New algebraic invariants for definable subsets in universal algebra

We consider problems of comparing universal algebras in respect of their conditional algebraic geometries. Such comparisons admit of a quite natural algebraic interpretation. Geometric scales for varieties of algebras constructed based on these relations are a natural tool for classifying the varieties of algebras, discriminator varieties in particular.     

Implicit algebraic geometry on categories of universal algebras

Using the notion of an implicit operation on universal algebras, we redefine basic notions of the algebraic geometry of universal algebras. The results obtained for the implicit algebraic geometry imply (as special cases) the known results on the conditional geometric and algebraic geometric comparability of algebras.     

Coherent algebraic sheaves in real algebraic geometry

Summary In the following paper we give examples of vector bundlesF→X defined on a regular, compact, connected affine varietyX such thatF has no algebraic structure.

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Riassunto Nel presente lavoro si danno esempi di fibrati vettorialiF→X, definiti su varietà affini, regolari, compatte, connesseX che non ammettono struttura algebrica.

     

The universal conservative algebra

Lyubertsy City, Moscow District. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 3, pp. 30–38, May–June, 1990.     

Deformation quantization in algebraic geometry

We study deformation quantizations of the structure sheaf O_X of a smooth algebraic variety X in characteristic 0. Our main result is that when X is D-affine, any formal Poisson structure on X determines a deformation quantization of O_X (canonically, up to gauge equivalence). This is an algebro-geometric analogue of Kontsevich's celebrated result.     

Spherical subcategories in algebraic geometry

We study objects in triangulated categories which have a two‐dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical. This general result is then applied to examples from algebraic geometry.     

Gromov-Witten invariants in algebraic geometry

Gromov-Witten invariants for arbitrary non-singular projective varieties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.] Oblatum 26-II-1996 & 27-VI-1996     

DG Poisson algebra and its universal enveloping algebra

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 show that A^ue has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over A is isomorphic to the category of DG modules over A^ue. Furthermore, we prove that the notion of universal enveloping algebra A^ue is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.     

Quantum lie algebra of currents — The universal algebraic structure of symmetries of completely integrable dynamical systems

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 6, pp. 764–768, November–December, 1988.