Ioannis Dokas - Profile on Academia.edu (original) (raw)

Papers by Ioannis Dokas

Homology, Homotopy and Applications, 2015

In this paper we study Duskin-Glenn torsor cohomology in the context of restricted Lie algebras. ... more In this paper we study Duskin-Glenn torsor cohomology in the context of restricted Lie algebras. In particular, we give an interpretation of the torsor cohomology groups which appear in Cegarra-Aznar's eight-term exact sequence. Thus, we prove a classification theorem for the second Quillen-Barr-Beck cohomology in terms of 2-fold extensions of restricted Lie algebras. This paper is dedicated to the memory of Jean-Louis Loday. I would like to express my appreciation to the Isaac Newton Institute for the excellent working conditions that I enjoyed there during the program "Grothendieck-Teichmüller groups, Deformations and Operads."

arXiv (Cornell University), Oct 13, 2011

We give an interpretation of the Brauer group of a purely inseparable extension of exponent 1, in... more We give an interpretation of the Brauer group of a purely inseparable extension of exponent 1, in terms of restricted Lie-Rinehart cohomology. In particular, we define and study the category p-LR(A) of restricted Lie-Rinehart algebras over a commutative algebra A. We define cotriple cohomology groups H p-LR (L, M ) for L ∈ p-LR(A) and M a Beck L-module. We classify restricted Lie-Rinehart extensions. Thus, we obtain a classification theorem for regular extensions considered by Hoshschild.

$Research paper thumbnail of On K\"ahler differentials of divided powers algebras$

The Quillen-Barr-Beck cohomology of augmented algebras with divided powers is defined as the deri... more The Quillen-Barr-Beck cohomology of augmented algebras with divided powers is defined as the derived functor of Beck derivations. The main theorem of this paper states that the Kähler differentials of an augmented algebra with divided powers in prime characteristic represents Beck derivations. We give a geometrical interpretation of this statement for the sheaf of relative differentials. As an application in modular Lie theory we prove that any special derivation of a divided powers algebra is a Beck derivation and we apply the theorem to Witt algebras.

Journal of Lie Theory, Nov 18, 2010

In prime characteristic we introduce the notion of restricted pre-Lie algebras. We prove in the p... more In prime characteristic we introduce the notion of restricted pre-Lie algebras. We prove in the pre-Lie context the analogue to Jacobson's theorem for restricted Lie algebras. In particular, we prove that any dendriform algebra over a field of positive characteristic is a restricted pre-Lie algebra. Thus we obtain that Rota-Baxter algebras and quasitriangular algebras are restricted pre-Lie algebras. Moreover, we prove that the free Γ(preLie)-algebra is a restricted pre-Lie algebra, where preLie denotes the pre-Lie operad. Finally, we define the notion of restricted enveloping dendriform algebra and we construct a left adjoint functor for the functor (-) p-preLie : Dend → p -preLie.

Communications in Algebra, 2006

In this paper we prove that in prime characteristic there is a functorp-Leib from the category of... more In this paper we prove that in prime characteristic there is a functorp-Leib from the category of diassociative algebras to the category of restricted Leibniz algebras, generalizing the functor from associative algebras to restricted Lie algebras. Moreover we define the notion of restricted universal enveloping diassociative algebra U dp(g) of a restricted Leibniz algebra g and we show that U dp is left adjoint to the functorp-Leib . We also construct the restricted enveloping algebra, which classifies the restricted Leibniz modules. In the last section we put a restricted pre-Lie structure on the tensor product of a Leibniz algebra by a Zinbiel algebra.

Journal of the Australian Mathematical Society, 2009

In this paper using the Quillen–Barr–Beck (co-)homology theory of universal algebras we define (c... more In this paper using the Quillen–Barr–Beck (co-)homology theory of universal algebras we define (co-)homology groups for commutative algebras with divided powers in prime characteristic. In particular, we determine for A a commutative 𝔽p-algebra with divided powers, the category of Beck A-modules and the group of Beck derivations. We construct the abelianization functor and we define (co-)homology. Moreover, we determine the cohomology in low dimensions and we interpret the first cohomology in terms of extensions.

Journal of Pure and Applied Algebra, 2004

In this paper we use Quillen-Barr-Beck's theory of (co-) homology of algebras in order to deÿne (... more In this paper we use Quillen-Barr-Beck's theory of (co-) homology of algebras in order to deÿne (co-) homology for the category RLie of restricted Lie algebras over a ÿeld k of characteristic p = 0. In contrast with the cases of groups, associative algebras and Lie algebras we do not obtain Hochschild (co-) homology shifted by 1. Precisely, we determine for L ∈ RLie the category of Beck L-modules and the group of Beck derivations of g ∈ RLie=L to a Beck L-module M. Moreover, we prove a classiÿcation theorem which gives a one-to-one correspondence between the one cohomology and the set of equivalent classes of p-extensions. Finally, a universal coe cient theorem is proved, relating the homology to the Hochschild homology via a short exact sequence. This shows that the new homology determines the Hochschild homology.

Glasgow Mathematical Journal, 2009

In this paper, we prove that any Zinbiel algebra can be endowed with the structure of commutative... more In this paper, we prove that any Zinbiel algebra can be endowed with the structure of commutative algebra with divided powers. We introduce the notion of universal enveloping Zinbiel algebra of a commutative algebra with divided powers algebras. We prove that the free divided powers algebra on a free module M, is the divided powers sub-algebra generated by M, of the divided powers algebra induced by the free Zinbiel algebra on M. Finally, we construct a basis for the enveloping Zinbiel algebra.

Given an exact sequence of restricted Lie algebras using Duskin's torsors theory, we establish an... more Given an exact sequence of restricted Lie algebras using Duskin's torsors theory, we establish an eight term exact sequence for Quillen-Barr-Beck cohomology of restricted Lie algebras. As an application we obtain for any extension of algebraic groups over an algebraic closed field of prime characteristic an eight term exact sequence for the corresponding restricted Lie algebras extension.