Central extensions of Lax operator algebras (original) (raw)
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Almost-graded central extensions of Lax operator algebras
Banach Center Publications, 2011
Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for gl(n), with the spectral parameter on a Riemann surface, were introduced by Krichever. In joint works of Krichever and Sheinman their algebraic structure was revealed and extended to more general groups. These algebras are almost-graded. In this article their definition is recalled and classification and uniqueness results for almost-graded central extensions for this new class of algebras are presented. The explicit forms of the defining cocycles are given. If the finite-dimensional Lie algebra on which the Lax operator algebra is based is simple then, up to equivalence and rescaling of the central element, there is a unique non-trivial almost-graded central extension. These results are joint work with Oleg Sheinman. This is an extended write-up of a talk presented at the 5 th
Multipoint Lax operator algebras: almost-graded structure and central extensions
Sbornik: Mathematics, 2014
Recently, Lax operator algebras appeared as a new class of higher genus current type algebras. Based on I. Krichever's theory of Lax operators on algebraic curves they were introduced by I. Krichever and O. Sheinman. These algebras are almost-graded Lie algebras of currents on Riemann surfaces with marked points (in-points, out-points, and Tyurin points). In a previous joint article of the author with Sheinman the local cocycles and associated almost-graded central extensions are classified in the case of one in-point and one out-point. It was shown that the almost-graded extension is essentially unique. In this article the general case of Lax operator algebras corresponding to several in-and out-points is considered. In a first step it is shown that they are almost-graded. The grading is given by the splitting of the marked points which are non-Tyurin points into in-and out-points. Next, classification results both for local and bounded cocycles are shown. The uniqueness theorem for almost-graded central extensions follows. For this generalization additional techniques are needed which are presented in this article.
Multipoint Lax operator algebras: almost graded structure and central extension
2016
Recently, Lax operator algebras appeared as a new class of higher genus current type algebras. Based on I. Krichever's theory of Lax operators on algebraic curves they were introduced by I. Krichever and O. Sheinman. These algebras are almost-graded Lie algebras of currents on Riemann surfaces with marked points (in-points, out-points, and Tyurin points). In a previous joint article of the author with Sheinman the local cocycles and associated almost-graded central extensions are classified in the case of one in-point and one out-point. It was shown that the almost-graded extension is essentially unique. In this article the general case of Lax operator algebras corresponding to several in-and out-points is considered. In a first step it is shown that they are almost-graded. The grading is given by the splitting of the marked points which are non-Tyurin points into in-and out-points. Next, classification results both for local and bounded cocycles are shown. The uniqueness theorem for almost-graded central extensions follows. For this generalization additional techniques are needed which are presented in this article. Contents 1. Introduction 2. The algebras 2.1. Lax operator algebras 2.2. Krichever-Novikov algebras of current type 3. The almost-graded structure 3.1. The statements 3.2. The function algebra A and the vector field algebra L 3.3. The proofs 4. Module structure 4.1. Lax operator algebras as modules over A 4.2. Lax operator algebras as modules over L 4.3. Module structure over D 1 and the algebra D 1 g 5. Cocycles 5.1. Geometric cocycles
Central extensions of some Lie algebras
Proceedings of the American Mathematical Society, 1998
We consider three Lie algebras: D e r C ( ( t ) ) Der \mathbb {C}((t)) , the Lie algebra of all derivations on the algebra C ( ( t ) ) \mathbb {C}((t)) of formal Laurent series; the Lie algebra of all differential operators on C ( ( t ) ) \mathbb {C}((t)) ; and the Lie algebra of all differential operators on C ( ( t ) ) ⊗ C n . \mathbb {C}((t))\otimes \mathbb {C}^n. We prove that each of these Lie algebras has an essentially unique nontrivial central extension.
Local cocycles and central extensions for multipoint algebras of Krichever-Novikov type
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000
Multi-point algebras of Krichever Novikov type for higher genus Riemann surfaces are generalisations of the Virasoro algebra and its related algebras. Complete existence and uniqueness results for local 2-cocycles defining almost-graded central extensions of the functions algebra, the vector field algebra, and the differential operator algebra (of degree ≤ 1) are shown. This is applied to the higher genus, multi-point affine algebras to obtain uniqueness for almost-graded central extensions of the current algebra of a simple finite-dimensional Lie algebra. An earlier conjecture of the author concerning the central extension of the differential operator algebra induced by the semi-infinite wedge representations is proved.
D G ] 2 6 Fe b 20 04 EXTENSIONS OF LIE ALGEBRAS
2008
We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out.
2000
We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. 2000 Mathematics Subject Classification. Primary 17B05, 17B56. Key words and phrases. Extensions of Lie algebras, cohomology of Lie algebras. P.W.M. was supported by 'Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 14195 MAT'. Typeset by A M S-T E X
A cohomology theory for Lie 2-algebras
2018
In this article, we introduce a new cohomology theory associated to a Lie 2-algebras. This cohomology theory is shown to extend the classical cohomology theory of Lie algebras; in particular, we show that the second cohomology group classifies an appropriate type of extensions.
From the Virasoro Algebra to Krichever–Novikov Type Algebras and Beyond
Harmonic and Complex Analysis and its Applications, 2013
Starting from the Virasoro algebra and its relatives the generalization to higher genus compact Riemann surfaces was initiated by Krichever and Novikov. The elements of these algebras are meromorphic objects which are holomorphic outside a finite set of points. A crucial and non-trivial point is to establish an almost-grading replacing the honest grading in the Virasoro case. Such an almost-grading is given by splitting the set of points of possible poles into two non-empty disjoint subsets. Krichever and Novikov considered the twopoint case. Schlichenmaier studied the most general multi-point situation with arbitrary splittings. Here we will review the path of developments from the Virasoro algebra to its higher genus and multi-point analogs. The starting point will be a Poisson algebra structure on the space of meromorphic forms of all weights. As sub-structures the vector field algebras, function algebras, Lie superalgebras and the related current algebras show up. All these algebras will be almost-graded. In detail almost-graded central extensions are classified. In particular, for the vector field algebra it is essentially unique. The defining cocycle are given in geometric terms. Some applications, including the semi-infinite wedge form representations are recalled. Finally, some remarks on the by Krichever and Sheinman recently introduced Lax operator algebras are made.