Homological models for semidirect products of finitely generated Abelian groups

Let G be a semidirect product of finitely generated Abelian groups. We provide a method for constructing an explicit contraction (special homotopy equivalence) from the reduced bar construction of the group ring of G, B¯¯¯¯(ZZ[G]) , to a much smaller DGA-module hG. Such a contraction is called a homological model for G and is used as the input datum in the methods described in Álvarez et al. (J Symb Comput 44:558–570, 2009; 2012) for calculating a generating set for representative 2-cocycles and n-cocycles over G, respectively. These computations have led to the finding of new cocyclic Hadamard matrices (Álvarez et al. in 2006)

Strongly minimal PD4-complexes

We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $\pi$ such that $c.d.\pi=2$ and $\pi$ has one end. Let $\beta=\beta_2(\pi;F_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on $(\pi,w)$) there are at most $2^\beta$ orbits of $k$-invariants determining "strongly minimal" complexes (i.e., those with homotopy intersection pairing $\lambda_X$ trivial). The homotopy type of a $PD_4$-complex $X$ with $\pi$ a $PD_2$-group is determined by $\pi$, $w$, $\lambda_X$ and the $v_2$-type of $X$. Our result also implies that Fox's 2-knot with metabelian group is determined up to TOP isotopy and reflection by its group.Comment: 17 page

Homology of iterated semidirect products of free groups

Let $G$ be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of $G$. This resolution is used to define representations of groups which act compatibly on $G$, generalizing classical constructions of Magnus, Burau, and Gassner. Our construction also yields algorithms for computing the homology of the Milnor fiber of a fiber-type hyperplane arrangement, and more generally, the homology of the complement of such an arrangement with coefficients in an arbitrary local system.Comment: 31 pages. AMSTeX v 2.1 preprint styl

Axiomatic framework for the BGG Category O

We introduce a general axiomatic framework for algebras with triangular decomposition, which allows for a systematic study of the Bernstein-Gelfand-Gelfand Category $\mathcal{O}$. The framework is stated via three relatively simple axioms; algebras satisfying them are termed "regular triangular algebras (RTAs)". These encompass a large class of algebras in the literature, including (a) generalized Weyl algebras, (b) symmetrizable Kac-Moody Lie algebras $\mathfrak{g}$, (c) quantum groups $U_q(\mathfrak{g})$ over "lattices with possible torsion", (d) infinitesimal Hecke algebras, (e) higher rank Virasoro algebras, and others. In order to incorporate these special cases under a common setting, our theory distinguishes between roots and weights, and does not require the Cartan subalgebra to be a Hopf algebra. We also allow RTAs to have roots in arbitrary monoids rather than root lattices, and the roots of the Borel subalgebras to lie in cones with respect to a strict subalgebra of the Cartan subalgebra. These relaxations of the triangular structure have not been explored in the literature. We then study the BGG Category $\mathcal{O}$ over an RTA. In order to work with general RTAs - and also bypass the use of central characters - we introduce conditions termed the "Conditions (S)", under which distinguished subcategories of Category $\mathcal{O}$ possess desirable homological properties, including: (a) being a finite length, abelian, self-dual category; (b) having enough projectives/injectives; or (c) being a highest weight category satisfying BGG Reciprocity. We discuss whether the above examples satisfy the various Conditions (S). We also discuss two new examples of RTAs that cannot be studied using previous theories of Category $\mathcal{O}$, but require the full scope of our framework. These include the first construction of algebras for which the "root lattice" is non-abelian.Comment: 59 pages, LaTeX. This paper supersedes (and goes far beyond) the older preprint arXiv:0811.2080 (31 pages), which has been completely rewritte

On the representability of actions for topological algebras

The actions of a group B on a group X correspond bijectively to the group homomorphisms B ⟶ Aut(X), proving that the functor “actions on X” is representable by the group of automorphisms of X. Making the detour through pseudotopological spaces, we generalize this result to the topological case, for quasi-locally compact groups and some other algebraic structures. We investigate next the case of arbitrary topological algebras for a semi-abelian theory and prove that the representability of topological actions reduces to the preservation of coproducts by the functor Act(−,X)