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1 -cocycles on the group of contactomorphisms on the supercircle S 1 | 3 generalizing the Schwarzian derivative

Boujemaa Agrebaoui, Raja Hattab (2016)

Czechoslovak Mathematical Journal

The relative cohomology H diff 1 ( 𝕂 ( 1 | 3 ) , 𝔬𝔰𝔭 ( 2 , 3 ) ; 𝒟 λ , μ ( S 1 | 3 ) ) of the contact Lie superalgebra 𝕂 ( 1 | 3 ) with coefficients in the space of differential operators 𝒟 λ , μ ( S 1 | 3 ) acting on tensor densities on S 1 | 3 , is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating 1 -cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative 1 -cocycle s ( X f ) = D 1 D 2 D 3 ( f ) α 3 1 / 2 , X f 𝕂 ( 1 | 3 ) which is invariant with respect to the conformal subsuperalgebra 𝔬𝔰𝔭 ( 2 , 3 ) of 𝕂 ( 1 | 3 ) . In this work we study the supergroup case. We give an explicit construction of 1 -cocycles of the group...

A Characterization of One-Element p-Bases of Rings of Constants

Piotr Jędrzejewicz (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

Let K be a unique factorization domain of characteristic p > 0, and let f ∈ K[x₁,...,xₙ] be a polynomial not lying in K [ x p , . . . , x p ] . We prove that K [ x p , . . . , x p , f ] is the ring of constants of a K-derivation of K[x₁,...,xₙ] if and only if all the partial derivatives of f are relatively prime. The proof is based on a generalization of Freudenburg’s lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.

A characterization of p-bases of rings of constants

Piotr Jędrzejewicz (2013)

Open Mathematics

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.

A G -minimal model for principal G -bundles

Shrawan Kumar (1982)

Annales de l'institut Fourier

Sullivan associated a uniquely determined D G A | Q to any simply connected simplicial complex E . This algebra (called minimal model) contains the total (and exactly) rational homotopy information of the space E . In case E is the total space of a principal G -bundle, ( G is a compact connected Lie-group) we associate a G -equivariant model U G [ E ] , which is a collection of “ G -homotopic” D G A ’s | R with G -action. U G [ E ] will, in general, be different from the Sullivan’s minimal model of the space E . U G [ E ] contains the total rational...

A note on characterizations of rings of constants with respect to derivations

Piotr Jędrzejewicz (2004)

Colloquium Mathematicae

Let A be a commutative algebra without zero divisors over a field k. If A is finitely generated over k, then there exist well known characterizations of all k-subalgebras of A which are rings of constants with respect to k-derivations of A. We show that these characterizations are not valid in the case when the algebra A is not finitely generated over k.

A Note on Elementary Derivations

Khoury, Joseph (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary: 14R10. Secondary: 14R20, 13N15.Let R be a UFD containing a field of characteristic 0, and Bm = R[Y1, . . . , Ym] be a polynomial ring over R. It was conjectured in [5] that if D is an R-elementary monomial derivation of B3 such that ker D is a finitely generated R-algebra then the generators of ker D can be chosen to be linear in the Yi ’s. In this paper, we prove that this does not hold for B4. We also investigate R-elementary derivations D of Bm...

A note on Frobenius divided modules in mixed characteristics

Pierre Berthelot (2012)

Bulletin de la Société Mathématique de France

If X is a smooth scheme over a perfect field of characteristic p , and if 𝒟 X ( ) is the sheaf of differential operators on X [7], it is well known that giving an action of 𝒟 X ( ) on an 𝒪 X -module is equivalent to giving an infinite sequence of 𝒪 X -modules descending via the iterates of the Frobenius endomorphism of X [5]. We show that this result can be generalized to any infinitesimal deformation f : X S of a smooth morphism in characteristic p , endowed with Frobenius liftings. We also show that it extends to adic...

A note on Poisson derivations

Jiantao Li (2018)

Czechoslovak Mathematical Journal

Free Poisson algebras are very closely connected with polynomial algebras, and the Poisson brackets are used to solve many problems in affine algebraic geometry. In this note, we study Poisson derivations on the symplectic Poisson algebra, and give a connection between the Jacobian conjecture with derivations on the symplectic Poisson algebra.

A note on rings of constants of derivations in integral domains

Piotr Jędrzejewicz (2011)

Colloquium Mathematicae

We observe that the characterization of rings of constants of derivations in characteristic zero as algebraically closed subrings also holds in positive characteristic after some natural adaptation. We also present a characterization of such rings in terms of maximality in some families of rings.

A note on semisimple derivations of commutative algebras

Andrzej Tyc (2005)

Colloquium Mathematicae

A concept of a slice of a semisimple derivation is introduced. Moreover, it is shown that a semisimple derivation d of a finitely generated commutative algebra A over an algebraically closed field of characteristic 0 is nothing other than an algebraic action of a torus on Max(A), and, using this, that in some cases the derivation d is linearizable or admits a maximal invariant ideal.

Affine rulings of weighted projective planes

Daniel Daigle (2001)

Annales Polonici Mathematici

It is explained that the following two problems are equivalent: (i) describing all affine rulings of any given weighted projective plane; (ii) describing all weighted-homogeneous locally nilpotent derivations of k[X,Y,Z]. Then the solution of (i) is sketched. (Outline of our joint work with Peter Russell.)

AK-invariant, some conjectures, examples and counterexamples

L. Makar-Limanov (2001)

Annales Polonici Mathematici

In my talk I am going to remind you what is the AK-invariant and give examples of its usefulness. I shall also discuss basic conjectures about this invariant and some positive and negative results related to these conjectures.

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