From Bialgebras to Operads. Quantum Line and Cooperad of Correlation Functions

  • Yu. N. Bespalov Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
Keywords: bialgebra, operad, unbiased tensor products, multitensor category, vertex algebra

Abstract

A q-line is a simple example of a braided Hopf algebra. This is just an algebra of polynomials kq[z] with primitive generator and q-deformed statistics.
The (co)action of a q-line on an algebra is a q-derivation. We construct an operad and a cooperad from a bialgebra. In the case of a q-line, this construction is related to the cooperad of correlation functions of I. Kriz et al., which describes vertex algebras.
Modules over the factor-algebra kq[z]/(z^N) are N-complexes. We consider a homotopical category of N-complexes as an example of the q-analog of Maltsiniotis’ strongly triangulated category.
The general constructions are considered in the context of iterated monoidal categories with unbiased lax tensor products described in the terms of the Gray tensor products of 2-fold categorical operads of sequential trees Tree.

References

R. Hortsch, I. Kriz, and A. Pultr, J. of Algebra 324, 1731 (2010).

https://doi.org/10.1016/j.jalgebra.2010.05.012

Yu. Bespalov, V.V. Lyubashenko, and O. Manzyuk, Pretriangulated A∞-Categories (Institute of Mathematics of the NAS of Ukraine, Kyiv, 2008).

S. Forcey, J. Siehler, and S. Sowers, J. of Homotopy and Rel. Str. 2, 1 (2007).

Y. Bespalov and B. Drabant, in Quantum Groups and Quantum Spaces, edited by W. Pusz, R. Budzy´nski, and S. Zakrzewski (PUBLISHER, Warsaw, 1997), Vol. 40, arXiv: q-alg/9608019.

M. Aguiar and S. Mahajan, Monoidal Functors, Species and Hopf Algebras (AMS, Providence, RI, 2010), http://www.math.tamu.edu/maguiar/a.pdf.

https://doi.org/10.1090/crmm/029

M.M. Kapranov, arXiv:q-alg/9611005 (1996).

M. Dubois-Violette, dN = 0, LPTHE-ORSAY 97/53, available from http://qcd.th.u-psud.fr, 1997, arXiv: qalg/9710021.

T. Leinster, Higher Operads, Higher Categories (Cambridge Univ. Press, Cambridge, 2003), arXiv:math/0305049.

G. Maltsiniotis, Cat´egories triangul´ees sup´erieures (2006), http://www.math.jussieu.fr/ maltsin.

Published
2018-10-11
How to Cite
Bespalov, Y. (2018). From Bialgebras to Operads. Quantum Line and Cooperad of Correlation Functions. Ukrainian Journal of Physics, 58(11), 1033. https://doi.org/10.15407/ujpe58.11.1033
Section
Archive