Formal Deformation Theory Research Articles

Braided Hochschild cohomology and Hopf actions

We showt hat the braided Hochschild cohomology, of an algebra in a suitably algebraic braided monoidal category, admits a graded ring structure under which it is braided commutative. We then give a canonical identification between the usual Hochschild cohomology ring of a smash product and the (derived) invariants of its braided Hochschild cohomology ring. We apply our results to identify the associative formal deformation theory of a smash product with its formal deformation theory as a module algebra over the given Hopf algebra (when the Hopf algebra is sufficiently semisimple). As a second application we deduce some structural results for the usual Hochschild cohomology of a smash product, and discuss specific implications for finite group actions on smooth affine schemes.

The construction and deformation of BiHom-Novikov algebras

BiHom-Novikov agebra is a generalized Hom-Novikov algebra endowed with two commuting multiplicative linear maps. The main purpose of this paper is to show that two classes of BiHom-Novikov algebras can be constructed from BiHom-commutative algebras together with derivations and BiHom-Novikov algebras with Rota-Baxter operators, respectively. We show that quadratic BiHom-Novikov algebras are associative algebras and the sub-adjacent BiHom-Lie algebras of BiHom-Novikov algebras are 2-step nilpotent. Moreover, we develop the 1-parameter formal deformation theory of BiHom-Novikov algebras.

Central extensions and deformations of Hom-Lie triple systems

ABSTRACTIn this paper, we study the cohomology theory of Hom-Lie triple systems generalizing the Yamaguti cohomology theory of Lie triple systems. We introduce the central extension theory for Hom-Lie triple systems and show that there is a one-to-one correspondence between equivalent classes of central extensions of Hom-Lie triple systems and the third cohomology group. We develop the 1-parameter formal deformation theory of Hom-Lie triple systems and prove that it is governed by the cohomology group.

Ternary Distributive Structures and Quandles

We introduce a notion of ternary distributive algebraic structure, give exam- ples, and relate it to the notion of a quandle. Classification is given for low order structures of this type. Constructions of such structures from 3-Lie algebras are provided. We also describe ternary distributive algebraic structures coming from groups and give examples from vector spaces whose bases are elements of a finite ternary distributive set. We intro- duce a cohomology theory that is analogous to Hochschild cohomology and relate it to a formal deformation theory of these structures.

Right simple singularities in positive characteristic

Abstract We classify isolated singularities f ∈ K [ [ x 1 , ... , x n ] ] ${f\in K[[x_1,\ldots , x_n]]}$ , which are simple, i.e. have no moduli, with respect to right equivalence, where K is an algebraically closed field of characteristic p > 0. For K = ℝ ${K=\mathbb {R}}$ or ℂ this classification was initiated by Arnol'd, resulting in the famous ADE-series. The classification with respect to contact equivalence for p > 0 was done by Greuel and Kröning with a result similar to Arnol'd's. It is surprising that with respect to right equivalence and any given p > 0 we have only finitely many simple singularities, i.e. there are only finitely many k such that Ak and Dk are right simple, all the others have moduli. We conjecture a similar finiteness result for singularities with an arbitrary number of moduli. A major point of this paper is the generalisation of the notion of modality to the algebraic setting, its behaviour under morphisms, and its relations to formal deformation theory. As an application we show that the modality is semicontinuous in any characteristic.

Cohomology and Formal Deformations of Alternative Algebras

The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of alternative algebras. A short review of basics on alternative algebras and their connections to some other algebraic structures is also provided.

Formal deformations of Dirac structures

In this paper we set-up a general framework for a formal deformation theory of Dirac structures. We give a parameterization of formal deformations in terms of two-forms obeying a cubic equation. The notion of equivalence is discussed in detail. We show that the obstruction for the construction of deformations order by order lies in the third Lie algebroid cohomology of the Dirac structure. However, the classification of inequivalent first order deformations is not given by the second Lie algebroid cohomology but turns out to be more complicated.

ANALYTIC APPROACH TO DEFORMATION OF RESOLUTION OF NORMAL ISOLATED SINGULARITIES: FORMAL DEFORMATIONS

We give an analytic approach to the versal deformation of a resolution of a germ of normal isolated singularities. In this paper, we treat only formal deformation theory and it is applied to complete the CR-description of the simultaneous resolution of a cone eve. a rational curve of degree n in P<TEX>$^{n}$</TEX> (n <TEX>$\leq$</TEX> 4).

A Darboux theorem for Hamiltonian operators in the formal calculus of variations

We prove a formal Darboux-type theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain an analogue of the Darboux normal form in this context. We include an exposition of the formal deformation theory of differential graded Lie algebras $\mathfrak {g}$ concentrated in degrees $[-1,\infty)$; the formal deformations of $\mathfrak {g}$ are parametrized by a 2-groupoid that we call the Deligne 2-groupoid of $\mathfrak {g}$, and quasi-isomorphic differential graded Lie algebras have equivalent Deligne 2-groupoids.

A Darboux theorem for Hamiltonian operators in the formal calculus of variations

We prove a formal Darboux-type theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain an analogue of the Darboux normal form in this context. We include an exposition of the formal deformation theory of differential graded Lie algebras $\mathfrak {g}$ concentrated in degrees $[-1,\infty)$; the formal deformations of $\mathfrak {g}$ are parametrized by a 2-groupoid that we call the Deligne 2-groupoid of $\mathfrak {g}$, and quasi-isomorphic differential graded Lie algebras have equivalent Deligne 2-groupoids.