Theory of Finite Pseudoalgebras

Abstract

Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional'' analogues of conformal algebras. They are defined as Lie algebras in a certain ``pseudotensor'' category instead of the category of vector spaces. A pseudotensor category (as introduced by Lambek, and by Beilinson and Drinfeld) is a category equipped with ``polylinear maps'' and a way to compose them. This allows for the definition of Lie algebras, representations, cohomology, etc. An instance of such a category can be constructed starting from any cocommutative (or more generally, quasitriangular) Hopf algebra $H$. The Lie algebras in this category are called Lie $H$-pseudoalgebras. The main result of this paper is the classification of all simple and all semisimple Lie $H$-pseudoalgebras which are finitely generated as $H$-modules. We also start developing the representation theory of Lie pseudoalgebras; in particular, we prove analogues of the Lie, Engel, and Cartan-Jacobson Theorems. We show that the cohomology theory of Lie pseudoalgebras describes extensions and deformations and is closely related to Gelfand-Fuchs cohomology. Lie pseudoalgebras are closely related to solutions of the classical Yang-Baxter equation, to differential Lie algebras (introduced by Ritt), and to Hamiltonian formalism in the theory of nonlinear evolution equations. As an application of our results, we derive a classification of simple and semisimple linear Poisson brackets in any finite number of indeterminates.

102 pages, 7 figures, AMS latex

References in corpus (2)

• Cohomology of Conformal Algebras
• $\Gamma$-Conformal Algebras

Cited by in corpus (30)

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• Lie conformal algebra cohomology and the variational complex
• The variational Poisson cohomology
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• Irreducible Modules over Finite Simple Lie Pseudoalgebras I. Primitive Pseudoalgebras of Type W and S
• On special identities for dialgebras
• Gelfand--Dorfman algebras, derived identities, and the Manin product of operads
• Jacobi Identity for Vertex Algebras in Higher Dimensions
• Simple associative conformal algebras of linear growth
• Universal enveloping Poisson conformal algebras
• On the Hochschild cohomologies of associative conformal algebras with a finite faithful representation
• On deformations of multidimensional Poisson brackets of hydrodynamic type
• On the Ado Theorem for finite Lie conformal algebras with Levi decomposition
• Lie conformal superalgebras and duality of modules over linearly compact Lie superalgebras
• Homogeneous conformal averaging operators on semisimple Lie algebras
• Irreducible Modules over Finite Simple Lie Pseudoalgebras II. Primitive Pseudoalgebras of Type K
• Computation of cohomology of Lie conformal and Poisson vertex algebras
• Calculus structure on the Lie conformal algebra complex and the variational complex
• A remark on simplicity of vertex algebras and Lie conformal algebras
• Finite vertex algebras and nilpotence
• Irreducible modules over finite simple Lie pseudoalgebras III. Primitive pseudoalgebras of type H
• Hochshild cohomology of the associative conformal algebra Cend_{1,x}
• Standard bases for the universal associative conformal envelopes of Kac--Moody conformal algebras
• Graded associative conformal algebras of finite type
• On the Dong Property for a binary quadratic operad
• Hochschild cohomology of the Weyl conformal algebra with coefficients in finite modules
• Rota-Baxter operators on $\mathrm{Cur}({\mathrm{sl}}_2(\mathbb{C}))$
• Conformal Yang-Baxter equation on $\mathrm{Cur}({\mathrm{sl}}_2(\mathbb{C}))$
• Finiteness and orbifold Vertex Operator Algebras
• Classification of simple differential Lie and Jordan (super)coalgebras of finite rank