Twisted dendriform algebras and the pre-Lie Magnus expansion

Abstract In this paper an application of the recently introduced pre-Lie Magnus expansion to Jackson’s q -integral and q -exponentials is presented. Twisted dendriform algebras, which are the natural algebraic framework for Jackson’s q -analogs, are introduced for that purpose. It is shown how the pre-Lie Magnus expansion is used to solve linear q -differential equations. We also briefly outline the theory of linear equations in twisted dendriform algebras.

[1]  Andrei A. Agrachev,et al.  Chronological algebras and nonstationary vector fields , 1981 .

[2]  A no‐go theorem for a Lie‐consistent q‐Campbell–Baker–Hausdorff expansion , 1994 .

[3]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[4]  J. Plebański,et al.  Combinatorial approach to Baker-Campbell-Hausdorff exponents , 1970 .

[5]  J. Littlewood,et al.  Notes on the theory of series (XXIV): a curious power-series , 1946, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  Li Guo,et al.  On Free Baxter Algebras: Completions and the Internal Construction , 2000 .

[7]  F. Chapoton.  Fractions de Bernoulli-Carlitz et op\'erateurs q-Zeta , 2009, 0909.1694.

[8]  A. Iserles,et al.  Lie-group methods , 2000, Acta Numerica.

[9]  Kurusch Ebrahimi-Fard,et al.  A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion , 2007 .

[10]  Pierre Cartier,et al.  On the structure of free baxter algebras , 1972 .

[11]  G. Baxter,et al.  AN ANALYTIC PROBLEM WHOSE SOLUTION FOLLOWS FROM A SIMPLE ALGEBRAIC IDENTITY , 1960 .

[12]  W. Assche,et al.  Leonhard Euler and a $q$-analogue of the logarithm , 2007, 0709.1788.

[13]  Li Guo,et al.  On Differential Rota-Baxter Algebras , 2007 .

[14]  Li Guo,et al.  Rota–Baxter algebras and dendriform algebras , 2008 .

[15]  A. Heller,et al.  The topology of discrete groups , 1980 .

[16]  Jean-Yves Thibon,et al.  A one-parameter family of dendriform identities , 2009, J. Comb. Theory, Ser. A.

[17]  Frederic Chapoton,et al.  Pre-Lie algebras and the rooted trees operad , 2000 .

[18]  S. Blanes,et al.  The Magnus expansion and some of its applications , 2008, 0810.5488.

[19]  R. Strichartz.  The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations , 1987 .

[20]  Frederic Chapoton.  Un théorème de Cartier–Milnor–Moore–Quillen pour les bigèbres dendriformes et les algèbres braces , 2000 .

[21]  Israel M. Gelfand,et al.  Noncommutative Symmetrical Functions , 1995 .

[22]  C. Nelson,et al.  On the two q -analogue logarithmic functions: ? , 1996, q-alg/9608015.

[23]  C. Quesne,et al.  Disentangling q-Exponentials: A General Approach , 2003, math-ph/0310038.

[24]  Gérard Henry Edmond Duchamp,et al.  Ordering relations for q-boson operators, continued fraction techniques and the q-CBH enigma , 1995 .

[25]  Kurusch Ebrahimi-Fard,et al.  A Magnus- and Fer-Type Formula in Dendriform Algebras , 2007, Found. Comput. Math..

[26]  Gian-Carlo Rota,et al.  Baxter algebras and combinatorial identities. II , 1969 .

[27]  Li Guo,et al.  Baxter Algebras and Shuffle Products , 2000 .

[28]  J. Katriel,et al.  A q-analogue of the Campbell-Baker-Hausdorff expansion , 1991 .

[29]  R. Wilcox.  Exponential Operators and Parameter Differentiation in Quantum Physics , 1967 .

[30]  Li Guo,et al.  On products and duality of binary, quadratic, regular operads , 2004 .

[31]  Kurusch Ebrahimi-Fard,et al.  Dendriform Equations , 2008, 0805.0762.

[32]  F. Atkinson,et al.  Some aspects of Baxter's functional equation , 1963 .

[33]  Fr'ed'eric Chapoton,et al.  A rooted-trees q-series lifting a one-parameter family of Lie idempotents , 2008, 0807.1830.