7,114 research outputs found
Decomposing locally compact groups into simple pieces
We present a contribution to the structure theory of locally compact groups.
The emphasis is on compactly generated locally compact groups which admit no
infinite discrete quotient. It is shown that such a group possesses a
characteristic cocompact subgroup which is either connected or admits a
non-compact non-discrete topologically simple quotient. We also provide a
description of characteristically simple groups and of groups all of whose
proper quotients are compact. We show that Noetherian locally compact groups
without infinite discrete quotient admit a subnormal series with all
subquotients compact, compactly generated Abelian, or compactly generated
topologically simple. Two appendices introduce results and examples around the
concept of quasi-product.Comment: Index added; minor change
Efficient quantum processing of ideals in finite rings
Suppose we are given black-box access to a finite ring R, and a list of
generators for an ideal I in R. We show how to find an additive basis
representation for I in poly(log |R|) time. This generalizes a recent quantum
algorithm of Arvind et al. which finds a basis representation for R itself. We
then show that our algorithm is a useful primitive allowing quantum computers
to rapidly solve a wide variety of problems regarding finite rings. In
particular we show how to test whether two ideals are identical, find their
intersection, find their quotient, prove whether a given ring element belongs
to a given ideal, prove whether a given element is a unit, and if so find its
inverse, find the additive and multiplicative identities, compute the order of
an ideal, solve linear equations over rings, decide whether an ideal is
maximal, find annihilators, and test the injectivity and surjectivity of ring
homomorphisms. These problems appear to be hard classically.Comment: 5 page
Indicators of Tambara-Yamagami categories and Gauss sums
We prove that the higher Frobenius-Schur indicators, introduced by Ng and
Schauenburg, give a strong enough invariant to distinguish between any two
Tambara-Yamagami fusion categories. Our proofs are based on computation of the
higher indicators as quadratic Gauss sums for certain quadratic forms on finite
abelian groups and relies on the classification of quadratic forms on finite
abelian groups, due to Wall.
As a corollary to our work, we show that the state-sum invariants of a
Tambara-Yamagami category determine the category as long as we restrict to
Tambara-Yamagami categories coming from groups G whose order is not a power of
2. Turaev and Vainerman proved this result under the assumption that G has odd
order and they conjectured that a similar result should hold for groups of even
order. We also give an example to show that the assumption that G does not have
a power of 2, cannot be completely relaxed.Comment: 29 page
Cohomology of local systems on loci of d-elliptic abelian surfaces
We consider the loci of d-elliptic curves in , and corresponding loci of
d-elliptic surfaces in . We show how a description of these loci as
quotients of a product of modular curves can be used to calculate cohomology of
natural local systems on them, both as mixed Hodge structures and -adic
Galois representations. We study in particular the case d=2, and compute the
Euler characteristic of the moduli space of n-pointed bi-elliptic genus 2
curves in the Grothendieck group of Hodge structures.Comment: 15 pages, complete re-write of earlier versio
Fast Quantum Fourier Transforms for a Class of Non-abelian Groups
An algorithm is presented allowing the construction of fast Fourier
transforms for any solvable group on a classical computer. The special
structure of the recursion formula being the core of this algorithm makes it a
good starting point to obtain systematically fast Fourier transforms for
solvable groups on a quantum computer. The inherent structure of the Hilbert
space imposed by the qubit architecture suggests to consider groups of order
2^n first (where n is the number of qubits). As an example, fast quantum
Fourier transforms for all 4 classes of non-abelian 2-groups with cyclic normal
subgroup of index 2 are explicitly constructed in terms of quantum circuits.
The (quantum) complexity of the Fourier transform for these groups of size 2^n
is O(n^2) in all cases.Comment: 16 pages, LaTeX2
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