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𝒟 -bundles and integrable hierarchies

David Ben-Zvi, Thomas Nevins (2011)

Journal of the European Mathematical Society

We study the geometry of 𝒟 -bundles—locally projective 𝒟 -modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev–Petviashvili (KP) and spin Calogero–Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of 𝒟 -bundles; in particular, we prove that the local structure of 𝒟 -bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP...

-graded Lie superalgebras of infinite depth and finite growth

Nicoletta Cantarini (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In 1998 Victor Kac classified infinite-dimensional -graded Lie superalgebras of finite depth. We construct new examples of infinite-dimensional Lie superalgebras with a -gradation of infinite depth and finite growth and classify -graded Lie superalgebras of infinite depth and finite growth under suitable hypotheses.

-homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si studiano gli omomorfismi reticolari ( -omomorfismi) di algebre di Lie sopra anelli commutativi con unità. Le algebre di Lie sopra un campo e le p -algebre di Lie non ammettono -omomorfismi propri. Si assegnano condizioni necessarie e sufficienti affinchè un'algebra di Lie periodica o mista possieda un « -omomorfismo su una catena di lunghezza n .

1 -cocycles on the group of contactomorphisms on the supercircle S 1 | 3 generalizing the Schwarzian derivative

Boujemaa Agrebaoui, Raja Hattab (2016)

Czechoslovak Mathematical Journal

The relative cohomology H diff 1 ( 𝕂 ( 1 | 3 ) , 𝔬𝔰𝔭 ( 2 , 3 ) ; 𝒟 λ , μ ( S 1 | 3 ) ) of the contact Lie superalgebra 𝕂 ( 1 | 3 ) with coefficients in the space of differential operators 𝒟 λ , μ ( S 1 | 3 ) acting on tensor densities on S 1 | 3 , is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating 1 -cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative 1 -cocycle s ( X f ) = D 1 D 2 D 3 ( f ) α 3 1 / 2 , X f 𝕂 ( 1 | 3 ) which is invariant with respect to the conformal subsuperalgebra 𝔬𝔰𝔭 ( 2 , 3 ) of 𝕂 ( 1 | 3 ) . In this work we study the supergroup case. We give an explicit construction of 1 -cocycles of the group...

A 2-category of chronological cobordisms and odd Khovanov homology

Krzysztof K. Putyra (2014)

Banach Center Publications

We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms along any 3-dimensional subcube is trivial. This allows us to create a chain complex whose homotopy type modulo certain relations is a link invariant. Both the original and the odd Khovanov...

A 4 3 -grading on a 56 -dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type E

Diego Aranda-Orna, Alberto Elduque, Mikhail Kochetov (2014)

Commentationes Mathematicae Universitatis Carolinae

We describe two constructions of a certain 4 3 -grading on the so-called Brown algebra (a simple structurable algebra of dimension 56 and skew-dimension 1 ) over an algebraically closed field of characteristic different from 2 . The Weyl group of this grading is computed. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types E 6 , E 7 and E 8 .

A characterization of coboundary Poisson Lie groups and Hopf algebras

Stanisław Zakrzewski (1997)

Banach Center Publications

We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known π + ). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the π + structure on SU(N) is described in terms of generators and relations as an example.

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