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1.
Let
be an algebraic algebra over an infinite field K and let
(
) be its group of units. We prove a stronger version of Hartley's conjecture for
, namely, if a Laurent polynomial identity (LPI, for short) f = 0 is satisfied in
(
), then
satisfies a polynomial identity (PI). We also show that if
is non-commutative, then
is a PI-ring, provided f = 0 is satisfied by the non-central units of
. In particular,
is locally finite and, thus, the Kurosh problem has a positive answer for K-algebras whose unit group is LPI. Moreover, f = 0 holds in
(
) if and only if the same identity is satisfied in
. The last fact remains true for generalized Laurent polynomial identities, provided that
is locally finite. 相似文献
2.
Let G be a unipotent algebraic subgroup of some defined over . We describe an algorithm for finding a finite set of generators of the subgroup . This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists. 相似文献
3.
Let G be a finite group. Efficient generation of nearly uniformly distributed random elements in G, starting from a given set of generators of G, is a central problem in computational group theory. In this paper we demonstrate a weakness in the popular “product replacement algorithm,” widely used for this purpose. The main results are the following. Let
be the set of generating k-tuples of elements of G. Consider the distribution of the first components of the k-tuples in
induced by the uniform distribution over
. We show that there exist infinite sequences of groups G such that this distribution is very far from uniform in two different senses: (1) its variation distance from uniform is >1−ε; and (2) there exists a short word (of length (loglog|G|)O(k)) which separates the two distributions with probability 1−ε. The class of groups we analyze is direct powers of alternating groups. The methods used include statistical analysis of permutation groups, the theory of random walks, the AKS sorting network, and a randomized simulation of monotone Boolean operations by group operations, inspired by Barrington's work on bounded-width branching programs. The problem is motivated by the product replacement algorithm which was introduced in [Comm. Algebra 23 (1995) 4931–4948] and is widely used. Our results show that for certain groups the probability distribution obtained by the product replacement algorithm has a bias which can be detected by a short straight line program. 相似文献
4.
Let X be a smooth toric variety. Cox introduced the homogeneous coordinate ring S of X and its irrelevant ideal
. Let A denote the ring of differential operators on Spec(S). We show that the category of
-modules on X is equivalent to a subcategory of graded A-modules modulo
-torsion. Additionally, we prove that the characteristic variety of a
-module is a geometric quotient of an open subset of the characteristic variety of the associated A-module and that holonomic
-modules correspond to holonomic A-modules. 相似文献
5.
Julien Bichon 《Journal of Algebra》2000,230(2):83
Let
be a (small) category and let F:
→
algf be a functor, where
algf is the category of finite-dimensional measured algebras over a field k (or Frobenius algebras). We construct a universal Hopf algebra Aaut(F) such that F factorizes through a functor
:
→
coalgf(Aaut(F)), where
coalgf(Aaut(F)) is the category of finite-dimensional measured Aaut(F)-comodule algebras. This general reconstruction result allows us to recapture a finite-dimensional Hopf algebra A from the category
coalgf(A) and the forgetful functor ω:
coalgf(A) →
algf: we have A Aaut(ω). Our universal construction is also done in a C*-algebra framework, and we get compact quantum groups in the sense of Woronowicz. 相似文献
6.
Let C
n and C
n be the varieties of all completely regular and of all completely simple semigroups, respectively, whose idempotent generated subsemigroups are periodic with period n. We use Ol'shanski
's theory of geometric group presentations to show that for large odd n these varieties (and similarly defined varieties of epigroups) do not have finitely axiomatizable equational theories. 相似文献
7.
Let
be the collection of all polynomials of degree at most n with real coefficients that have at most m distinct complex zeros. We prove thatfor every
. This is far away from what we expect. We conjecture that the Markov factor 32·8mn above may be replaced by cmn with an absolute constant c>0. We are not able to prove this conjecture at the moment. However, we think that our result above gives the best-known Markov-type inequality for
on a finite interval when mc log n. 相似文献
8.
Pu Zhang 《Journal of Algebra》2002,250(2):709
Generalized Green classes are introduced; some basic properties of members in a generalized Green class are studied. Finally, we apply the results to
(Λ), the Ringel–Hall algebra of a finite-dimensional hereditary algebra Λ over a finite field. In particular, it is proved that
(Λ) belongs to a suitable generalized Green class, and that there is direct decomposition of spaces
(Λ) =
(Λ) J, where
(Λ) is the composition algebra of Λ and J is a twisted Hopf ideal of
(Λ), which is exactly the orthogonal complement of
(Λ). 相似文献
9.
Paul Levy 《Journal of Algebra》2002,250(2):473
Let K be an algebraically closed field of positive characteristic and let G be a reductive group over K with Lie algebra
. This paper will show that under certain mild assumptions on G, the commuting variety
(
) is an irreducible algebraic variety. 相似文献
10.
Raymundo Bautista William Crawley-Boevey Tiangang Lei Yingbo Zhang 《Journal of Algebra》2000,230(2):665
In the present paper we prove that a certain subcategory
of the module category over some infinite-dimensional algebra R has almost split sequences and strongly homogeneous property; i.e., for each indecomposable module M in
, there is an almost split sequence starting and also ending at M. It is also proved that except for a trivial case,
is of wild representation type. 相似文献
11.
We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system is such that for each k, σk has a constant sign on its compact support consisting of an interval , on which almost everywhere, and a discrete set without accumulation points in . If denotes the smallest interval containing , we assume that Δk∩Δk+1=0/, k=1,…,m−1. The second Nikishin system is a perturbation of the first by means of rational functions rk, k=1,…,m, whose zeros and poles lie in . 相似文献
12.
Given an (n+1)-dimensional space
of piecewise smooth functions in which each basis has a non-vanishing Wronskian, and its dual space
*, a canonical bilinear form is defined on
×
*, which provides a simple characterization of a contact of order rn. An intrinsic reproducing function is introduced, leading to Marsden-type identities. In the case of Chebyshev spaces connected with totally positive matrices, the bilinear form yields a general notion of blossom which can be extended to Chebyshev splines. 相似文献
13.
Let , with
-1=x0n<x1n<<xnn<xn+1,n=1