Commutative Differential Algebras Research Articles

Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras

We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of O-operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras and the set of coherent derivations on an ASI bialgebra composes a Lie algebra which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras.

Polynomial identities in Novikov algebras

In this paper, we study Novikov algebras satisfying nontrivial identities. We show that a Novikov algebra over a field of zero characteristic that satisfies a nontrivial identity satisfies some unexpected “universal” identities, in particular, right associator nilpotence, and right nilpotence of the commutator ideal. This, in particular, implies that a Novikov algebra over a field of zero characteristic satisfies a nontrivial identity if and only if it is Lie-solvable. We also establish that any system of identities of Novikov algebras over a field of zero characteristic follows from finitely many of them, and that the same holds over any field for multilinear Novikov identities. Some analogous simpler statements are also proved for commutative differential algebras.

Poisson Dixmier-Moeglin equivalence from a topological point of view

A complex affine Poisson algebra A is said to satisfy the Poisson Dixmier-Moeglin equivalence if the Poisson cores of maximal ideals of A are precisely those Poisson prime ideals that are locally closed in the Poisson prime spectrum P.spec A and if, moreover, these Poisson prime ideals are precisely those whose extended Poisson centers are exactly the complex numbers. In this paper, we provide some topological criteria for the Poisson Dixmier-Moeglin equivalence for A in terms of the poset (P.spec A, ⊆) and the symplectic leaf or core stratification on its maximal spectrum. In particular, we prove that the Zariski topology of the Poisson prime spectrum and of each symplectic leaf or core can detect the Poisson Dixmier-Moeglin equivalence for any complex affine Poisson algebra. Moreover, we generalize the weaker version of the Poisson Dixmier-Moeglin equivalence for a complex affine Poisson algebra proved in [J. Bell, S. Launois, O. L. Sanchez and B. Moosa, Poisson algebras via model theory and differential-algebraic geometry, J. Eur. Math. Soc. (JEMS) 19 (2017), 2019–2049] to the general context of a commutative differential algebra.

Effective computation of degree bounded minimal models of GCDAs

Given a finitely presented Graded Commutative Differential Algebra (GCDA), we present a method to compute its minimal model, together with a map that is a quasi-isomorphism up to a given degree. The method works by adding generators one by one. We also provide a specific implementation of the method. We also provide two criteria for i-formality, one necessary and one sufficient.

Prime Lie algebras satisfying the standard Lie identity of degree 5

For every commutative differential algebra one can define the Lie algebra of special derivations. It is known for years that not every Lie algebra can be embedded into the Lie algebra of special derivations of some differential algebra. More precisely, the Lie algebra of special derivations of a commutative algebra always satisfies the standard Lie identity of degree 5. The problem of existence of such embedding is a long-standing problem (see [2,4,3]), which is closely related to the Lie algebra of vector fields on the affine line (see [2]). It was solved by Razmyslov in [2] for simple Lie algebras satisfying this identity (see also [3, Th. 16]). We extend this result to prime (and semiprime) Lie algebras over a field of zero characteristic satisfying the standard Lie identity of degree 5. As an application, we prove that for any semiprime Lie algebra the standard identity St5 implies all other identities of the Lie algebra of polynomial vector fields on the affine line.We also generalize some previous results about primeness of the Lie algebra of special derivations of a prime differential algebra to the case of non-unitary differential algebra.

Differential étale extensions and differential modules over differential rings

This paper studies differential square zero extensions and differential modules of a commutative differential algebra R over a differential field F where the field of constants of F is algebraically closed and of characteristic 0. All elements of R and the differential R modules and algebras considered are assumed to satisfy linear homogeneous differential equations over F. For R differentially simple, we describe the invectives, and, using that all considered differential modules are R flat, provide a criterion for all square zero extensions to be differentially split.

Differential (Lie) algebras from a functorial point of view

It is well-known that any associative algebra becomes a Lie algebra under the commutator bracket. This relation is actually functorial, and this functor, as any algebraic functor, is known to admit a left adjoint, namely the universal enveloping algebra of a Lie algebra. This correspondence may be lifted to the setting of differential (Lie) algebras. In this contribution it is shown that, also in the differential context, there is another, similar, but somewhat different, correspondence. Indeed any commutative differential algebra becomes a Lie algebra under the Wronskian bracket W(a,b)=ab′−a′b. It is proved that this correspondence again is functorial, and that it admits a left adjoint, namely the differential enveloping (commutative) algebra of a Lie algebra. Other standard functorial constructions, such as the tensor and symmetric algebras, are studied for algebras with a given derivation.

Differential (Lie) algebras from a functorial point of view

Abstract It is well-known that any associative algebra becomes a Lie algebra under the commutator bracket. This relation is actually functorial, and this functor, as any algebraic functor, is known to admit a left adjoint, namely the universal enveloping algebra of a Lie algebra. This correspondence may be lifted to the setting of differential (Lie) algebras. In this contribution it is shown that, also in the differential context, there is another, similar, but somewhat different, correspondence. Indeed any commutative differential algebra becomes a Lie algebra under the Wronskian bracket W ( a , b ) = a b ′ − a ′ b . It is proved that this correspondence again is functorial, and that it admits a left adjoint, namely the differential enveloping (commutative) algebra of a Lie algebra. Other standard functorial constructions, such as the tensor and symmetric algebras, are studied for algebras with a given derivation.

Wronskian Envelope of a Lie Algebra

The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.

Real homotopy theory of semi-algebraic sets

We complete the details of a theory outlined by Kontsevich and Soibelman that associates to a semi-algebraic set a certain graded commutative differential algebra of "semi-algebraic differential forms" in a functorial way. This algebra encodes the real homotopy type of the semi-algebraic set in the spirit of the DeRham algebra of differential forms on a smooth manifold. Its development is needed for Kontsevich's proof of the formality of the little cubes operad.

Invariant tensors and partial differential equations

We consider tensors with coefficients in a commutative differential algebra A. Using the Lie derivative, we introduce the notion of a tensor invariant under a derivation on an ideal of A. Each system of partial differential equations generates an ideal in some differential algebra. This makes it possible to study invariant tensors on such an ideal. As examples we consider the equations of gas dynamics and magnetohydrodynamics.

The Prime Spectrum of Commutative Differential Algebras

For a commutative differential algebra A , if p 1 and p 2 are prime ideals with p 1 ⊂ p 2, p 1 ≠ p 2, and D( p 1) not contained in p 2, then there exists a prime ideal p 3 ⊂ p 2, with p 3 ⊄ p 1 and p 1 ⊄ p 3 and { x/ D kx ∈ p 1 for all k ≥ 0} ⊂ p 3. This yields an alternative proof that there is no non-trivial derivation on C k ([0,1]) and gives a generalized existence theorem for polarized prime ideals of C∞([0,1], C).

On commutative differential algebras

On commutative differential algebras

Continuous Cohomology and Real Homotopy Type

Various aspects of homotopy theory in the category of simplicial spaces are developed. Topics covered include continuous cohomology, continuous de Rham cohomology, the Kan extension condition, the homotopy relation, fibrations, the Serre spectral sequence, real homotopy type and its relation to graded commutative differential algebras over the reals.

Continuous cohomology and real homotopy type

Various aspects of homotopy theory in the category of simplicial spaces are developed. Topics covered include continuous cohomology, continuous de Rham cohomology, the Kan extension condition, the homotopy relation, fibrations, the Serre spectral sequence, real homotopy type and its relation to graded commutative differential algebras over the reals.

Commutative differential algebras with an algebraic element

Commutative differential algebras with an algebraic element