arXiv subjects

Niels Kowalzig

Publications and source records attributed to Niels Kowalzig.

23 records · Page 2Linked to original sources

arXiv2012

Batalin-Vilkovisky Structures on Ext and Tor

This article studies the algebraic structure of homology theories defined by a left Hopf algebroid U over a possibly noncommutative base algebra A, such as for example Hochschild, Lie algebroid (in particular Lie algebra and Poisson), or group and etale groupoid (co)homology. Explicit formulae for the canonical Gerstenhaber algebra structure on Ext_U(A,A) are given. The main technical result constructs a Lie derivative satisfying a generalised Cartan-Rinehart homotopy formula whose essence is that Tor^U(M,A) becomes for suitable right U-modules M a Batalin-Vilkovisky module over Ext_U(A,A), or in the words of Nest, Tamarkin, Tsygan and others, that Ext_U(A,A) and Tor^U(M,A) form a differential calculus. As an illustration, we show how the well-known operators from differential geometry in the classical Cartan homotopy formula can be obtained. Another application consists in generalising Ginzburg's result that the cohomology ring of a Calabi-Yau algebra is a Batalin-Vilkovisky algebra to twisted Calabi-Yau algebras.

math.KT↗

arXiv2010

Cyclic structures in algebraic (co)homology theories

This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-Drinfel'd modules.

math.KT↗

arXiv2010

The cyclic theory of Hopf algebroids

We give a systematic description of the cyclic cohomology theory of Hopf algebroids in terms of its associated category of modules. Then we introduce a dual cyclic homology theory by applying cyclic duality to the underlying cocyclic object. We derive general structure theorems for these theories in the special cases of commutative and cocommutative Hopf algebroids. Finally, we compute the cyclic theory in examples associated to Lie-Rinehart algebras and étale groupoids.

math.KT↗

arXiv2009

Duality and products in algebraic (co)homology theories

The origin and interplay of products and dualities in algebraic (co)homology theories is ascribed to a $\times_A$-Hopf algebra structure on the relevant universal enveloping algebra. This provides a unified treatment for example of results by Van den Bergh about Hochschild (co)homology and by Huebschmann about Lie-Rinehart (co)homology.

math.QA↗

arXiv2004

Phase Space Reduction of Star Products on Cotangent Bundles

In this paper we construct star products on Marsden-Weinstein reduced spaces in case both the original phase space and the reduced phase space are (symplectomorphic to) cotangent bundles. Under the assumption that the original cotangent bundle $T^*Q$ carries a symplectique structure of form $ω_{B_0}=ω_0 + π^*B_0$ with $B_0$ a closed two-form on $Q$, is equipped by the cotangent lift of a proper and free Lie group action on $Q$ and by an invariant star product that admits a $G$-equivariant quantum momentum map, we show that the reduced phase space inherits from $T^*Q$ a star product. Moreover, we provide a concrete description of the resulting star product in terms of the initial star product on $T^*Q$ and prove that our reduction scheme is independent of the characteristic class of the initial star product. Unlike other existing reduction schemes we are thus able to reduce not only strongly invariant star products. Furthermore in this article, we establish a relation between the characteristic class of the original star product and the characteristic class of the reduced star product and provide a classification up to $G$-equivalence of those star products on $(T^*Q,ω_{B_0})$, which are invariant with respect to a lifted Lie group action. Finally, we investigate the question under which circumstances `quantization commutes with reduction' and show that in our examples non-trivial restrictions arise.

math.SG↗