The Hochschild cohomology of the algebra of differential operators tangent to a central arrangement of lines

Given a central arrangement of lines A in a 2-dimensional vector space V over a field of characteristic zero, we study the algebra D (A) of differential operators on V which are logarithmic along A. Among other things we determine the Hochschild cohomology of D (A) as a Gerstenhaber algebra, establish a connection between that cohomology and the de Rham cohomology of the complement M(A) of the arrangement, determine the isomorphism group of D (A) and classify the algebras of that form up to isomorphism.Fil: Kordon, Francisco. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Suárez-Alvarez, Mariano. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Sistemas Hamiltonianos

Los sistemas hamiltonianos definidos en variedades son sistemas dinámicos que nacen para formalizar la descripción de problemas de la mecánica clásica. Integrar un sistema hamiltoniano es, moralmente, conseguir ecuaciones no diferenciales para sus trayectorias. En este artículo hacemos una introducción a estos temas y en la última sección damos condiciones suficientes para integrar sistemas hamiltonianos y dar descripciones geométricas cualitativas de su dinámica.Fil: Kordon, Francisco. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin

Hochschild cohomology of algebras of differential operators associated with hyperplane arrangements

Dado un arreglo de hiperplanos A en un espacio vectorial V sobre un cuerpo de característica cero, estudiamos el álgebra Diff(A) de operadores diferenciales enV tangentes a los hiperplanos de A desde el punto de vista del álgebra homológica. Hacemos un estudio detallado de este álgebra para el caso de un arreglo central de rectas en un espacio vectorial de dimensión 2. Entre otras cosas, determinamos la cohomología de Hochschild HH(Diff(A)) como álgebra de Gerstenhaber, establecemos un vínculo entre ésta y la cohomología de de Rham del complemento M(A) del arreglo, determinamos el grupo de isomorfismos de Diff(A), clasificamos las álgebras de esta forma a menos de isomorfismo y estudiamos las deformaciones formales de Diff(A). Mostramos que en el contexto general de un arreglo de hiperplanos de dimensión arbitraria el álgebra Diff(A) es isomorfa al álgebra envolvente del par de Lie–Rinehart formado por el álgebra de funciones coordenadas del espacio vectorial y el álgebra de Lie de derivaciones tangentes al arreglo. El cálculo de la cohomología de Hochschild de Diff(A) puede ser ubicado entonces en el contexto del cálculo de la del álgebra envolvente U de un par de Lie–Rinehart (S; L): damos un método para hacer esto en el caso en que L es un S-módulo proyectivo. Concretamente, presentamos una sucesión espectral que converge a HH(U ) cuya segunda página involucra la cohomología de Lie–Rinehart del par (S; L) y la cohomología de Hochschild de S a valores en U .Given a free hyperplane arrangement A in a vector space V over a field of characteristic zero, we study the algebra Diff(A) of differential operators on V which are tangent to the hyperplanes of A from the point of view of homological algebra. We make a thorough study of this algebra for the case of a central arrangement of lines in a vector space of dimension 2. Among other things, we determine the Hochschild cohomology HH(Diff(A)) as a Gerstenhaber algebra, establish a connection between that cohomology and the de Rham cohomology of the complement M(A) of the arrangement, determine the isomorphism group of Diff(A), classify the algebras of that form up to isomorphism and study the formal deformations of Diff(A). We show that in the general setting of a free arrangement of hyperplanes of arbitrary dimension the algebra Diff(A) is isomorphic to the enveloping algebra of the Lie–Rinehart pair formed by the algebra of coordinates functions on the vector space and the Lie algebra of derivations tangent to the arrangement. The computation of the Hochschild cohomology of Diff(A) can be then put in the context of computing that of the enveloping algebra U of a Lie–Rinehart pair (S; L): we provide a method to do this if L is S-projective. Concretely, we present a spectral sequence which converges to HH(U ) and whose second page involves the Lie—Rinehart cohomology of the pair and the Hochschild cohomology of S with values on U .Fil: Kordon, Francisco. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

CENTER AND LIE ALGEBRA OF OUTER DERIVATIONS FOR ALGEBRAS OF DIFFERENTIAL OPERATORS ASSOCIATED TO HYPERPLANE ARRANGEMENTS

We compute the center and the Lie algebra of outer derivations of a familiy of algebras of differential operators associated to hyperplane arrangements of the affine space A 3. The results are completed for 4-braid arrangements and for reflection arrangements associated to the wreath product of a cyclic group with the symmetric group S 3. To achieve this we use tools from homological algebra and Lie-Rinehart algebras of differential operators

LIE-RINEHART AND HOCHSCHILD COHOMOLOGY FOR ALGEBRAS OF DIFFERENTIAL OPERATORS

International audienceLet (S, L) be a Lie-Rinehart algebra such that L is S-projective and let U be its universal enveloping algebra. In this paper we present a spectral sequence which converges to the Hochschild cohomology of U with values on a U-bimodule M and whose second page involves the Lie-Rinehart cohomology of the algebra and the Hochschild cohomology of S with values on M. After giving a convenient description of the involved algebraic structures we use the spectral sequence to compute explicitly the Hochschild cohomology of the algebra of differential operators tangent to a central arrangement of three lines

Embryonal mass and hormone-associated effects of pregnancy inducing a differential growth of four murine tumors

A differential effect of pregnancy on the growth of subcutaneous implants of four murine tumors has been observed. Two tumors lacking receptors for progesterone and estrogen [methylcholanthrene-induced fibrosarcoma (MC-C) and spontaneous lymphoid leukemia (LB)] exhibited slow kinetics throughout the course of pregnancy, although inhibition was stronger beyond day 10. On the other hand, one of two tumors bearing receptors for progesterone and estrogen [medroxyprogesterone (MPA)-induced mammary adenocarcinoma (C7HI)] exhibited three phases: up to days 8-10 of gestation the tumor grew faster than in virgins, between days 8-10 and 15 it reached a plateau, and beyond day 15 a sharp reduction in tumor mass was observed. The other tumor [mouse mammary tumor virus (MMTV)-induced mammary carcinoma(T2280)] behaved as a typical pregnancy-dependent tumor (i.e., it grew in pregnant but not in virgin mice, regressed soon after delivery, and reassumed its growth at the middle of a second round of pregnancy). Neither MPA nor estrogen affected MC-C and LB tumor growth. On the other hand, MPA-treated mice enhanced C7HI tumor and reciprocally C7HI tumor-bearing mice treated with estrogen strongly inhibited tumor growth. As for T2280, neither MPA nor estrogen alone could promote tumor growth and, in consequence, no tumor developed. However, when MPA plus estrogen was administered in a schedule simulating the successive appearance of these hormones in pregnancy, T2280 grew even faster than in pregnant mice. When the four tumors were implanted in mice bearing grafts of embryonal tissues (teratomas), all of them were inhibited. This antitumor effect was similar to that observed in pregnancy when tumors unresponsive to progesterone and estrogen were tested. On the other hand, with tumors bearing progesterone and estrogen receptors, differences in tumor growth were detected in pregnant and teratoma-bearing mice. This suggested the existence during pregnancy of two factors potentially acting on tumor growth. First, a progesterone and estrogen-mediated hormonal component, which would exert either inhibitory or stimulatory effects only evidenced with tumors bearing hormonal receptors. Secondly, an antitumor effect proportional to the growing embryonal mass, inhibiting all tumors independently of their origin or hormone responsiveness. This antitumor effect could be attributed to a beat-resistant serum factor (1,000-1,200 Da molecular weight) presumably associated with the pathway of the arachidonic acid metabolism. The interplay between the hormonal component and the serum factor associated with embryonal mass could account for some of the largely heterogeneous and otherwise unexplained effects of pregnancy on tumor growth reported in the literature and illustrated by the four tumors studied here.Fil: Bustuoabad, Oscar David. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Medicina Experimental. Academia Nacional de Medicina de Buenos Aires. Instituto de Medicina Experimental; ArgentinaFil: di Gianni, Pedro D.. Academia Nacional de Medicina de Buenos Aires; ArgentinaFil: Franco, Marcela. Academia Nacional de Medicina de Buenos Aires; ArgentinaFil: Kordon, Edith Claudia. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Academia Nacional de Medicina de Buenos Aires; ArgentinaFil: Vanzulli, Silvia I.. Academia Nacional de Medicina de Buenos Aires; ArgentinaFil: Meiss, Roberto P.. Academia Nacional de Medicina de Buenos Aires; ArgentinaFil: Grion, Lorena C.. Academia Nacional de Medicina de Buenos Aires; ArgentinaFil: Diaz, Graciela Susana. Academia Nacional de Medicina de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental. Fundación de Instituto de Biología y Medicina Experimental. Instituto de Biología y Medicina Experimental; ArgentinaFil: Nosetto, Sergio H.. Academia Nacional de Medicina de Buenos Aires; ArgentinaFil: Hockl, Pablo Francisco. Academia Nacional de Medicina de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental. Fundación de Instituto de Biología y Medicina Experimental. Instituto de Biología y Medicina Experimental; ArgentinaFil: Lombardi, María Gabriela. Academia Nacional de Medicina de Buenos Aires; ArgentinaFil: Pasqualini, Christiane Dosne. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Academia Nacional de Medicina de Buenos Aires; ArgentinaFil: Ruggiero, Raul Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Academia Nacional de Medicina de Buenos Aires; Argentin