From Bialgebras to Operads. Quantum Line and Cooperad of Correlation Functions
Keywords:bialgebra, operad, unbiased tensor products, multitensor category, vertex algebra
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.
R. Hortsch, I. Kriz, and A. Pultr, J. of Algebra 324, 1731 (2010).
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.
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.
How to Cite
License to Publish the Paper
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.