413 research outputs found
Cohomology of idempotent braidings, with applications to factorizable monoids
We develop new methods for computing the Hochschild (co)homology of monoids
which can be presented as the structure monoids of idempotent set-theoretic
solutions to the Yang--Baxter equation. These include free and symmetric
monoids; factorizable monoids, for which we find a generalization of the
K{\"u}nneth formula for direct products; and plactic monoids. Our key result is
an identification of the (co)homologies in question with those of the
underlying YBE solutions, via the explicit quantum symmetrizer map. This
partially answers questions of Farinati--Garc{\'i}a-Galofre and Dilian Yang. We
also obtain new structural results on the (co)homology of general YBE
solutions
Classification of Reductive Monoid Spaces Over an Arbitrary Field
In this semi-expository paper we review the notion of a spherical space. In
particular we present some recent results of Wedhorn on the classification of
spherical spaces over arbitrary fields. As an application, we introduce and
classify reductive monoid spaces over an arbitrary field.Comment: This is the final versio
Euclidean Quadratic Forms and ADC Forms I
Motivated by classical results of Aubry, Davenport and Cassels, we define the
notion of a Euclidean quadratic form over a normed integral domain and an ADC
form over an integral domain. The aforementioned classical results generalize
to: Euclidean forms are ADC forms. We then initiate the study and
classification of these two classes of quadratic forms, especially over
discrete valuation rings and Hasse domains.Comment: 26 page
From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis
The novelty of the Jean Pierre Badiali last scientific works stems to a
quantum approach based on both (i) a return to the notion of trajectories
(Feynman paths) and (ii) an irreversibility of the quantum transitions. These
iconoclastic choices find again the Hilbertian and the von Neumann algebraic
point of view by dealing statistics over loops. This approach confers an
external thermodynamic origin to the notion of a quantum unit of time (Rovelli
Connes' thermal time). This notion, basis for quantization, appears herein as a
mere criterion of parting between the quantum regime and the thermodynamic
regime. The purpose of this note is to unfold the content of the last five
years of scientific exchanges aiming to link in a coherent scheme the Jean
Pierre's choices and works, and the works of the authors of this note based on
hyperbolic geodesics and the associated role of Riemann zeta functions. While
these options do not unveil any contradictions, nevertheless they give birth to
an intrinsic arrow of time different from the thermal time. The question of the
physical meaning of Riemann hypothesis as the basis of quantum mechanics, which
was at the heart of our last exchanges, is the backbone of this note.Comment: 13 pages, 2 figure
Twisting structures and strongly homotopy morphisms
In an application of the notion of twisting structures introduced by Hess and
Lack, we define twisted composition products of symmetric sequences of chain
complexes that are degreewise projective and finitely generated. Let Q be a
cooperad and let BP be the bar construction on the operad P. To each morphism
of cooperads g from Q to BP is associated a P-co-ring, K(g), which generalizes
the two-sided Koszul and bar constructions. When the co-unit from K(g) to P is
a quasi-isomorphism, we show that the Kleisli category for K(g) is isomorphic
to the category of P-algebras and of their morphisms up to strong homotopy, and
we give the classifying morphisms for both strict and homotopy P-algebras.
Parametrized morphisms of (co)associative chain (co)algebras up to strong
homotopy are also introduced and studied, and a general existence theorem is
proved. In the appendix, we study the particular case of the two-sided Koszul
resolution of the associative operad.Comment: 54 page
Toroidal crossings and logarithmic structures
We generalize Friedman's notion of d-semistability, which is a necessary
condition for spaces with normal crossings to admit smoothings with regular
total space. Our generalization deals with spaces that locally look like the
boundary divisor in Gorenstein toroidal embeddings. In this situation, we
replace d-semistability by the existence of global log structures for a given
gerbe of local log structures. This leads to cohomological descriptions for the
obstructions, existence, and automorphisms of log structures. We also apply
toroidal crossings to mirror symmetry, by giving a duality construction
involving toroidal crossing varieties whose irreducible components are toric
varieties. This duality reproduces a version of Batyrev's construction of
mirror pairs for hypersurfaces in toric varieties, but it applies to a larger
class, including degenerate abelian varieties.Comment: 34 pages, 1 figure, notational changes, to appear in Adv. Mat
The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy
For two given -terms and , the word problem for
-terms over a variety asks whether
in all monoids in . We show that the
word problem for -terms over each level of the Trotter-Weil Hierarchy
is decidable. More precisely, for every fixed variety in the Trotter-Weil
Hierarchy, our approach yields an algorithm in nondeterministic logarithmic
space (NL). In addition, we provide deterministic polynomial time algorithms
which are more efficient than straightforward translations of the
NL-algorithms. As an application of our results, we show that separability by
the so-called corners of the Trotter-Weil Hierarchy is witnessed by
-terms (this property is also known as -reducibility). In
particular, the separation problem for the corners of the Trotter-Weil
Hierarchy is decidable
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