Completions of symplectic reflection algebras

In this paper, we study the structure of completions of symplectic reflection algebras. Our results provide a reduction to smaller algebras. We apply this reduction to the study of two-sided ideals and Harish–Chandra bimodules.     

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras.We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.     

Affine cellular algebras and Morita equivalences

In this paper we prove that Morita equivalences under particular conditions are compatible with affine cellular structures.     

Lusztig's isomorphism theorem for cellular algebras

In this paper, we first introduce a notion of semisimple system with parameters, then we establish Lusztig's isomorphism theorem for any cellular semisimple system with parameters. As an application, we obtain Lusztig's isomorphism theorem for Ariki-Koike algebras, cyclotomic q-Schur algebras and Birman-Murakami-Wenzl algebras. Second, using the results for certain Ariki-Koike algebras, we prove an analogue of Lusztig's isomorphism theorem for the cyclotomic Hecke algebras of type G(p,p,n) (which are not known to be cellular in general). These generalize earlier results of [G. Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981) 490-498.] on such isomorphisms for Iwahori-Hecke algebras associated to finite Weyl groups.     

On cell modules of symmetric cellular algebras

Let A be a symmetric cellular algebra with cell datum (??, M, C, i) and let ${\Lambda_1=\{\lambda \in \Lambda_0 \mid W(\lambda) \, {\rm is \, simple}\}}$ . We prove that ??₁ consists of two parts: one gives a lower bound for the cardinality of the set of cell modules with zero bilinear forms and the other parametrizes all the projective cell modules. Moreover, it is proved in Li (arxiv: math0911.3524, 2009) that the dual basis of ${\{C_{S, T}^{\lambda} \mid \lambda \in \Lambda, S,T \in M(\lambda)\}}$ is again cellular. In this paper, we will study the cell modules defined by dual basis. In particular, we study the dual basis of the Murphy basis.     

Hecke algebras of finite type are cellular

Let be the one-parameter Hecke algebra associated to a finite Weyl group W, defined over a ground ring in which “bad” primes for W are invertible. Using deep properties of the Kazhdan–Lusztig basis of and Lusztig’s a-function, we show that has a natural cellular structure in the sense of Graham and Lehrer. Thus, we obtain a general theory of “Specht modules” for Hecke algebras of finite type. Previously, a general cellular structure was only known to exist in types A _n and B _n .     

Projective cell modules of Frobenius cellular algebras

In this note we give a criterion of projectiveness of the simple cell modules over finite dimensional Frobenius cellular algebras.     

Completions of pro-spaces

For every ring R, we present a pair of model structures on the category of pro-spaces. In the first, the weak equivalences are detected by cohomology with coefficients in R. In the second, the weak equivalences are detected by cohomology with coefficients in all R-modules (or equivalently by pro-homology with coefficients in R). In the second model structure, fibrant replacement is essentially just the Bousfield-Kan R-tower. When the first homotopy category is equivalent to a homotopy theory defined by Morel but has some convenient categorical advantages.Mathematical Subject Classification (1991): 55P60, 55N10 18G55, 55U35This work was partially supported by a National Science Foundation Postdoctoral Research Fellowship. The author acknowledges useful conversations with Bill Dwyer and Daniel Biss. The author thanks the referee for several corrections and excellent suggestions.     

Completions of -algebras

A μ-algebra is a model of a first-order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f,μ_x.f) where μ_x.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications.Standard μ-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any nontrivial quasivariety of μ-algebras contains a μ-algebra that has no embedding into a complete μ-algebra.We then focus on modal μ-algebras, i.e. algebraic models of the propositional modal μ-calculus. We prove that free modal μ-algebras satisfy a condition–reminiscent of Whitman’s condition for free lattices–which allows us to prove that (i) modal operators are adjoints on free modal μ-algebras, (ii) least prefixed points of Σ₁-operations satisfy the constructive relation μ_x.f=_n≥0fⁿ(). These properties imply the following statement: the MacNeille–Dedekind completion of a free modal μ-algebra is a complete modal μ-algebra and moreover the canonical embedding preserves all the operations in the class of the fixed point alternation hierarchy.     

On the projective-injective modules over cellular algebras

We show that the projective module over a cellular algebra is injective if and only if the socle of coincides with the top of , and this is also equivalent to the condition that the th socle layer of is isomorphic to the th radical layer of for each positive integer . This eases the process of determining the Loewy series of the projective-injective modules over cellular algebras.

     

Completions of fans

In a finite-dimensional real vector space furnished with a rational structure with respect to a subfield of the field of real numbers, every (simplicial) rational semifan is contained in a complete (simplicial) rational semifan. In this paper this result is proved constructively on use of techniques from polyhedral geometry.     

Completions of orthomodular lattices

If K is a variety of orthomodular lattices generated by a finite orthomodular lattice the MacNeille completion of every algebra in K again belongs to K.