Formal Deformations Research Articles

Deformations and homotopy theory for Rota–Baxter family algebras

The concept of Rota–Baxter family algebra is a generalization of Rota–Baxter algebra. It appears naturally in the algebraic aspects of renormalizations in quantum field theory. Rota–Baxter family algebras are closely related to dendriform family algebras. In this paper, we first construct an [Formula: see text]-algebra whose Maurer–Cartan elements correspond to Rota–Baxter family algebra structures. Using this characterization, we define the cohomology of a given Rota–Baxter family algebra. As an application of our cohomology, we study formal and infinitesimal deformations of a given Rota–Baxter family algebra. Next, we define the notion of a homotopy Rota–Baxter family algebra structure on a given [Formula: see text]-algebra. We end this paper by considering the homotopy version of dendriform family algebras and their relations with homotopy Rota–Baxter family algebras.

Period integrals of smooth projective complete intersections as exponential periods

Let X be a smooth projective complete intersection over Q of dimension n−k in the projective space PQn defined by the zero locus of f_(x_)=(f1(x_),⋯,fk(x_)), for given positive integers n and k. For a given primitive homology cycle [γ]∈Hn−k(X(C),Z)0, the period integral is defined to be a linear map from the primitive de Rham cohomology group HdR,primn−k(X(C);Q) to C given by [ω]↦∫γω. The goal of this article is to interpret this period integral as Feynman's path integral of 0-dimensional quantum field theory with the action functional S=∑ℓ=1kyℓfℓ(x_) (in other words, the exponential period with the action functional S) and use this interpretation to develop a formal deformation theory of period integrals of X, which can be viewed as a modern deformation theoretic treatment of the period integrals based on the Maurer-Cartan equation of a dgla (differential graded Lie algebra).

Deformations and abelian extensions of compatible pre-Lie algebras

In this paper, first, we give the notion of a compatible pre-Lie algebra and its representation. We study the relation between compatible Lie algebras and compatible pre-Lie algebras. We also construct a new bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie algebra which controls deformations of a compatible pre-Lie algebra. Then, we introduce a cohomology of a compatible pre-Lie algebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie algebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group H2(g;g) is trivial, then the compatible pre-Lie algebra is rigid. Finally, we give a cohomology of a compatible pre-Lie algebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie algebras using this cohomology. We show that abelian extensions are classified by the second cohomology group.

Formal deformations, cohomology theory and L∞[1]-structures for differential Lie algebras of arbitrary weight

We introduced a cohomology theory for differential Lie algebras of arbitrary weight which generalised a previous work of Jiang and Sheng. The underlying L∞[1]-structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between L∞[1]-structures for absolute and relative differential Lie algebras is established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras.

The cohomology and deformations of [formula omitted]-operators on BiHom-associative algebras

We first generalize the cohomology of O-operators on BiHom-associative algebras by construct a graded Lie-algebra, in which the Maurer-Cartan elements are characterized by the given O-operator, and show that the cohomology represents the Hochschild cohomology of a certain BiHom-associative algebra with coefficients in a bimodule. Next, we study the linear and formal deformations of O-operators on BiHom-associative algebras, which are controlled by the Hochschild cohomology. Finally, as applications, we introduce the deformations of BiHom-associative r-matrices and infinitesimal BiHom-bialgebras on certain regular BiHom-associative algebras.

Commutative Poisson algebras from deformations of noncommutative algebras

It is well-known that a formal deformation of a commutative algebra A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} leads to a Poisson bracket on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} and that the classical limit of a derivation on the deformation leads to a derivation on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}, which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. The deformation leads in this case to a Poisson algebra structure on Π(A):=Z(A)×(A/Z(A))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A}){:}{=}Z(\\mathcal {A})\ imes (\\mathcal {A}/Z(\\mathcal {A}))$$\\end{document} and to the structure of a Π(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A})$$\\end{document}-Poisson module on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. The limiting derivations are then still derivations of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}, but with the Hamiltonian belong to Π(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A})$$\\end{document}, rather than to A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra.

Matching relative Rota-Baxter algebras, matching dendriform algebras and their cohomologies

The notion of matching Rota-Baxter algebras was recently introduced by Gao, Guo, and Zhang motivated by the study of algebraic renormalization of regularity structures. In this paper, we introduce matching relative Rota-Baxter algebras that are closely related to matching dendriform algebras. We define the cohomology of a matching relative Rota-Baxter algebra using the classical Hochschild cohomology and a new cohomology induced by the matching operators. As an application, we show that our cohomology governs the formal deformation theory of the matching relative Rota-Baxter algebra. Finally, we define the cohomology of a matching dendriform algebra and study homotopy matching dendriform algebras.

Jet schemes over auto-arc spaces and deformations of locally complete intersection varieties

We consider reduced generalized jet schemes LZ(Y) along Z of a infinitesimal deformation Y of a complete intersection variety X over the same fat point Z. By definition, these reduced generalized jet schemes LZ(Y) live over the so-called reduced auto-arc spaces AZ, which were originally introduced by H. Schoutens and later studied by the author. Some well-known results of M. Mustaţǎ on classical jet schemes of locally complete intersections carry over to this case. For example, as a consequence of miracle flatness and some generalizations of the aforementioned results of M. Mustaţǎ, it is shown that the reduced generalized jet space, denoted Ln(Xn), of a flat deformation Xn→Dn along ▪ is usually a global flat deformation over Akn of the classical jet scheme Ln(X) provided the base scheme X is a locally complete intersection and the fibers are all of minimal dimension. Similar interesting relativized versions of those aforementioned results are also explored. Also, the initiation of the so-called regulated defect of a formal deformation in analogy to log canonical threshold is introduced. In the end, the situation is studied when the base scheme X is an algebraic curve, and the notions of so-called strong/weak deformations is introduced in this context. Finally, a so-called deformed motivic volume is defined.

Deformation cohomology of morphisms of Lie-Yamaguti algebras

We study cohomology of morphisms of Lie-Yamaguti algebras. As an application, we establish that this cohomology ‘controls’ the formal deformations. Additionally, we demonstrate its connection to the abelian extensions of morphisms of Lie-Yamaguti algebras.

Holomorphic support functions for uniformly pseudoconvex hypersurfaces, with an application to CR maps

We construct holomorphic support functions for smooth weakly pseudoconvex hypersurfaces with Levi form of constant rank. These are then applied to show that formal holomorphic curves which are tangential to infinite order to such a hypersurface must be formally contained in its Levi foliation. As a consequence, we obtain a holomorphic deformation theorem for nowhere smooth CR maps into smooth pseudoconvex hypersurfaces with one-dimensional Levi foliation, strengthening a very general result of Lamel and Mir about formal deformations in this particular case.

Deformations and cohomology theory of Ω-Rota-Baxter algebras of arbitrary weight

In this paper, we introduce the concepts of relative and absolute Ω-Rota-Baxter algebras of weight λ, which can be considered as a family algebraic generalization of relative and absolute Rota-Baxter algebras of weight λ. We study the deformations of relative and absolute Ω-Rota-Baxter algebras of arbitrary weight. Explicitly, we construct an L∞[1]-algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative Ω-Rota-Baxter algebra structures of weight λ. For a relative Ω-Rota-Baxter algebra of weight λ, the corresponding twisted L∞[1]-algebra controls its deformations, which leads to the cohomology theory of it, and this cohomology theory can interpret the formal deformations of the relative Ω-Rota-Baxter algebra. Moreover, we also obtain the corresponding results for absolute Ω-Rota-Baxter algebras of weight λ from the relative version.

Twisted relative Rota-Baxter operators on Leibniz conformal algebras

In this paper, we first investigate some properties of relative Rota-Baxter operators on Leibniz conformal algebras with respect to representations and their connections with Leibniz dendriform conformal algebras. Next, we introduce the notion of a twisted relative Rota-Baxter operator and construct a conformal NS-Leibniz algebra structure related to twisted relative Rota-Baxter operators. Furthermore, we define the cohomology of a twisted relative Rota-Baxter operator with coefficients in a suitable representation. Finally, we consider formal deformations of twisted relative Rota-Baxter operators from cohomological points of view.

On deformations of Rota–Baxter algebra morphisms

Rota–Baxter algebras are important in probability, combinatorics, associative Yang–Baxter equation and splitting of algebras. This paper studies the formal deformations of Rota–Baxter algebra morphisms. As a consequence, we develop a cohomology theory of Rota–Baxter algebra morphisms to interpret the lower degree cohomology groups as formal deformations. Finally, we prove the cohomology comparison theorem of Rota–Baxter algebra morphisms, i.e. the cohomology of a morphism of Rota–Baxter algebras is isomorphic to the cohomology of an auxiliary Rota–Baxter algebra.

Representations and formal deformations of Leibniz H-pseudoalgebras

In this paper, we first give some constructions of right (resp. left) Leibniz H-pseudoalgebras and explore the characterization of symmetric Leibniz H-pseudoalgebras. Furthermore, we introduce the notion of relative Rota-Baxter H-operator and show that a right (resp. left) Leibniz H-pseudoalgebra together with a relative Rota-Baxter H-operator gives rise to a representation of related right (resp. left) Leibniz H-pseudoalgebra. Finally, we describe the central extension and the formal deformation in terms of the corresponding second cohomology group.

L ∞ -structures and cohomology theory of compatible O-operators and compatible dendriform algebras

The notion of O-operator is a generalization of the Rota–Baxter operator in the presence of a bimodule over an associative algebra. A compatible O-operator is a pair consisting of two O-operators satisfying a compatibility relation. A compatible O-operator algebra is an algebra together with a bimodule and a compatible O-operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O-operators and compatible O-operator algebras as Maurer–Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible O-operators and compatible O-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible O-operators.

Pre-Leibniz algebras

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras.

Deformations and abelian extensions of compatible pre-Lie superalgebras

In this paper, we give cohomologies and deformations theory, as well as abelian extensions for compatible pre-Lie superalgebras. Explicitly, we first introduce the notation of a compatible pre-Lie superalgebra and its representation. We also construct a new bidfferential graded Lie superalgebra whose Maurer–Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie superalgebra which controls deformations of a compatible pre-Lie superalgebra. Then, we introduce a cohomology of a compatible pre-Lie superalgebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie superalgebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group [Formula: see text] is trivial, then the compatible pre-Lie superalgebra is rigid. Additionally, we give a cohomology of a compatible pre-Lie superalgebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie superalgebras using this cohomology. We show that abelian extensions are classified by the second cohomology group. Finally, we study the relation between compatible Lie superalgebras and compatible pre-Lie superalgebras and we give some examples of compatible pre-Lie superalgebras.

Cohomology of modified Rota–Baxter Leibniz algebra of weight λ

Rota–Baxter operators have been paid much attention in the last few decades as they have many applications in mathematics and physics. In this paper, our object of study is modified Rota–Baxter operators on Leibniz algebras. We investigate modified Rota–Baxter Leibniz algebras from the cohomological point of view. We study a one-parameter formal deformation theory of modified Rota–Baxter Leibniz algebras and define the associated deformation cohomology that controls the deformation. Finally, as an application, we characterize equivalence classes of abelian extensions in terms of second cohomology groups.

Nijenhuis-operator on Hom-Lie conformal algebras

The aim of this paper is to study the Nijenhuis operators on Hom-Lie conformal algebras. We construct a graded Lie algebra whose Maurer-Cartan elements are given by Nijenhuis operators with the aid of Frölicher-Nijenhuis bracket. After constructing graded Lie algebra, we define the cohomology associated with a Nijenhuis operator. We further introduce Hom-NS Lie conformal algebra as an algebraic structure behind Nijenhuis operators on Hom-Lie conformal algebra. We provide various examples of Hom-NS Lie conformal algebras. Finally, as an application to cohomology, we study formal deformations of Nijenhuis operators.

Cohomology and Deformations of Relative Rota–Baxter Operators on Lie-Yamaguti Algebras

In this paper, we establish the cohomology of relative Rota–Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then, we use this type of cohomology to characterize deformations of relative Rota–Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota–Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order n deformation of a relative Rota–Baxter operator can be extended to an order n+1 deformation if and only if the obstruction class in the second cohomology group is trivial.