elementary sorting operators acting on permutations. The
two others are the monoid algebras of nondecreasing functions and nondecreasing
parking functions. For these three towers, we describe the structure of simple
and indecomposable projective modules, together with the Cartan map. The
Grothendieck algebras and coalgebras given respectively by the induction
product and the restriction coproduct are also given explicitly. This yields
some new interpretations of the classical bases of quasi-symmetric and
noncommutative symmetric functions as well as some new bases.Comment: 12 pages. Presented at FPSAC'06 San-Diego, June 2006 (minor
explanation improvements w.r.t. the previous version
De Vries powers: a generalization of Boolean powers for compact Hausdorff spaces
We generalize the Boolean power construction to the setting of compact
Hausdorff spaces. This is done by replacing Boolean algebras with de Vries
algebras (complete Boolean algebras enriched with proximity) and Stone duality
with de Vries duality. For a compact Hausdorff space and a totally ordered
algebra , we introduce the concept of a finitely valued normal function
. We show that the operations of lift to the set of all
finitely valued normal functions, and that there is a canonical proximity
relation on . This gives rise to the de Vries power
construction, which when restricted to Stone spaces, yields the Boolean power
construction.
We prove that de Vries powers of a totally ordered integral domain are
axiomatized as proximity Baer Specker -algebras, those pairs ,
where is a torsion-free -algebra generated by its idempotents that is a
Baer ring, and is a proximity relation on . We introduce the
category of proximity Baer Specker -algebras and proximity morphisms between
them, and prove that this category is dually equivalent to the category of
compact Hausdorff spaces and continuous maps. This provides an analogue of de
Vries duality for proximity Baer Specker -algebras.Comment: 34 page
Zero-product balanced algebras
We say that an algebra is zero-product balanced if ab⊗c and a⊗bc agree modulo tensors of elements with zero-product. This is closely related to but more general than the notion of a zero-product determined algebra introduced and developed by Brešar, Villena and others. Every surjective, zero-product preserving map from a zero-product balanced algebra is automatically a weighted epimorphism, and this implies that zero-product balanced algebras are determined by their linear and zero-product structure. Further, the commutator subspace of a zero-product balanced algebra can be described in terms of square-zero elements. We show that a commutative, reduced algebra is zero-product balanced if and only if it is generated by idempotents. It follows that every commutative, zero-product balanced algebra is spanned by nilpotent and idempotent elements
M\"obius Functions and Semigroup Representation Theory II: Character formulas and multiplicities
We generalize the character formulas for multiplicities of irreducible
constituents from group theory to semigroup theory using Rota's theory of
M\"obius inversion. The technique works for a large class of semigroups
including: inverse semigroups, semigroups with commuting idempotents,
idempotent semigroups and semigroups with basic algebras. Using these tools we
are able to give a complete description of the spectra of random walks on
finite semigroups admitting a faithful representation by upper triangular
matrices over the complex numbers. These include the random walks on chambers
of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and
Diaconis. Applications are also given to decomposing tensor powers and exterior
products of rook matrix representations of inverse semigroups, generalizing and
simplifying earlier results of Solomon for the rook monoid.Comment: Some minor typos corrected and references update
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