Projective limits of Banach-Lie groups

In this paper we characterize commutative Fréchet-Lie groups using the exponential map. In particular we prove that if a commutative Fréchet-Lie groupG has an exponential map, which is a local diffeomorphism, thenG is the limit of a projective system of Banach-Lie groups.     

Projective functors for algebraic groups

We investigate the regular subquotient category introduced by Soergel in [W. Soergel, On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra 152 (1–3) (2000) 311–335]. A detailed study of projective functors enables one to relate those categories for semi-simple algebraic group G and its subgroup schemata G_rT. As an application we derive some information about the characters of tilting modules for G.     

Projective limits of nilpotent Hall groups

By analogy with the notion of a pronilpotent W-group, we define the notion of a pronilpotent Lie algebra and establish a one-to-one correspondence between pronilpotent W-groups and pronilpotent Lie algebras in the case where W is a field of zero characteristic; also, we establish a connection between free and projective groups of a given manifold. We prove that, for some manifolds, free and projective groups coincide; we investigate conditions under which a subgroup is free in an absolutely free pronilpotent W-group. In the class of pronilpotent groups, we introduce and discuss the notion of a free product; we construct an example which shows that an analog of the Kurosh theorem on subgroups of a free product does not hold even for finitely generated subgroups. A series of results stated in this paper was announced in [35–38]. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 49, Algebra and Geometry, 2007.     

Projective Schur groups of Henselian fields

One of the open questions that has emerged in the study of the projective Schur group of a field F is whether or not is an algebraic relative Brauer group over F, i.e. does there exist an algebraic extension L/F such that ? We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, no counterexample is known. In this paper we prove that is an algebraic relative Brauer group for all Henselian valued fields F of equal characteristic whose residue field is a local or global field. For this, we first show how is determined by for an equicharacteristic Henselian field with arbitrary residue field k.     

Projective discrete modules over profinite groups

We show that the category of discrete modules over an infinite profinite group has no non-zero projective objects and does not satisfy Ab4*. We also prove the same types of results in a generalized setting using a ring with linear topology.     

Jacobians of reflection groups

Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This result generalizes verbatim to fields whose characteristic is prime to the order of the group. Our main theorem gives a generalization of Steinberg's result for groups with a polynomial ring of invariants over arbitrary fields using a ramification formula of Benson and Crawley-Boevey.

     

Gray codes for reflection groups

LetG be a finite group generated by reflections. It is shown that the elements ofG can be arranged in a cycle (a Gray code) such that each element is obtained from the previous one by applying one of the generators. The case G =A ₁ ⁿ yields a conventional binary Gray code. These generalized Gray codes provide an efficient way to run through the elements of any finite reflection group.     

Reflection subgroups of reflection groups

Let G be a discrete group generated by reflections in hyperbolic or Euclidean space, and let H G be a finite index reflection subgroup. Suppose that the fundamental chamber of G is a finite volume polytope with k facets. We prove that the fundamental chamber of H has at least k facets.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 90–92, 2004Original Russian Text Copyright © by A. A. Felikson and P. V. Tumarkin     

Reducibility of finite reflection groups

A finite (pseudo-)reflection group G naturally gives rise to a hyperplane arrangement,i.e.,its reflection arrangement.We show that G is reducible if and only if its reflection arrangement is reducible.     

Quotients of infinite reflection groups

Mathematische Annalen -