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Abelian deformations of ordinary algebras of functions are studied. The role of Harrison cohomology in classifying such deformations is illustrated in the context of simple examples chosen for their relevance to physics. It is well known…
Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely…
In this paper, we consider compatible Hom-associative algebras as a twisted version of compatible associative algebras. Compatible Hom-associative algebras are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie…
A Poisson geometry arising from maximal commutative subalgebras is studied. A spectral sequence convergent to Hochschild homology with coefficients in a bimodule is presented. It depends on the choice of a maximal commutative subalgebra…
Motivated by the Poisson Dixmier-Moeglin equivalence problem, a systematic study of commutative unitary rings equipped with a {\em biderivation}, namely a binary operation that is a derivation in each argument, is here begun, with an eye…
We compute the space of Poisson traces on a classical W-algebra modulo an arbitrary central character, i.e., linear functionals on such an algebra invariant under Hamiltonian derivations. This space identifies with the top cohomology of the…
In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We…
We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations…
This paper investigates the Poisson geometry associated to a cluster algebra over the complex numbers, and its relationship to compatible torus actions. We show, under some assumptions, that each Noetherian cluster algebra has only finitely…
The geometrical description of deformation quantization based on quantum duality principle makes it possible to introduce deformed Lie-Poisson structure. It serves as a natural analogue of classical Lie bialgebra for the case when the…
We construct a canonical map from the Poisson vertex algebra cohomology complex to the differential Harrison cohomology complex, which restricts to an isomorphism on the top degree. This is an important step in the computation of Poisson…
A polar hypersurface P of a complex analytic hypersurface germ, f=0, can be investigated by analyzing the invariance of certain Newton polyhedra associated to the image of P, with respect to suitable coordinates, by certain morphisms…
Let X be a an affine smooth symplectic variety over $\mathbb{Z}/p\mathbb{Z},$ and A be its deformation quantization over the p-adic integers. We prove that for all $n\geq 1,$ the Hochschild cohomogy of $A/p^nA$ is isomorphic to the de…
We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several…
In this paper we construct a combinatorial algorithm of resolution of singularities for binomial ideals, over a field of arbitrary characteristic. This algorithm is applied to any binomial ideal. This means ideals generated by binomial…
We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential…
Let~$X=\Po/\Gamma$ be an~$n$-punctured sphere, $n>3$. We introduce and study~$n-3$ deformation operators on the space of modular forms~$M_*(\Gamma)$ based on the classical theory of uniformizing differential equations and accessory…
The nilpotent cone of a reductive Lie algebra has a desingularization given by thecotangent bundle of the flag variety. Analogously, the nullcone of a cartesianpower of the algebra has a desingularization given by a vector bundle over…
We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough…
Let X(\Sigma) be a smooth projective toric variety for a complex torus T_\C. In this paper, a real T_\C-invariant Poisson structure \Pi_\Sigma is constructed on the complex manifold X(\Sigma), the symplectic leaves of which are the…