Free integro-differential algebras and Grobner Shirshov bases

Abstract The notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such equations. In both cases, free objects are crucial for analyzing the underlying algebraic structures, e.g. of the (matrix) functions. In this paper we apply the method of Grobner–Shirshov bases to construct the free (noncommutative) integro-differential algebra on a set. The construction is from the free Rota–Baxter algebra on the free differential algebra on the set modulo the differential Rota–Baxter ideal generated by the noncommutative integration by parts formula. In order to obtain a canonical basis for this quotient, we first reduce to the case when the set is finite. Then in order to obtain the monomial order needed for the Composition-Diamond lemma, we consider the free Rota–Baxter algebra on the truncated free differential algebra. A Composition-Diamond lemma is proved in this context, and a Grobner–Shirshov basis is found for the corresponding differential Rota–Baxter ideal.

[1]  J. Ritt,et al.  Differential Equations from the Algebraic Standpoint , 1933 .

[2]  L. A. Bokut,et al.  Embeddings into simple associative algebras , 1976 .

[3]  Yuqun Chen,et al.  Gröbner–Shirshov bases and their calculation , 2013, 1303.5366.

[4]  Yongshan Chen,et al.  Composition–Diamond lemma for tensor product of free algebras☆ , 2008, 0804.2115.

[5]  Li Guo,et al.  Differential type operators and Gröbner-Shirshov bases , 2012, J. Symb. Comput..

[6]  L. A. Bokut,et al.  Gröbner-Shirshov bases for Rota-Baxter algebras , 2009, 0908.2281.

[7]  J. Ritt,et al.  Differential Equations from the Algebraic Standpoint , 1933, Nature.

[8]  Chang Liu,et al.  Term rewriting and all that , 2000, SOEN.

[9]  Li Guo,et al.  On integro-differential algebras , 2012, 1212.0266.

[10]  Super, quantum and non-commutative species , 2005, math/0509674.

[11]  Li Guo,et al.  An introduction to Rota-Baxter algebra , 2012 .

[12]  Jianjun Qiu,et al.  Grobner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras ∗† , 2008, 0805.0640.

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

[14]  Li Guo,et al.  Relative locations of subwords in free operated semigroups and Motzkin words , 2014, 1401.7385.

[15]  Tobias Nipkow,et al.  Term rewriting and all that , 1998 .

[16]  Li Guo,et al.  Constructions of Free Commutative Integro-Differential Algebras , 2012, AADIOS.

[17]  Leonid A. Bokut,et al.  Selected works of A.I. Shirshov , 2009 .

[18]  Markus Rosenkranz,et al.  Solving and factoring boundary problems for linear ordinary differential equations in differential algebras , 2008, J. Symb. Comput..

[19]  Gilbert Labelle,et al.  Combinatorial species and tree-like structures , 1997, Encyclopedia of mathematics and its applications.

[20]  Li Guo,et al.  Operated semigroups, Motzkin paths and rooted trees , 2007, 0710.0429.

[21]  H. Hironaka.  Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II , 1964 .

[22]  Li Guo,et al.  Mixable shuffles, quasi-shuffles and Hopf algebras , 2006 .

[23]  E. Kolchin.  Differential Algebra and Algebraic Groups , 2012 .

[24]  Michèle Soria,et al.  Algorithms for combinatorial structures: Well-founded systems and Newton iterations , 2011, J. Comb. Theory, Ser. A.

[25]  G. Bergman.  The diamond lemma for ring theory , 1978 .

[26]  Gilbert Labelle.  On combinatorial differential equations , 1986 .

[27]  C. Deavours.  The Quaternion Calculus , 1973 .

[28]  Li Guo,et al.  Construction of free commutative integro-differential algebras by the method of Gr\"{o}bner-Shirshov bases , 2013, 1302.0041.

[29]  L. A. Bokut,et al.  Gr\"obner-Shirshov bases for categories , 2011, 1101.1563.