arXiv subjects

Piyush Shroff

Publications and source records attributed to Piyush Shroff.

6 recordsLinked to original sources

arXiv2017

Oriented Hypergraphic Matrix-tree Type Theorems and Bidirected Minors via Boolean Order Ideals

Restrictions of incidence-preserving path maps produce an oriented hypergraphic All Minors Matrix-tree Theorems for Laplacian and adjacency matrices. The images of these maps produce a locally signed graphic, incidence generalization, of cycle covers and basic figures that correspond to incidence-k-forests. When restricted to bidirected graphs the natural partial ordering of maps results in disjoint signed boolean lattices whose minor calculations correspond to principal order ideals. As an application, (1) the determinant formula of a signed graphic Laplacian is reclaimed and shown to be determined by the maximal positive-circle-free elements, and (2) spanning trees are equivalent to single-element order ideals.

math.CO↗

arXiv2017

Order dividing bijective function from non-cyclic to cyclic groups of same finite order

In this article we give an order-dividing bijective function between cyclic and non cyclic groups of finite order. In particular, we prove that there exists a bijective function from D_{2n} to Z_{2n} for any natural integer n; and from Z_p x Z_k to Z_{pk} when p is an odd prime and k is not a multiple of p.

math.GR↗

arXiv2015

PBW deformations of skew polynomial rings and their group extensions

We examine PBW deformations of finite group extensions of skew polynomial rings, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

math.RA↗

arXiv2014

Quantum Drinfeld Orbifold Algebras

Quantum Drinfeld orbifold algebras are the generalizations of Drinfeld orbifold algebras, which are obtained by replacing polynomial rings by quantum polynomial rings. Shepler and Witherspoon in their paper, give necessary and sufficient conditions on a defining parameters to obtain Drinfeld orbifold algebras. In this article we generalize their result. It also simultaneously generalizes the result of Levandovskyy and Shepler about quantum Drinfeld Hecke algebras.

math.RA↗

arXiv2012

Finite Generation of the Cohomology of Quotients of PBW Algebras

In this article we prove finite generation of the cohomology of quotients of a PBW algebra A by relating it to the cohomology of quotients of a quantum symmetric algebra S which is isomorphic to the associated graded algebra of A. The proof uses a spectral sequence argument and a finite generation lemma adapted from Friedlander and Suslin.

math.RA↗

arXiv2010

Hochschild cohomology of group extensions of quantum symmetric algebras

Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule algebra. When this bimodule algebra is a finite group extension (under a diagonal action) of a quantum symmetric algebra, we give explicitly the graded vector space structure. This yields a complete description of the Hochschild cohomology ring of the corresponding skew group algebra.

math.RA↗