Let ℚ p be the field of p-adic numbers and ???? a split connected reductive group scheme over ℤ p. In this work we will introduce a sheaf of twisted arithmetic differential operators on the formal flag variety of ????, associated to a general character. In particular, we will generalize the results of [21], concerning the ???? †-affinity of the smooth formal flag variety of ????, of certain sheaves of p-adically complete twisted arithmetic differential operators associated to an algebraic character, and the results of [24] concerning the calculation of the global sections.

This paper is focused on some algebraic and combinatorial properties of a TMTO (Time–Memory Trade-Off) for a chosen plaintext attack against a cryptosystem with a perfect secrecy property. TMTO attacks aim to retrieve the preimage of a given one-way function more efficiently than an exhaustive search and with less memory than a dictionary attack. TMTOs for chosen plaintext attacks against cryptosystems with a perfect secrecy property are associated with some directed graphs, which can be defined by suitable collections of multisets called Brauer configurations. Such configurations induce so-called Brauer configuration algebras, the algebraic and combinatorial invariant analysis of which is said to be a Brauer analysis. In this line, this paper proposes formulas for dimensions of Brauer configuration algebras (and their centers) induced by directed graphs defined by TMTO attacks. These results are used to provide some set-theoretical solutions for the Yang–Baxter equation.

Let F be a field of zero characteristic. We define a dialgebra structure over the polynomial algebra $$F[x] \otimes F[x]$$ F [ x ] ⊗ F [ x ] , and we give a complete classification of the Lie algebra of derivations "Equation missing" and the space of diderivations "Equation missing". We also show that the results of this work are closely related to those obtained in Lin and Zhang (Commun Algebra 38(9):3417–3447, 2010) and Restrepo-Sánchez (Inmersión de dialgebras en dialgebras separables, adición de unidades barra y biderivaciones, Tesis de Maestría, Universidad de Antioquia, Medellin, 2010).

Localisation de représentations localement analytiques admissibles: Soit G un schéma en groupes réductif, connexe et déployé sur l’anneau d’entiers d’une extension finie du corps de nombres p-adiques. Un théorème important dans la théorie des groupes c’est le théorème de localisation, ce qui a été démontré par A. Beilinson et J. Bernstein, et par J.L. Brylinsky et M. Kashiwara. Il s’agit d’un résultat de D-affinité pour la variété de drapeaux de la fibre générique de G. En caractéristique mixte un progrès important se trouve dans les travaux de C. Huyghe et T. Schmidt. Ils donnent une réponse partielle en considérant des caractères algébriques. Les premières quatre chapitres de cette thèse sont consacrés à étendre cette correspondance (le théorème de localisation arithmétique) pour des caractères arbitraires. Dans les chapitres cinq et six, nous traiterons l’objectif principal de cette thèse qui concerne les représentations localement analytiques. Nous montrerons que pour un caractère algébrique, qui est de plus dominant et régulier, la catégorie des représentations admissibles localement analytiques, à caractère central, c’est équivalente à une catégorie de modules arithmétiques coadmissibles et équivariants sur la famille des modèles formels de la variété de drapeaux rigide.

We construct and analyze a family of support-defined Brauer-type configurations canonically associated with the Boolean geometry underlying the Grassmann algebra. The construction is governed by an x-support map on monomial labels, which identifies the vertex set with the Boolean lattice $\mathcal{P}\left(\right[n\left]\right)$. This identification yields a Boolean support quiver isomorphic to the directed Hasse diagram of $\mathcal{P}\left(\right[n\left]\right)$, equivalently, to an oriented hypercube. We then equip the family with a canonical cyclic ordering at each vertex and obtain a genuine connected reduced Brauer configuration in the standard sense, together with its associated Brauer configuration algebra and its standard Brauer quiver. A ghost-variable mechanism is introduced to obtain a connected realization without altering any support-controlled invariants. We prove that polygon membership, valencies, multiplicities, Boolean stratification, and the support quiver are invariant under support-preserving ghost relabelings. We also give an explicit description of the standard Brauer quiver and show that it is different from the Boolean support quiver. On the algebraic side, we derive closed formulas for the center dimension, the algebra dimension, and the normalization constant of the induced weighted distribution. On the probabilistic side, we distinguish the vertex entropy from the layer entropy, establish an exact decomposition of the former by Hamming layers, and show that the layer distribution is asymptotically concentrated on the middle layers, while extremal vertices and any fixed maximal path contribute a negligible fraction of the total weight. As a consequence, the layer entropy satisfies a logarithmic asymptotic law. We also investigate geometric consequences of the Boolean model transported through the support identification. Coordinate projections produce a rigidity phenomenon for antipodal pairs, providing a combinatorial analogue of Greenberger–Horne–Zeilinger (GHZ)-type fragility, whereas the first Boolean layer exhibits a persistence property analogous to W-type robustness. Together, these results exhibit a concrete bridge between Grassmann combinatorics, Brauer configuration theory, hypercube geometry, and entropy asymptotics.

We compare Cuntz–Krieger and Brauer relations inside the same path algebra when every configuration vertex has valency two. Choosing one arrow from each successor pair produces a directed graph, and the Cuntz–Krieger quotient is its Leavitt path algebra. The Brauer quotient is the usual Brauer configuration algebra; it belongs to the Brauer graph class when all polygons are 2-gons. We also determine the two mixed quotients. One can be described for relaxed data and disappears under the standard Brauer axioms, while the other is always zero. In the converse direction, any finite directed graph can be recovered from a standard Brauer configuration and a cut, with isolated vertices accounting for scalar summands. The choice of cut affects more than growth: for diamond-tail and complete binary-tree configurations it changes the Grothendieck group. In the balanced cases we obtain closed spectral formulas and a determinant recurrence for the orders of the finite groups that occur.