Differential (Lie) algebras from a functorial point of view

Poinsot, Laurent

doi:10.1016/j.aam.2015.09.003

Cited by 15 publications

(12 citation statements)

References 14 publications

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“…This mathematical subject has been developed into a broad realm such as differential Galois theory [20,32], differential algebraic geometry and differential algebraic groups [17]. In recent years, researchers began to investigate noncommutative differential algebras in order to broaden the scope of the theory to include path algebras, for instance, and to have a more meaningful differential Lie algebra theory [23,24] and also from an operadic point of view [19,9]. There are also some recent work dealing with other algebraic structures endowed with derivations [29,8,31].…”

Section: Introductionmentioning

confidence: 99%

Gröbner-Shirshov bases and linear bases for free differential type algebras over algebras

Zhou¹,

Qin²,

Zhou³

2021

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We study a question which can be roughly stated as follows: Given a (unital or nonunital) algebra A together with a Gröbner-Shirshov basis G, consider the free operated algebra B over A, such that the operator satisfies some polynomial identities Φ which are Gröbner-Shirshov in the sense of Guo et al., when does the union Φ ∪ G will be an operated Gröbner-Shirshov basis for B? We answer this question in the affirmative under a mild condition in our previous work with Wang. When this condition is satisfied, Φ ∪ G is an operated Gröbner-Shirshov basis for B and as a consequence, we also get a linear basis of B. However, the condition could not be applied directly to differential type algebras introduced by Guo, Sit and Zhang, including usual differential algebras.This paper solves completely this problem for differential type algebras. Some new monomial orders are introduced which, together with some known ones, permit the application of the previous result to most of differential type algebras, thus providing new operated GS bases and linear bases for these differential type algebras. Versions are presented both for unital and nonunital algebras. However, a class of examples are also presented, for which the natural expectation in the question is wrong and these examples are dealt with by direct inspection.

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Section: Introductionmentioning

confidence: 99%

Gröbner-Shirshov bases and linear bases for free differential type algebras over algebras

Zhou¹,

Qin²,

Zhou³

2021

Preprint

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“…For instance, in connection with combinatorics, differential structures were found on heap ordered trees [14] and on decorated rooted trees [15]. More recently, derivations on other algebraic structures have been initiated, including for path algebras [16], Lie algebras and Archimedean d-rings [23,26,27]. The operad of differential associative algebras was studied in [24].…”

Section: Introductionmentioning

confidence: 99%

On differential lattices

Gan¹,

Guo²

2021

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This paper studies the differential lattice, defined to be a lattice L equipped with a map d : L → L that satisfies a lattice analog of the Leibniz rule for a derivation. Isomorphic differential lattices are studied and classifications of differential lattices are obtained for some basic lattices. Several families of derivations on a lattice are explicitly constructed, giving realizations of the lattice as lattices of derivations. Derivations on a finite distributive lattice are shown to have a natural structure of lattice. Moreover, derivations on a complete infinitely distributive lattice form a complete lattice. For a general lattice, it is conjectured that its poset of derivations is a lattice that uniquely determines the given lattice. Contents 1. Introduction 1 2. Differential lattices and basic properties 3 3. Isomorphic classes of differential lattices 6 3.1. Isomorphic differential lattices 6 3.2. Classification of differential lattices 9 4. The lattices of derivations 13 4.1. Lattice structures for derivations on distributive lattices 13 4.2. Lattice structures on inner and other special derivations 16 4.3. Lattice structures for derivations on specific lattices 18 References 21

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“…This mathematical branch has received ample development in the work [37,41,46] and has broad applications to other areas such as arithmetic geometry, logic, computer science and mathematical physics [15,19,43,52,53] etc. In recent years, researchers began to investigate noncommutative differential algebras in order to broaden the scope of the theory to include path algebras, for instance, and to have a more meaningful differential Lie algebra theory [29,44,45] and also from an operadic point of view [40] Another important class of algebras with operators are Rota-Baxter algebras. These algebras (previously known as Baxter algebras) originated with the work of Baxter [7] on probability theory.…”

Section: Introductionmentioning

confidence: 99%

Free objects and Gröbner-Shirshov bases in operated contexts

Qi,

Qin,

Wang

et al. 2021

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This paper investigates algebraic objects equipped with an operator, such as operated monoids, operated algebras etc. Various free object functors in these operated contexts are explicitly constructed. For operated algebras whose operator satisfies a set Φ of relations (usually called operated polynomial identities (aka. OPIs)), Guo defined free objects, called free Φ-algebras, via universal algebra. Free Φ-algebras over algebras are studied in details. A mild sufficient condition is found such that Φ together with a Gröbner-Shirshov basis of an algebra A form a Gröbner-Shirshov basis of the free Φ-algebra over algebra A in the sense of Guo et al.. Ample examples for which this condition holds are provided, such as all Rota-Baxter type OPIs, a class of differential type OPIs, averaging OPIs and Reynolds OPI. ContentsIntroduction 1. Free object functors I 1.1. The bottom face 1.2. The upper face 2. Free object functors II 2.1. Free object functor from sets to operated sets 2.2. Free object functor from semigroups to operated semigroups 2.3. Free object functor from monoids to operated monoids 2.4. Free object functor from operated sets to operated semigroups 2.5. Free object functor from operated semigroups to operated monoids 2.6. Composites of free object functors 3. Free object functors III 3.1. Free object functor from vector spaces to operated vector spaces 3.2. Free object functor from algebras to operated algebras 3.3. Free object functor from unital algebras to unital operated algebras 3.4. Free object functor from operated vector spaces to operated algebras 3.5. Free object functor from operated algebras to unital operated algebras 3.6. Compositions of free object functors 4. Free Φ-algebras via universal algebra 5. Gröbner-Shirshov bases for free Φ-algebras over algebras 6. Examples of operated GS bases for free Φ-algebras over Algebras 6.1. Preliminaries about rewriting systems 6.2. Single OPI of Rota-Baxter type 6.3. Single OPI of differential type

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Differential (Lie) algebras from a functorial point of view

Cited by 15 publications

References 14 publications

Gröbner-Shirshov bases and linear bases for free differential type algebras over algebras

Gröbner-Shirshov bases and linear bases for free differential type algebras over algebras

On differential lattices

Free objects and Gröbner-Shirshov bases in operated contexts

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