14,607 research outputs found
On Bousfield's problem for solvable groups of finite Pr\"ufer rank
For a group and we denote by the -completion of We study the map
where We prove that is an epimorphism
for a finitely generated solvable group of finite Pr\"ufer rank. In
particular, Bousfield's -localisation of such groups coincides with the
-completion for Moreover, we prove that
is an epimorphism for any if is a
finitely presented group of the form where is the infinite
cyclic group and is a -module
On the Hochschild cohomology ring of the quaternion group of order eight in characteristic two
Let be an algebraically closed field of characteristic two and let
be the quaternion group of order . We determine the Gerstenhaber Lie algebra
structure and the Batalin-Vilkovisky structure on the Hochschild cohomology
ring of the group algebra
Higher Jacobi identities
By definition the identities and
hold in any Lie algebra. It is
easy to check that the identity
holds in any Lie algebra as well. We investigate sets of permutations that give
identities of this kind. In particular, we construct a family of such subsets
of the symmetric group and hence, a family of identities
that hold in any Lie algebra
A higher limit approach to homology theories
A lot of well-known functors such as group homology, cyclic homology of
algebras can be described as limits of certain simply defined functors over
categories of presentations. In this paper, we develop technique for the
description of the higher limits over categories of presentations and show that
certain homological functors can be described in this way. In particular, we
give a description of Hochschild homology and the derived functors of tensor,
symmetric and exterior powers in the sense of Dold and Puppe as higher limits.Comment: 25 page
Hyper-sparsity of the density matrix in a wavelet representation
O(N) methods are based on the decay properties of the density matrix in real
space, an effect sometimes refered to as near-sightedness. We show, that in
addition to this near-sightedness in real space there is also a
near-sightedness in Fourier space. Using a basis set with good localization
properties in both real and Fourier space such as wavelets, one can exploit
both localization properties to obtain a density matrix which exhibits
additional sparseness properties compared to the scenario where one has a basis
set with real space localization only. We will call this additional sparsity
hyper-sparsity. Taking advantage of this hyper-sparsity, it is possible to
represent very large quantum mechanical systems in a highly compact way. This
can be done both for insulating and metallic systems and for arbitrarily
accurate basis sets. We expect that hyper-sparsity will pave the way for O(N)
calculations of large systems requiring many basis functions per atom, such as
Density Functional calculations.Comment: 4 color figures, 3 normal figure
Generalized Jacobi identities and Jacobi elements of the group ring of the symmetric group
By definition the identities and hold in any Lie algebra. It is
easy to check that the identity holds in any Lie algebra as
well. I. Alekseev in his recent work introduced the notion of Jacobi subset of
the symmetric group . It is a subset of that gives an identity of
this kind. We introduce a notion of Jacobi element of the group ring
and describe them on the language of equations on
coefficients. Using this description we obtain a purely combinatorial necessary
and sufficient condition for a subset to be Jacobi
Mod-2 (co)homology of an abelian group
It is known that for a prime there is the following natural
description of the homology algebra of an abelian group and for finitely generated abelian
groups there is the following description of the cohomology algebra of
We prove that there are no such descriptions for that `depend' only on
and but we provide natural descriptions of
and that `depend' on and a linear map
Moreover, we prove that there is a filtration by
subfunctors on whose quotients are
and that for finitely generated
abelian groups there is a natural filtration on whose
quotients are $ \Lambda^{n-2i}((A/2)^\vee)\otimes {\sf Sym}^i(({}_2A)^\vee).
On Bousfield problem for the class of metabelian groups
The homological properties of localizations and completions of metabelian
groups are studied. It is shown that, for or and
a finitely presented metabelian group , the natural map from to its
-completion induces an epimorphism of homology groups . This
answers a problem of A.K. Bousfield for the class of metabelian groups.Comment: 31 page
A finite Q-bad space
We prove that for a free noncyclic group , is an uncountable -vector space. Here is the
-completion of . This answers a problem of A.K. Bousfield for the
case of rational coefficients. As a direct consequence of this result it
follows that, a wedge of circles is -bad in the sense of
Bousfield-Kan. The same methods as used in the proof of the above results allow
to show that, the homology is not divisible
group, where is the integral pronilpotent completion of
Higher limits, homology theories and fr-codes
This text is based on lectures given by authors in summer 2015. It contains
an introduction to the theory of limits over the category of presentations,
with examples of different well-known functors like homology or derived
functors of non-additive functors in a form of derived limits. The theory of
so-called -codes also is developed. This is a method how different
functors from the category of groups to the category of abelian groups, such as
group homology, tensor products of abelianization, can be coded as sentences in
the alphabet with two symbols and .Comment: 23 page
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