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Juan J. Guccione

Publications and source records attributed to Juan J. Guccione.

36 records · Page 2Linked to original sources
arXiv2011

Cyclic homology of braided Hopf crossed products

Let k be a field, A a unitary associative k-algebra and V a k-vector space endowed with a distinguished element 1_V. We obtain a mixed complex, simpler that the canonical one, that gives the Hochschild, cyclic, negative and periodic homology of a crossed product E:=A#_f V, in the sense of Brzezinski. We actually work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A that satisfies suitable hypothesis and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. Then, when E is a cleft braided Hopf crossed product, we obtain a simpler mixed complex, that also gives the Hochschild, cyclic, negative and periodic homology of E.

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arXiv2010

Universal deformation formulas and braided module algebras

We study formal deformations of a crossed product $S(V)#_f G$, of a polynomial algebra with a group, induced from a universal deformation formula introduced by Witherspoon. These deformations arise from braided actions of Hopf algebras generated by automorphisms and skew derivations. We show that they are nontrivial in the characteristic free context, even if $G$ is infinite, by showing that their infinitesimals are not coboundaries. For this we construct a new complex which computes the Hochschild cohomology of $S(V)#_f G$.

math.RA↗
arXiv2008

A characterization of quiver algebras based on double derivations

Let k a characteristic zero field. We give a characterization for the finite quiver k-algebras, based on double derivations. More precisely, we prove that if an associative and unitary k-algebra have a family of double derivations satisfying suitable conditions, then it is (canonically isomorphic to) a quiver algebra. This is the non-commutative version of a result of D. Wright.

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arXiv2008

Cyclic homology of crossed products

We obtain a mixed complex, simpler that the canonical one, given the Hochschild, cyclic, negative and periodic homology of a crossed product E=A#fH, where H is an arbitrary Hopf algebra and f is a convolution invertible cocycle with values in A. Actually, we work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A which is stable under the action of H, and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. As an application we obtain two spectral sequences converging to the cyclic homology of E relative to K. The first one in the general setting and the second one (which generalizes those previously found by several authors) when f takes its values in K.

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arXiv2007

Twisted planes

Let k be a commutative ring. We find and characterize a new family of twisted planes (i. e. associative unitary k-algebra structures on the k-module k[X,Y], having k[X] and and k[Y] as subalgebras).Similar results are obtained for the k-module of two variables power series k[[X,Y]].

math.RA↗
arXiv2007

Braided Sweedler cohomology

We introduced a braided Sweedler cohomology, which is adequate to work with the H-braided cleft extensions studied in [J. A. Guccione and J. J. Guccione, Theory of braided Hopf crossed products, Journal of Algebra, Vol 261 (2003) 54-101]

math.RA↗
arXiv2007

The cohomology of monogenic extensions in the noncommutative setting

We extend the notion of monogenic extension to the noncommutative setting, and we study the Hochschild cohomology ring of such an extension. As an aplication we complete the computation of the cohomology ring of the rank one Hopf algebras beggined in [S. M. Burciu and S. J. Witherspoon, Hochschild cohomology of smash products and rank one Hopf algebras].

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arXiv2007

The cyclic homology of monogenic extensions in the noncommutative setting

We study the Hochschild and cyclic homologies of noncommutative monogenic extensions. As an aplication we compute the Hochschild and cyclic homologies of the rank~1 Hopf algebras introduced by L. Krop and D. Radford in [Finite dimensional Hopf algebras of rank 1 in characteristic 0, Journal of Algebra 302, no. 1, 214-230} (2006)].

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arXiv2006

Relative cyclic homology of square zero extensions

Let k be a characteristic zero field, C a k-algebra and M a square zero two sided ideal of C. We obtain a new mixed complex, simpler that the canonical one, giving the Hochschild and cyclic homologies of C relative to M. This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like to the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra. We also give new proofs of two theorems of Goodwillie, obtaining an improvement of one of them.

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arXiv2003

Semiquasitriangular Hopf algebras

We say that a Hopf algebra H is semicocommutative if the right adjoint coaction factorizes through the tensor product of H with the center of H. For instance the commutative and the cocommutative Hopf algebras are semicocommutative. The quasitriangular Hopf algebras generalize the cocommutative Hopf algebras. In this paper we introduce and begin the study of a similar generalization for the semicocommutative ones. These algebras, which we call semiquasitriangular Hopf algebras have many of the basic properties of the quasitriangular ones. In particular, they have associated braided categories of representations in a natural way.

math.QA↗
arXiv2002

Hochschild cohomology of Frobenius algebras

Let k be a field and let A be a Frobenius algebra over k. Assume that the Nakayama automorphism of A associated to a Frobenius homomorphism of A has finite order m, and k has a m-th primitive root of unity. Then, A has a natural Z/mZ-gradation. Consider the decomposition of the Hochschild cohomology HH*(A), of A with coefficients in A, induced by this gradation. We prove that just the 0-degree component of HH*(A) is non trivial. Moreover, we prove that if A is a strongly Z/mZ-graded algebra, then Z/mZ acts on the Hochschild cohomology HH*(A_0), of the 0-degree component of A, and HH*(A) is the set of invariants of this action.

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arXiv2002

Hochschild homology of iterate skew polynomial rings

We study the Hochschild homology of the iterated skew polynomial rings introduced by D. Jordan in ``A simple localization of the quantized Weyl algebra''. First, we obtain a complex, smaller than the canonical one of Hochschild, given the homology of such an algebra, and then, we study this complex in order to compute the homology of some families of algebras. In particular we compute the homology of some quantum groups, in the generic case.

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arXiv2001

Cohomology of trivial extensions of Frobenius algebras

We obtain a decomposition for the Hochschild cochain complex of a split algebra and we study some properties of the cohomology of each term of this decomposition. Then, we consider the case of trivial extensions, specially of Frobenius algebras. In particular, we determine completely the cohomology of the trivial extension of a finite dimensional Hopf algebra. Finally, as an application, we obtain a result about the Hochschild cohomology of Frobenius algebras.

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arXiv2001

Hochschild cohomology of triangular matrix algebras

We prove the existence of two long exact sequences relating the Hochschild cohomology of a triangular matrix algebra with the Hochschild homology of its component subalgebras. We also study the structure of the maps of the first sequence.

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arXiv2001

Hochschild (co)homology of Hopf crossed products

Let A a k-algebra, H a Hopf algebra, E = A#H a general crossed product and M an E-bimodule. We obtain a complex simpler than the canonical one, giving the Hochschild homology of E with coefficients in M. This complex is eqquiped with a natural filtration. We prove that the associated spectral sequence coincides with that obtained by either, the Hochschild-Serre direct method or the Cartan-Leray-Grothendieck method. We also get similar results for the cohomology.

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