量子代数/算子代数/交换代数/Rings代数学术速递[1.10]
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math.QA量子代数，共计0篇
math.OA算子代数，共计1篇
math.AC交换代数，共计4篇
math.RARings代数，共计5篇
1.math.QA量子代数:
2.math.OA算子代数:
【1】 de Finetti-type theorems on quasi-local algebras and infinite Fermi tensor products
标题：关于拟局部代数和无限费米张量积的de Finetti型定理
链接：https://arxiv.org/abs/2201.02488
备注：31 pages
摘要：Local actions of $\mathbb{P}_\mathbb{N}$, the group of finite permutations on
$\mathbb{N}$, on quasi-local algebras are defined and proved to be
$\mathbb{P}_\mathbb{N}$-abelian. It turns out that invariant states under local
actions are automatically even, and extreme invariant states are strongly
clustering. Tail algebras of invariant states are shown to obey a form of the
Hewitt and Savage theorem, in that they coincide with the fixed-point von
Neumann algebra. Infinite graded tensor products of $C^*$-algebras, which
include the CAR algebra, are then addressed as particular examples of
quasi-local algebras acted upon $\mathbb{P}_\mathbb{N}$ in a natural way.
Extreme invariant states are characterized as infinite products of a single
even state, and a de Finetti theorem is established. Finally, infinite products
of factorial even states are shown to be factorial by applying a twisted
version of the tensor product commutation theorem, which is also derived here.
3.math.AC交换代数:
【1】 Neighborly partitions and the numerators of Rogers-Ramanujan identities
标题：邻域划分与Rogers-Ramanujan恒等式的分子
链接：https://arxiv.org/abs/2201.02481
摘要：We prove two partition identities which are dual to the Rogers-Ramanujan
identities. These identities are inspired by (and proved using) a
correspondence between three kinds of objects: a new type of partitions
(neighborly partitions), monomial ideals and some infinite graphs.
【2】 On the theory of generalized Ulrich modules
标题：关于广义Ulrich模的理论
链接：https://arxiv.org/abs/2201.02398
备注：17 pages
摘要：In this paper we further develop the theory of generalized Ulrich modules
over Cohen-Macaulay local rings introduced in 2014 by Goto, Ozeki, Takahashi,
Watanabe and Yoshida. The term {\it generalized} refers to the fact that Ulrich
modules are taken with respect to a zero-dimensional ideal which is not
necessarily the maximal ideal, the latter situation corresponding to the
classical theory from the 80's; despite the apparent naivety of the idea, this
passage adds considerable depth to the theory and enlarges its horizon of
applications. First, we address the problem of when the Hom functor preserves
the Ulrich property, and in particular we study relations with semidualizing
modules. Second, we explore horizontal linkage of Ulrich modules, which we use
to provide a characterization of Gorensteiness. Finally, we investigate
connections between Ulrich modules and modules with minimal multiplicity,
including characterizations in terms of relative reduction numbers as well as
the Castelnuovo-Mumford regularity of certain blowup modules.
【3】 On M-O.Ore determinants
标题：关于M-O矿的决定因素
链接：https://arxiv.org/abs/2201.02361
摘要：The existence of certain Fq-spaces of differential forms of the projective
line over a field K containing Fq leads us to prove an identity linking the
determinant of the Moore matrix of n indeterminates with the determinant of the
Moore matrix of the cofactors of its first row. These same spaces give an
interpretation of Elkies pairing in terms of residues of differential forms.
This pairing puts in duality the Fq-vector space of the roots of a Fq-linear
polynomial and that of the roots of its reversed polynomial.
【4】 Stream/block ciphers, difference equations and algebraic attacks
标题：流/挡路密码、差分方程和代数攻击
链接：https://arxiv.org/abs/2003.14215
备注：26 pages, to appear in Journal of Symbolic Computation
摘要：In this paper we model a class of stream and block ciphers as systems of
(ordinary) explicit difference equations over a finite field. We call this
class "difference ciphers" and we show that ciphers of application interest, as
for example systems of LFSRs with a combiner, Trivium and Keeloq, belong to the
class. By using Difference Algebra, that is, the formal theory of difference
equations, we can properly define and study important properties of these
ciphers, such as their invertibility and periodicity. We describe then general
cryptanalytic methods for difference ciphers that follow from these properties
and are useful to assess the security. We illustrate such algebraic attacks in
practice by means of the ciphers Bivium and Keeloq.
4.math.RARings代数:
【1】 Graded irreducible representations of Leavitt path algebras: a new type and complete classification
标题：Leavitt路代数的分次不可约表示：一种新的完全分类
链接：https://arxiv.org/abs/2201.02446
摘要：We present a new class of graded irreducible representations of a Leavitt
path algebra. This class is new in the sense that its representation space is
not isomorphic to any of the existing simple Chen modules. The corresponding
graded simple modules complete the list of Chen modules which are graded,
creating an exhaustive class: the annihilator of any graded simple module is
equal to the annihilator of either a graded Chen module or a module of this new
type.
Our characterization of graded primitive ideals of a Leavitt path algebra in
terms of the properties of the underlying graph is the main tool for proving
the completeness of such classification. We also point out a problem with the
characterization of primitive ideals of a Leavitt path algebra in [K. M.
Rangaswamy, Theory of prime ideals of Leavitt path algebras over arbitrary
graphs, J. Algebra 375 (2013), 73 -- 90].
【2】 Not all nilpotent monoids are finitely related
标题：并非所有的幂零幺半群都是有限相关的
链接：https://arxiv.org/abs/2201.02375
摘要：A finite semigroup is finitely related (has finite degree) if its term
functions are determined by a finite set of finitary relations. For example, it
is known that all nilpotent semigroups are finitely related. A nilpotent monoid
is a nilpotent semigroup with adjoined identity. We show that every
$4$-nilpotent monoid is finitely related. We also give an example of a
$5$-nilpotent monoid that is not finitely related. This is the first known
example where adjoining an identity to a finitely related semigroup yields a
semigroup which is not finitely related. We also provide examples of finitely
related semigroups which have subsemigroups, homomorphic images, and in
particular Rees quotients, that are not finitely related.
【3】 Automorphisms of left Ideal relation graph over full matrix ring
标题：全矩阵环上左理想关系图的自同构
链接：https://arxiv.org/abs/2201.02345
摘要：The left-ideal relation graph on a ring $R$, denoted by
$\overrightarrow{\Gamma_{l-i}}(R)$, is a directed graph whose vertex set is all
the elements of $R$ and there is a directed edge from $x$ to a distinct $y$ if
and only if the left ideal generated by $x$, written as $[x]$, is properly
contained in the left ideal generated by $y$. In this paper, the automorphisms
of $\overrightarrow{\Gamma_{l-i}}(R)$ are characterized, where $R$ is the ring
of all $n \times n$ matrices over a finite field $F_q$. The undirected left
relation graph, denoted by $\Gamma_{l-i}(M_n(F_q))$, is the simple graph whose
vertices are all the elements of $R$ and two distinct vertices $x, y$ are
adjacent if and only if either $[x] \subset [y]$ or $[y] \subset [x]$ is
considered. Various graph theoretic properties of $\Gamma_{l-i}(M_n(F_q))$
including connectedness, girth, clique number, etc. are studied.
【4】 Torsion and torsion-free classes from objects of finite type in Grothendieck categories
标题：Grothendieck范畴中有限型对象的挠类和无挠类
链接：https://arxiv.org/abs/2201.02224
备注：27 pages
摘要：In an arbitrary Grothendieck category, we find necessary and sufficient
conditions for the class of $\text{FP}_n$-injective objects to be a torsion
class. By doing so, we propose a notion of $n$-hereditary categories. We also
define and study the class of $\text{FP}_n$-flat objects in Grothendieck
categories with a generating set of small projective objects, and provide
several equivalent conditions for this class to be torsion-free. In the end, we
present several applications and examples of $n$-hereditary categories in the
contexts modules over a ring, chain complexes of modules and categories of
additive functors from an additive category to the category of abelian groups.
Concerning the latter setting, we find a characterization of when these functor
categories are $n$-hereditary in terms of the domain additive category.
【5】 Stream/block ciphers, difference equations and algebraic attacks
标题：流/挡路密码、差分方程和代数攻击
链接：https://arxiv.org/abs/2003.14215
备注：26 pages, to appear in Journal of Symbolic Computation
摘要：In this paper we model a class of stream and block ciphers as systems of
(ordinary) explicit difference equations over a finite field. We call this
class "difference ciphers" and we show that ciphers of application interest, as
for example systems of LFSRs with a combiner, Trivium and Keeloq, belong to the
class. By using Difference Algebra, that is, the formal theory of difference
equations, we can properly define and study important properties of these
ciphers, such as their invertibility and periodicity. We describe then general
cryptanalytic methods for difference ciphers that follow from these properties
and are useful to assess the security. We illustrate such algebraic attacks in
practice by means of the ciphers Bivium and Keeloq.
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