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We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of coadjoint orbits of a semisimple algebraic group to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras and conjecture that in that case the image consists of the primitive ideals corresponding to one-dimensional representations of W-algebras. Along the way, we get several new results on the Lusztig-Spaltenstein induction for coadjoint orbits. [ABSTRACT FROM AUTHOR]

We construct a differential graded Lie algebra g controlling the Poisson deformations of an affine Poisson variety. We analyze g in the case of affine Gorenstein toric Poisson varieties. Moreover, explicit description of the second and third Hochschild cohomology groups is given for three-dimensional affine Gorenstein toric varieties. [ABSTRACT FROM AUTHOR]

Using 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. [ABSTRACT FROM AUTHOR]

In this paper, first we study infinitesimal deformations of a Lie algebra with a representation and introduce the notion of a Nijenhuis pair, which gives a trivial deformation of a Lie algebra with a representation. Then we introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on a Lie algebra with a representation, which is a generalization of the r − n structure (r-matrix-Nijenhuis structure) introduced by Ravanpak, Rezaei-Aghdam, and Haghighatdoost. We show that a Kupershmidt-(dual-)Nijenhuis structure gives rise to a hierarchy of Kupershmidt operators. Finally, we define a Rota-Baxter-Nijenhuis structure to be a Kupershmidt-Nijenhuis structure on a Lie algebra with respect to the adjoint representation and study the relation between Rota-Baxter-Nijenhuis structures and r-matrix-Nijenhuis structures. [ABSTRACT FROM AUTHOR]

We classify kinematical Lie algebras in dimension D + 1 for D > 3 up to isomorphism. This is part of a series of papers applying deformation theory to the classification of kinematical Lie algebras in arbitrary dimension. This is approached via the classification of deformations of the relevant static kinematical Lie algebra. We also classify the deformations of the universal central extension of the static kinematical Lie algebra in dimension D + 1 for D > 3. In addition, we determine which of these Lie algebras admit an invariant inner product. [ABSTRACT FROM AUTHOR]

We prove that when Kontsevich's deformation quantization is applied on weight homogeneous Poisson structures, the operators in the ∗-product formula are weight homogeneous. In the linear Poisson case for a semisimple Lie algebra the Poisson manifold is . As an application we provide an isomorphism between the Cattaneo-Felder-Torossian reduction algebra and the -algebra . We also show that in the -algebra setting, is polynomial. Finally, we compute generators of as a deformation of . [ABSTRACT FROM AUTHOR]

We give explicit formulas proving that the following Lie (super)algebras are restricted: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, simple subquotients over an algebraically closed field of characteristic 3 of these (super)algebras (under certain conditions), and deformed divergence-free Lie superalgebras of a certain type with any finite number of indeterminates in any characteristic. [ABSTRACT FROM AUTHOR]

All classes of integrable cocycles inH2(L,L) are obtained for Lie algebra of typeG2over an algebraically closed field of characteristic 2. It is proved that there exist only two orbits of classes of integrable cocycles with respect to automorphism group. The global deformation is shown to exist for any nontrivial class of integrable cocycles. These deformations are isomorphic to one of the two algebras of Cartan type, one of which beingS(3:1,ω) while the otherH(4:1,ω). [ABSTRACT FROM PUBLISHER]

The aim of this article is to discuss-Jordan Lie triple systems, which are a generalization of Lie triple systems. Cohomological characterizations of-Jordan Lie triple systems are established, then deformations, Nijenhuis operators, Abelian extensions and-extensions of-Jordan Lie triple systems are studied using cohomology. [ABSTRACT FROM PUBLISHER]

In this article we consider the centre of the reduced enveloping algebra of the Lie algebra of a reductive algebraic group in very good characteristic p > 2 . The Harish-Chandra centre maps to the centre of each reduced enveloping algebra and, using a combination of induction and deformation arguments, we describe precisely for which p -characters this map is surjective: it is if and only if the chosen character is regular. This provides the converse to a theorem of Mirković and Rumynin. [ABSTRACT FROM AUTHOR]

The Lie algebras of order F have important applications for the fractional supersymmetry, and on the other hand the filiform Lie (super)algebras have very important properties into the Lie Theory. Thus, the aim of this work is to study filiform Lie algebras of order F which were introduced in Navarro (2014). In this work we obtain new families of filiform Lie algebras of order 3, in which the complexity of the problem rises considerably respecting to the cases considered in Navarro (2014). [ABSTRACT FROM AUTHOR]

Let $${\left\{\mathsf{A}_j|j=0,1,\ldots,rank(g)\right\}}$$ be the fundamental generators of the generalized q-Onsager algebra $${{\cal O}_{q}({\widehat{g}})}$$ introduced in Baseilhac and Belliard (Lett Math Phys 93:213-228, ), where $${\widehat{g}}$$ is a simply laced affine Lie algebra. New relations between certain monomials of the fundamental generators-indexed by the integer $${r\in\mathbb{Z}^{+}}$$ -are conjectured. These relations can be seen as deformed analogs of Lusztig's rth higher order q-Serre relations associated with $${{\cal U}_q({\widehat g})}$$ , which are recovered as special cases. The relations are proven for $${r\leq 5}$$ . For r generic, several supporting evidences are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider deformations of Lie 2-algebras via the cohomology theory. We prove that a 1-parameter infinitesimal deformation of a Lie 2-algebra g corresponds to a 2-cocycle of g with the coefficients in the adjoint representation. A Nijenhuis operator on a Lie 2-algebra is introduced to describe trivial deformations. We also study abelian extensions of Lie 2-algebras from the viewpoint of deformations of semidirect product Lie 2-algebras. [ABSTRACT FROM AUTHOR]

In the present paper we describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any n -dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra F n 3 ( 0 ) . We establish that in the same way any n -dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras F n 1 , F n 2 and F n 3 ( α ) . Moreover, we describe the linear integrable deformations of the above-mentioned algebras with a fixed basis of H L 2 in the set of all n -dimensional Leibniz algebras. Among these deformations one new rigid algebra has been found. [ABSTRACT FROM AUTHOR]

The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F=3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, "Cohomologie des alg`ebres de Lie nilpotentes. Application à l'étude de la variété des algèbres de Lie nilpotentes," Bull. Soc. Math. France 98, 81-116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases. [ABSTRACT FROM AUTHOR]

By the Mather-Yau theorem, a complex hypersurface germ ν with isolated singularity is completely determined by its moduli algebra A(ν). The proof of the theorem does not provide an explicit procedure for recovering ν from A(ν), and finding such a procedure is a long-standing open problem. In this paper, we present an explicit way for reconstructing ν from A(ν) up to biholomorphic equivalence under the assumption that the singularity of ν is homogeneous, in which case A(ν) coincides with the Milnor algebra of ν. [ABSTRACT FROM AUTHOR]

We obtain algebraic Frobenius manifolds from classical W-algebras associated to subregular nilpotent elements in simple Lie algebras of type Dr (where r is even) and Er. The resulting Frobenius manifolds are certain subvarieties in the total spaces of semiuniversal deformation of simple hypersurface singularities of the same types. [ABSTRACT FROM AUTHOR]

We compute the restricted infinitesimal deformations of the restricted simple Lie algebras over an algebraically closed field of characteristic p ≥ 5. [ABSTRACT FROM AUTHOR]

Abstract: We propose a new approach to study coideal algebras. It is well-known that Manin triples (or equivalently Lie bi-algebra structures) are the requirement to deform Lie algebras and to obtain quantum groups. In this paper, introducing some particular automorphisms of Manin triples, we define new structures that we call Lie bi-ideal structures. A link with coisotropic subalgebras is explained. We show that their deformation provide coideal algebras. As examples, we recover from our general construction the twisted Yangians , the q-Onsager algebra and the augmented q-Onsager algebra. As an important by-product, we find a new presentation for the twisted Yangians. [Copyright &y& Elsevier]

We classify the nilpotent orbits in a simple Lie algebra for which the restriction of the adjoint quotient map to a Slodowy slice is the universal Poisson deformation of its central fibre. This generalises work of Brieskorn and Slodowy on subregular orbits. In particular, we find in this way new singular symplectic hypersurfaces of dimension four and six. [ABSTRACT FROM PUBLISHER]

Abstract: We show that the algebras describing blocks of the category of cuspidal weight (resp. generalized weight) -modules are one-parameter (resp. multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley–Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and Schur algebras and in the study of the Milnor fiber of Kleinian singularities. [Copyright &y& Elsevier]

We describe the Lie algebra deformations of D=4 Maxwell superalgebra that was recently introduced as the symmetry algebra of a kappa-symmetric massless superparticle in a supersymmetric constant electromagnetic backgrounds. Further we introduce the D=3 Maxwell superalgebra and present all its possible deformations. Finally, the deformed superalgebras are used to derive via a contraction procedure the complete set of Casimir operators for D=4 and D=3 Maxwell superalgebras. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend to Hom-algebra structures the theory of 1-parameter formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis and Richardson. In this paper, formal deformations of Hom-associative and Hom-Lie algebras are studied. The first groups of a deformation cohomology are constructed and several examples of deformations are given. We also provide families of Hom-Lie algebras deforming Lie algebra 2() and describe as formal deformations the q-deformed Witt algebra and Jackson 2(). [ABSTRACT FROM AUTHOR]

Abstract: Let be a Lie bialgebra and let V be a finite-dimensional -module. We study deformations of the symmetric algebra of V which are equivariant with respect to an action of the quantized enveloping algebra , resp. . We investigate, in particular, such quantizations obtained from the quantization of certain Lie bialgebra structures on the semidirect product of and V. We classify these structure in the important special case, when is complex, simple, with quasitriangular Lie bialgebra structure and V is a simple g-module. We then introduce a more general notion, co-Poisson module algebras and their quantizations, to further address the problem and show that many known examples of quantized symmetric algebras can be described in this language. [Copyright &y& Elsevier]

Abstract: We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartan-type: the Witt–Jacobson and the Special Lie algebras. [Copyright &y& Elsevier]

Nonlinear extensions of the two-mode squeezed vacuum states (NTMSVS's) are constructed and special cases of these states are discussed. We have constructed the NTMSVS's realization of SU(1,1) Lie algebra. Two cases of the definition are considered for unitary and non-unitary deformation operator functions. Some nonclassical properties of these states are discussed. [ABSTRACT FROM AUTHOR]

In this article we apply a method devised in Hartwig, Larsson, and Silvestrov (2006) and Larsson and Silvestrov (2005a) to the simple 3-dimensional Lie algebra 2(). One of the main points of this deformation method is that the deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when our deformation scheme is applied to 2() we can, by choosing parameters suitably, deform 2() into the Heisenberg Lie algebra and some other 3-dimensional Lie algebras in addition to more exotic types of algebras, this being in stark contrast to the classical deformation schemes where 2() is rigid. [ABSTRACT FROM AUTHOR]

In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever–Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac–Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess–Zumino–Witten–Novikov models appearing in the quantization of Conformal Field Theory. [ABSTRACT FROM AUTHOR]

In this paper, we initiate a general classification for Lie algebras of order 3 and we give all Lie algebras of order 3 based on sl(2,C) and iso(1, 3) the Poincaré algebra in four dimensions. We then set the basis of the theory of the deformations (in the Gerstenhaber sense) and contractions for Lie algebras of order 3. [ABSTRACT FROM AUTHOR]

In this paper, we study deformations of coisotropic submanifolds in a symplectic manifold. First we derive the equation that governs C∞ deformations of coisotropic submanifolds and define the corresponding C∞-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. This is a non-commutative and non-linear generalization of the well-known description of the local deformation space of Lagrangian submanifolds as the set of graphs of closed one forms in the Darboux-Weinstein chart of a given Lagrangian submanifold. We then introduce the notion of strong homotopy Lie algebroid (or L∞ -algebroid) and associate a canonical isomorphism class of strong homotopy Lie algebroids to each pre-symplectic manifold ( Y,ω) and identify the formal deformation space of coisotropic embeddings into a symplectic manifold in terms of this strong homotopy Lie algebroid. The formal moduli space then is provided by the gauge equivalence classes of solutions of a version of the Maurer-Cartan equation (or the master equation) of the strong homotopy Lie algebroid, and plays the role of the classical part of the moduli space of quantum deformation space of coisotropic A-branes. We provide a criterion for the unobstructedness of the deformation problem and analyze a family of examples that illustrates that this deformation problem is obstructed in general and heavily depends on the geometry and dynamics of the null foliation. [ABSTRACT FROM AUTHOR]

The formal rigidity of the Witt and Virasoro algebras was first established by Fialowski ['Deformations of some infinite-dimensional Lie algebras,' J. Math. Phys. 31, 1340-1343 (1990)]. The proof was based on some earlier results of the author and Goncharowa, and was not presented there. In this paper, we give an elementary proof of these facts. [ABSTRACT FROM AUTHOR]

We construct the deformation functor associated with a pair of morphisms of differential graded Lie algebras, and use it to study infinitesimal deformations of holomorphic maps of compact complex manifolds. In particular, using L∞ structures, we give an explicit description of the differential graded Lie algebra that controls this problem. [ABSTRACT FROM PUBLISHER]