Articles de revues : « Formal deformations »

1

Fialowski, Alice, et Michael Penkava. « On singular formal deformations ». Archiv der Mathematik 106, n^o 5 (12 mars 2016) : 431–38. http://dx.doi.org/10.1007/s00013-016-0894-2.

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Blanc, Anthony, Ludmil Katzarkov et Pranav Pandit. « Generators in formal deformations of categories ». Compositio Mathematica 154, n^o 10 (30 août 2018) : 2055–89. http://dx.doi.org/10.1112/s0010437x18007303.

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In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a$k$-linear$\infty$-category for a field$k$. Our main result states that if${\mathcal{C}}$is a$k$-linear$\infty$-category which has a compact generator whose groups of self-extensions vanish for sufficiently high positive degrees, then every formal deformation of${\mathcal{C}}$has zero curvature and moreover admits a compact generator.

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Keller, Frank, et Stefan Waldmann. « Formal deformations of Dirac structures ». Journal of Geometry and Physics 57, n^o 3 (février 2007) : 1015–36. http://dx.doi.org/10.1016/j.geomphys.2006.08.005.

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Grinberg, M., et D. Kazhdan. « Versal deformations of formal arcs ». Geometric and Functional Analysis 10, n^o 3 (septembre 2000) : 543–55. http://dx.doi.org/10.1007/pl00001628.

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Huebschmann, Johannes. « The formal Kuranishi parameterization via the universal homological perturbation theory solution of the deformation equation ». Georgian Mathematical Journal 25, n^o 4 (1 décembre 2018) : 529–44. http://dx.doi.org/10.1515/gmj-2018-0054.

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AbstractUsing homological perturbation theory, we develop a formal version of the miniversal deformation associated with a deformation problem controlled by a differential graded Lie algebra over a field of characteristic zero. Our approach includes a formal version of the Kuranishi method in the theory of deformations of complex manifolds.

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BOURQUI, DAVID, et JULIEN SEBAG. « DEFORMATIONS OF DIFFERENTIAL ARCS ». Bulletin of the Australian Mathematical Society 94, n^o 3 (16 août 2016) : 405–10. http://dx.doi.org/10.1017/s0004972716000459.

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Let$k$be field of characteristic zero. Let$f\in k[X,Y]$be a nonconstant polynomial. We prove that the space of differential (formal) deformations of any formal general solution of the associated ordinary differential equation$f(y^{\prime },y)=0$is isomorphic to the formal disc$\text{Spf}(k[[Z]])$.

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Elhamdadi, Mohamed, et Abdenacer Makhlouf. « Cohomology and Formal Deformations of Alternative Algebras ». Journal of Generalized Lie Theory and Applications 5 (2011) : 1–10. http://dx.doi.org/10.4303/jglta/g110105.

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Chouhy, Sergio. « On geometric degenerations and Gerstenhaber formal deformations ». Bulletin of the London Mathematical Society 51, n^o 5 (24 juillet 2019) : 787–97. http://dx.doi.org/10.1112/blms.12277.

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Karabegov, Alexander. « Infinitesimal Deformations of a Formal Symplectic Groupoid ». Letters in Mathematical Physics 97, n^o 3 (10 mai 2011) : 279–301. http://dx.doi.org/10.1007/s11005-011-0495-8.

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DEMCHENKO, OLEG, et ALEXANDER GUREVICH. « GROUP ACTION ON THE DEFORMATIONS OF A FORMAL GROUP OVER THE RING OF WITT VECTORS ». Nagoya Mathematical Journal 235 (20 décembre 2017) : 42–57. http://dx.doi.org/10.1017/nmj.2017.43.

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A recent result by the authors gives an explicit construction for a universal deformation of a formal group $\unicode[STIX]{x1D6F7}$ of finite height over a finite field $k$ . This provides in particular a parametrization of the set of deformations of $\unicode[STIX]{x1D6F7}$ over the ring ${\mathcal{O}}$ of Witt vectors over $k$ . Another parametrization of the same set can be obtained through the Dieudonné theory. We find an explicit relation between these parameterizations. As a consequence, we obtain an explicit expression for the action of $\text{Aut}_{k}(\unicode[STIX]{x1D6F7})$ on the set of ${\mathcal{O}}$ -deformations of $\unicode[STIX]{x1D6F7}$ in the coordinate system defined by the universal deformation. This generalizes a formula of Gross and Hopkins and the authors’ result for one-dimensional formal groups.

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Miyajima, Kimio. « ANALYTIC APPROACH TO DEFORMATION OF RESOLUTION OF NORMAL ISOLATED SINGULARITIES : FORMAL DEFORMATIONS ». Journal of the Korean Mathematical Society 40, n^o 4 (1 juillet 2003) : 709–25. http://dx.doi.org/10.4134/jkms.2003.40.4.709.

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Tang, Rong, Yunhe Sheng et Yanqiu Zhou. « Deformations of relative Rota–Baxter operators on Leibniz algebras ». International Journal of Geometric Methods in Modern Physics 17, n^o 12 (4 septembre 2020) : 2050174. http://dx.doi.org/10.1142/s0219887820501741.

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In this paper, we introduce the cohomology theory of relative Rota–Baxter operators on Leibniz algebras. We use the cohomological approach to study linear and formal deformations of relative Rota–Baxter operators. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order [Formula: see text] deformations of a relative Rota–Baxter operator are also characterized in terms of the cohomology theory.

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Stancu, Alin. « On some constructions of nil-clean, clean and exchange rings ». Journal of Algebra and Its Applications 14, n^o 07 (24 avril 2015) : 1550101. http://dx.doi.org/10.1142/s0219498815501017.

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In this paper, we discuss several constructions that lead to new examples of nil-clean, clean and exchange rings. Extensions by ideals contained in the Jacobson radical is the common theme of these constructions. A characterization of the idempotents in the algebra defined by a 2-cocycle is given and used to prove some of the algebra's properties (the infinitesimal deformation case). From infinitesimal deformations, we go to full deformations and prove that any formal deformation of a clean (exchange) ring is itself clean (exchange). Examples of nil-clean, clean and exchange rings, arising from poset algebras are also discussed.

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Griffith, Phillip. « Induced formal deformations and the Cohen-Macaulay property ». Transactions of the American Mathematical Society 353, n^o 1 (13 juin 2000) : 77–93. http://dx.doi.org/10.1090/s0002-9947-00-02513-7.

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Lazarev, A. « Deformations of formal groups and stable homotopy theory ». Topology 36, n^o 6 (novembre 1997) : 1317–31. http://dx.doi.org/10.1016/s0040-9383(96)00051-1.

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GUERRINI, L. « FORMAL AND ANALYTIC DEFORMATIONS FROM WITT TO VIRASORO ». Reviews in Mathematical Physics 14, n^o 03 (mars 2002) : 303–16. http://dx.doi.org/10.1142/s0129055x02001181.

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We introduce a new family [Formula: see text] of deformations of the Witt algebra [Formula: see text], F varying in the space of all polynomials with vanishing constant terms, and show the existence of an isomorphism of its formal and analytic completions with those of the Witt algebra. Central extensions of this algebra are considered and the existence of an isomorphism between their formal and analytic completions with those of the Virasoro algebra is proved.

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Pichereau, Anne. « Formal deformations of Poisson structures in low dimensions ». Pacific Journal of Mathematics 239, n^o 1 (1 janvier 2009) : 105–33. http://dx.doi.org/10.2140/pjm.2009.239.105.

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Ma, Yao, Liangyun Chen et Jie Lin. « One-parameter formal deformations of Hom-Lie-Yamaguti algebras ». Journal of Mathematical Physics 56, n^o 1 (janvier 2015) : 011701. http://dx.doi.org/10.1063/1.4905733.

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Brown, Richard A. « Generalized group presentation and formal deformations of CW complexes ». Transactions of the American Mathematical Society 334, n^o 2 (1 février 1992) : 519–49. http://dx.doi.org/10.1090/s0002-9947-1992-1153010-3.

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Fialowski, Alice, et Michael Penkava. « Formal Deformations, Contractions and Moduli Spaces of Lie Algebras ». International Journal of Theoretical Physics 47, n^o 2 (28 juillet 2007) : 561–82. http://dx.doi.org/10.1007/s10773-007-9481-4.

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Green, Barry. « Realizing deformations of curves using Lubin-Tate formal groups ». Israel Journal of Mathematics 139, n^o 1 (décembre 2004) : 139–48. http://dx.doi.org/10.1007/bf02787544.

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Saha, Ripan. « Equivariant associative dialgebras and its one-parameter formal deformations ». Journal of Geometry and Physics 146 (décembre 2019) : 103491. http://dx.doi.org/10.1016/j.geomphys.2019.103491.

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Lecomte, P. B. A., et C. Roger. « Formal deformations of the associative algebra of smooth matrices ». Letters in Mathematical Physics 15, n^o 1 (janvier 1988) : 55–63. http://dx.doi.org/10.1007/bf00416572.

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Das, Apurba. « Cohomology and deformations of weighted Rota–Baxter operators ». Journal of Mathematical Physics 63, n^o 9 (1 septembre 2022) : 091703. http://dx.doi.org/10.1063/5.0093066.

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Weighted Rota–Baxter operators on associative algebras are closely related to modified Yang–Baxter equations, splitting of algebras, and weighted infinitesimal bialgebras and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by λ-weighted relative Rota–Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal, and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation, which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota–Baxter operators in the Lie case and find a connection with the case of associative algebras.

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Zhu, Yifei. « Norm coherence for descent of level structures on formal deformations ». Journal of Pure and Applied Algebra 224, n^o 10 (octobre 2020) : 106382. http://dx.doi.org/10.1016/j.jpaa.2020.106382.

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ZHAO, WENHUA. « DEFORMATIONS AND INVERSION FORMULAS FOR FORMAL AUTOMORPHISMS IN NONCOMMUTATIVE VARIABLES ». International Journal of Algebra and Computation 17, n^o 02 (mars 2007) : 261–88. http://dx.doi.org/10.1142/s0218196707003676.

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Let z = (z1, z2,…, zn) be noncommutative free variables and t a formal parameter which commutes with z. Let k be any unital integral domain of any characteristic and Ft(z) = z - Ht(z) with Ht(z) ∈ k[[t]]〈〈z〉〉×n and the order o(Ht(z))≥ 2. Note that Ft(z) can be viewed as a deformation of the formal map F(z):= z - Ht=1(z) when it makes sense (for example, when Ht(z) ∈ k[t]〈〈z〉〉×n). The inverse map Gt(z) of Ft(z) can always be written as Gt(z) = z+Mt(z) with Mt(z) ∈ k[[t]]〈〈z〉〉×n and o(Mt(z)) ≥ 2. In this paper, we first derive the PDEs satisfied by Mt(z) and u(Ft), u(Gt) ∈ k[[t]]〈〈z〉〉 with u(z) ∈ k〈〈z〉〉 in the general case as well as in the special case when Ht(z) = tH(z) for some H(z) ∈ k〈〈z〉〉×n. We also show that the elements above are actually characterized by certain Cauchy problems of these PDEs. Secondly, we apply the derived PDEs to prove a recurrent inversion formula for formal maps in noncommutative variables. Finally, for the case char. k = 0, we derive an expansion inversion formula by the planar binary rooted trees.

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Bremer, Christopher L., et Daniel S. Sage. « Isomonodromic Deformations of Connections with Singularities of Parahoric Formal Type ». Communications in Mathematical Physics 313, n^o 1 (27 mai 2012) : 175–208. http://dx.doi.org/10.1007/s00220-012-1493-0.

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BIELIAVSKY, PIERRE, et PHILIPPE BONNEAU. « ON THE GEOMETRY OF THE CHARACTERISTIC CLASS OF A STAR PRODUCT ON A SYMPLECTIC MANIFOLD ». Reviews in Mathematical Physics 15, n^o 02 (avril 2003) : 199–215. http://dx.doi.org/10.1142/s0129055x0300159x.

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The characteristic class of a star product on a symplectic manifold appears as the class of a deformation of a given symplectic connection, as described by Fedosov. In contrast, one usually thinks of the characteristic class of a star product as the class of a deformation of the Poisson structure (as in Kontsevich's work). In this paper, we present, in the symplectic framework, a natural procedure for constructing a star product by directly quantizing a deformation of the symplectic structure. Basically, in Fedosov's recursive formula for the star product with zero characteristic class, we replace the symplectic structure by one of its formal deformations in the parameter ℏ. We then show that every equivalence class of star products contains such an element. Moreover, within a given class, equivalences between such star products are realized by formal one-parameter families of diffeomorphisms, as produced by Moser's argument.

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Basdouri, Khaled, et Salem Omri. « Cohomology and deformation of 𝔞𝔣𝔣(1|1) acting on differential operators ». International Journal of Geometric Methods in Modern Physics 15, n^o 05 (2 avril 2018) : 1850072. http://dx.doi.org/10.1142/s021988781850072x.

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We consider the [Formula: see text]-module structure on the spaces of differential operators acting on the spaces of weighted densities. We compute the second differential cohomology of the Lie superalgebra [Formula: see text] with coefficients in differential operators acting on the spaces of weighted densities. We classify formal deformations of the [Formula: see text]-module structure on the superspaces of symbols of differential operators. We prove that any formal deformation of a given infinitesimal deformation of this structure is equivalent to its infinitesimal part. This work is the simplest superization of a result by Basdouri [Deformation of [Formula: see text]-modules of pseudo-differential operators and symbols, J. Pseudo-differ. Oper. Appl. 7(2) (2016) 157–179] and application of work by Basdouri et al. [First cohomology of [Formula: see text] and [Formula: see text] acting on linear differential operators, Int. J. Geom. Methods Mod. Phys. 13(1) (2016)].

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GUERRINI, LUCA. « FORMAL AND ANALYTIC RIGIDITY OF THE WITH ALGEBRA ». Reviews in Mathematical Physics 11, n^o 03 (mars 1999) : 303–20. http://dx.doi.org/10.1142/s0129055x99000118.

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A family of deformations [Formula: see text] of the Witt algebra [Formula: see text] parametrized by the space ℰ of even polynomials with vanishing constant terms is defined. The existence of an isomorphism [Formula: see text], where [Formula: see text] refers to suitable completions of [Formula: see text], is proved. A relation between [Formula: see text] and Krichever–Novikov algebras of genus 0 and 1 is given.

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FIALOWSKI, ALICE, et MARTIN SCHLICHENMAIER. « GLOBAL DEFORMATIONS OF THE WITT ALGEBRA OF KRICHEVER–NOVIKOV TYPE ». Communications in Contemporary Mathematics 05, n^o 06 (décembre 2003) : 921–45. http://dx.doi.org/10.1142/s0219199703001208.

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By considering non-trivial global deformations of the Witt (and the Virasoro) algebra given by geometric constructions it is shown that, despite their infinitesimal and formal rigidity, they are globally not rigid. This shows the need of a clear indication of the type of deformations considered. The families appearing are constructed as families of algebras of Krichever–Novikov type. They show up in a natural way in the global operator approach to the quantization of two-dimensional conformal field theory. In addition, a proof of the infinitesimal and formal rigidity of the Witt algebra is presented.

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FRONSDAL, CHRISTIAN. « DEFORMATION QUANTIZATION : IS C1 NECESSARILY SKEW ? » International Journal of Modern Physics B 16, n^o 14n15 (20 juin 2002) : 1925–30. http://dx.doi.org/10.1142/s0217979202011640.

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Deformation quantization (of a commutative algebra) is based on the introduction of a new associative product, expressed as a formal series, [Formula: see text]. In the case of the algebra of functions on a symplectic space the first term in the perturbation is often identified with the antisymmetric Poisson bracket. There is a wide-spread belief that every associative *-product is equivalent to one for which C1(f,g) is antisymmetric and that, in particular, every abelian deformation is trivial. This paper shows that this is far from being the case and illustrates the existence of abelian deformations by physical examples.

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Bäck, P. « Notes on formal deformations of quantum planes and universal enveloping algebras ». Journal of Physics : Conference Series 1194 (avril 2019) : 012011. http://dx.doi.org/10.1088/1742-6596/1194/1/012011.

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Baklouti, A., N. Elaloui et I. Kedim. « The Selberg–Weil–Kobayashi rigidity theorem : The rank one solvable case ». International Journal of Mathematics 27, n^o 10 (septembre 2016) : 1650085. http://dx.doi.org/10.1142/s0129167x16500853.

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A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that any discontinuous group [Formula: see text] for a homogeneous space [Formula: see text] is abelian and at most of rank 2. Then we discuss an analog of the Selberg–Weil–Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the [Formula: see text]-action on [Formula: see text] is not always effective, and thus the space of group theoretic deformations (formal deformations) [Formula: see text] could be larger than geometric deformation spaces. We determine [Formula: see text] and also its quotient modulo uneffective parts when the rank [Formula: see text]. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.

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Khalfoun, Hafedh, Nizar Ben Fraj et Meher Abdaoui. « Cohomology of 𝔞𝔣𝔣(m|1) acting on the space of superpseudodifferential operators on the supercircle S1|m ». Asian-European Journal of Mathematics 11, n^o 04 (août 2018) : 1850057. http://dx.doi.org/10.1142/s1793557118500572.

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We investigate the first differential cohomology space associated with the embedding of the affine Lie superalgebra [Formula: see text] on the [Formula: see text]-dimensional supercircle [Formula: see text] in the Lie superalgebra [Formula: see text] of superpseudodifferential operators with smooth coefficients, where [Formula: see text]. Following Ovsienko and Roger, we give explicit expressions of the basis cocycles. We study the deformations of the structure of the [Formula: see text]-module [Formula: see text]. We prove that any formal deformation is equivalent to its infinitesimal part.

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Müller, Gerd. « Deformations of reductive group actions ». Mathematical Proceedings of the Cambridge Philosophical Society 106, n^o 1 (juillet 1989) : 77–88. http://dx.doi.org/10.1017/s0305004100067992.

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Consider actions of a reductive complex Lie group G on an analytic space germ (X, 0). In a previous paper [16] we proved that such an action is determined uniquely (up to conjugation with an automorphism of (X, 0)) by the induced action of G on the tangent space of (X, 0). Here it will be shown that every deformation of such an action, parametrized holomorphically by a reduced analytic space germ, is trivial, i.e. can be obtained from the given action by conjugation with a family of automorphisms of (X, 0) depending holomorphically on the parameter. (For a more precise formulation in terms of actions on analytic ℂ-algebras, see Theorem 2 below. An analogue for actions on formal ℂ-algebras is given there too.)

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Collin, Annabelle, Sébastien Imperiale, Philippe Moireau, Jean-Frédéric Gerbeau et Dominique Chapelle. « Apprehending the effects of mechanical deformations in cardiac electrophysiology : A homogenization approach ». Mathematical Models and Methods in Applied Sciences 29, n^o 13 (2 décembre 2019) : 2377–417. http://dx.doi.org/10.1142/s0218202519500490.

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We follow a formal homogenization approach to investigate the effects of mechanical deformations in electrophysiology models relying on a bidomain description of ionic motion at the microscopic level. To that purpose, we extend these microscopic equations to take into account the mechanical deformations, and proceed by recasting the problem in the framework of classical two-scale homogenization in periodic media, and identifying the equations satisfied by the first coefficients in the formal expansions. The homogenized equations reveal some interesting effects related to the microstructure — and associated with a specific cell problem to be solved to obtain the macroscopic conductivity tensors — in which mechanical deformations play a nontrivial role, i.e. they do not simply lead to a standard bidomain problem posed in the deformed configuration. We then present detailed numerical illustrations of the homogenized model with coupled cardiac electrical–mechanical simulations — all the way to ECG simulations — albeit without taking into account the abundantly-investigated effect of mechanical deformations in ionic models, in order to focus here on other effects. And in fact our numerical results indicate that these other effects are numerically of a comparable order, and therefore cannot be disregarded.

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Li, Qiang, et Lili Ma. « 1-parameter formal deformations and abelian extensions of Lie color triple systems ». Electronic Research Archive 30, n^o 7 (2022) : 2524–39. http://dx.doi.org/10.3934/era.2022129.

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<abstract><p>The purpose of this paper is to discuss Lie color triple systems. The cohomology theory of Lie color triple systems is established, then 1-parameter formal deformations and abelian extensions of Lie color triple systems are studied using cohomology.</p></abstract>

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Choie, YoungJu, François Dumas, François Martin et Emmanuel Royer. « Formal deformations of the algebra of Jacobi forms and Rankin–Cohen brackets ». Comptes Rendus. Mathématique 359, n^o 4 (17 juin 2021) : 505–21. http://dx.doi.org/10.5802/crmath.193.

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Kawamata, Yujiro. « On non-commutative formal deformations of coherent sheaves on an algebraic variety ». EMS Surveys in Mathematical Sciences 8, n^o 1 (31 août 2021) : 237–63. http://dx.doi.org/10.4171/emss/49.

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Liu, Shanshan, Abdenacer Makhlouf et Lina Song. « The full cohomology, abelian extensions and formal deformations of Hom-pre-Lie algebras ». Electronic Research Archive 30, n^o 8 (2022) : 2748–73. http://dx.doi.org/10.3934/era.2022141.

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<abstract><p>The main purpose of this paper is to provide a full cohomology of a Hom-pre-Lie algebra with coefficients in a given representation. This new type of cohomology exploits strongly the Hom-type structure and fits perfectly with simultaneous deformations of the multiplication and the homomorphism defining a Hom-pre-Lie algebra. Moreover, we show that its second cohomology group classifies abelian extensions of a Hom-pre-Lie algebra by a representation.</p></abstract>

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Lychev, S. A., K. G. Koifman et A. V. Digilov. « NONLINEAR DYNAMIC EQUATIONS FOR ELASTIC MICROMORPHIC SOLIDS AND SHELLS. PART I ». Vestnik of Samara University. Natural Science Series 27, n^o 1 (29 novembre 2021) : 81–103. http://dx.doi.org/10.18287/2541-7525-2021-27-1-81-103.

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The present paper develops a general approach to deriving nonlinear equations of motion for solids whose material points possess additional degrees of freedom. The essential characteristic of this approach is theaccount of incompatible deformations that may occur in the body due to distributed defects or in the result of the some kind of process like growth or remodelling. The mathematical formalism is based on least action principle and Noether symmetries. The peculiarity of such formalism is in formal description of reference shape of the body, which in the case of incompatible deformations has to be regarded either as a continual family of shapes or some shape embedded into non-Euclidean space. Although the general approach yields equations for Cosserat-type solids, micromorphic bodies and shells, the latter differ significantly in the formal description of enhanced geometric structures upon which the action integral has to be defined. Detailed discussion of this disparity is given.

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GUERRINI, L. « COMPLETIONS OF 2-TORSION KN-ALGEBRAS OF GENUS 1 ». Reviews in Mathematical Physics 13, n^o 02 (février 2001) : 253–66. http://dx.doi.org/10.1142/s0129055x01000648.

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Krichever–Novikov algebras [Formula: see text] of genus 1 with markings which are two 2-torsion points are related to a family [Formula: see text] of deformations of the Witt algebra [Formula: see text], where f varies in the space of even polynomials with vanishing constant terms. An isomorphism between the formal (resp. analytic) completion of these KN-algebras with those of the Witt algebra is proved. Central extensions of these algebras are also defined and their formal completion is proved to be isomorphic to that of the Virasoro algebra Vir.

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Demchenko, Oleg, et Alexander Gurevich. « p-adic period map for the moduli space of deformations of a formal group ». Journal of Algebra 288, n^o 2 (juin 2005) : 445–62. http://dx.doi.org/10.1016/j.jalgebra.2004.12.017.

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Lecomte, P. B. A., D. Melotte et C. Roger. « Explicit form and convergence of 1-differential formal deformations of the poisson Lie algebra ». Letters in Mathematical Physics 18, n^o 4 (novembre 1989) : 275–85. http://dx.doi.org/10.1007/bf00405259.

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Lecomte, P. B. A. « Application of the cohomology of graded Lie algebras to formal deformations of Lie Algebras ». Letters in Mathematical Physics 13, n^o 2 (février 1987) : 157–66. http://dx.doi.org/10.1007/bf00955206.

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BYTSENKO, A. A. « BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN SIGMA MODELS ». International Journal of Modern Physics A 26, n^o 22 (10 septembre 2011) : 3769–80. http://dx.doi.org/10.1142/s0217751x11054231.

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The closed string correlators can be constructed from the open ones using topological string theories as a model. The space of physical closed string states is isomorphic to the Hochschild cohomology of (A,Q) (operator Q of ghost number one), - this statement has been verified by means of computation of the Hochschild cohomology of the category of D -branes. We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A -model, and a Calabi-Yau manifold with boundaries in the B -model. We show that equivalent classes of deformations are describing by a Hochschild cohomology theory of the DG-algebra, [Formula: see text], [Formula: see text], which is defined to be the cohomology of (-1)nQ+d Hoch . Here [Formula: see text] is the initial non-deformed BRST operator while ∂ deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We assume that if in the theory exists a single D -brane then all the information associated with deformations is encoded in an associative algebra A equipped with a differential [Formula: see text]. In addition equivalence classes of deformations of these data are described by a Hochschild cohomology of (A,Q), an important geometric invariant of the (anti)holomorphic structure on X. We also discuss the identification of the harmonic structure (HT•(X); HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.

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BYTSENKO, A. A. « BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN SIGMA MODELS ». International Journal of Modern Physics : Conference Series 03 (janvier 2011) : 75–86. http://dx.doi.org/10.1142/s2010194511001164.

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The closed string correlators can be constructed from the open ones using topological string theories as a model. The space of physical closed string states is isomorphic to the Hochschild cohomology of (A, Q) (operator Q of ghost number one), - this statement has been verified by means of computation of the Hochschild cohomology of the category of D-branes. We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. We show that equivalent classes of deformations are describing by a Hochschild cohomology theory of the DG-algebra [Formula: see text], [Formula: see text], which is defined to be the cohomology of (-1)n Q + d Hoch . Here [Formula: see text] is the initial non-deformed BRST operator while ∂deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We assume that if in the theory exists a single D-brane then all the information associated with deformations is encoded in an associative algebra A equipped with a differential [Formula: see text]. In addition equivalence classes of deformations of these data are described by a Hochschild cohomology of (A, Q), an important geometric invariant of the (anti)holomorphic structure on X. We also discuss the identification of the harmonic structure (HT•(X); HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.

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BYTSENKO, A. A., M. CHAICHIAN, A. TUREANU et F. L. WILLIAMS. « BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN TOPOLOGICAL FIELD THEORIES ». International Journal of Modern Physics A 28, n^o 16 (28 juin 2013) : 1350069. http://dx.doi.org/10.1142/s0217751x13500693.

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We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi–Yau manifold with boundaries in the B-model. A relevant concept in the vertex operator algebra and the BRST cohomology is that of the elliptic genera (the one-loop string partition function). We show that the elliptic genera can be written in terms of spectral functions of the hyperbolic three-geometry (which inherits the cohomology structure of BRST-like operator). We show that equivalence classes of deformations are described by a Hochschild cohomology theory of the DG-algebra [Formula: see text], which is defined to be the cohomology of (-1)n Q + d Hoch . Here, [Formula: see text] is the initial nondeformed BRST operator while ∂ deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We discuss the identification of the harmonic structure (HT•(X);HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.

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Guan, Baoling, Liangyun Chen et Yao Ma. « On the Deformations and Derivations ofn-Ary Multiplicative Hom-Nambu-Lie Superalgebras ». Advances in Mathematical Physics 2014 (2014) : 1–9. http://dx.doi.org/10.1155/2014/381683.

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We introduce the relevant concepts ofn-ary multiplicative Hom-Nambu-Lie superalgebras and construct three classes ofn-ary multiplicative Hom-Nambu-Lie superalgebras. As a generalization of the notion of derivations forn-ary multiplicative Hom-Nambu-Lie algebras, we discuss the derivations ofn-ary multiplicative Hom-Nambu-Lie superalgebras. In addition, the theory of one parameter formal deformation ofn-ary multiplicative Hom-Nambu-Lie superalgebras is developed by choosing a suitable cohomology.

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