scholarly journals Nonabelian holomorphic Lie algebroid extensions

2015 ◽  
Vol 26 (05) ◽  
pp. 1550040 ◽  
Author(s):  
Ugo Bruzzo ◽  
Igor Mencattini ◽  
Vladimir N. Rubtsov ◽  
Pietro Tortella
Keyword(s):  
Line Bundle ◽  
Spectral Sequence ◽  
Lie Algebras ◽  
Hodge Structure ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Serre Spectral Sequence ◽  
Atiyah Algebroid

We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild–Serre spectral sequence for Lie algebras to the holomorphic Lie algebroid setting. As an application, we show that the hypercohomology of the Atiyah algebroid of a line bundle has a natural Hodge structure.

scholarly journals Koszul complexes and spectral sequences associated with Lie algebroids

2020 ◽  
Author(s):  
Ugo Bruzzo ◽  
Vladimir N. Rubtsov
Keyword(s):  
Spectral Sequence ◽  
Complex Manifold ◽  
Global Section ◽  
Koszul Complex ◽  
Inner Product ◽  
Closed Field ◽  
Spectral Sequences ◽  
Lie Algebroid ◽  
Atiyah Algebroid ◽  
Regular Scheme

AbstractWe study some spectral sequences associated with a locally free $${{\mathscr {O}}}_X$$ O X -module $${{\mathscr {A}}}$$ A which has a Lie algebroid structure. Here X is either a complex manifold or a regular scheme over an algebraically closed field k. One spectral sequence can be associated with $${{\mathscr {A}}}$$ A by choosing a global section V of $${{\mathscr {A}}}$$ A , and considering a Koszul complex with a differential given by inner product by V. This spectral sequence is shown to degenerate at the second page by using Deligne’s degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid $${{{\mathscr {D}}}_{{{\mathscr {E}}}}}$$ D E of a holomolorphic vector bundle $${{\mathscr {E}}}$$ E on a complex manifold. If V is a differential operator on $${{\mathscr {E}}}$$ E with scalar symbol, i.e, a global section of $${{{\mathscr {D}}}_{{{\mathscr {E}}}}}$$ D E , we associate with the pair $$({{\mathscr {E}}},V)$$ ( E , V ) a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted ($${{\mathscr {E}}}=0$$ E = 0 ) case.

scholarly journals DIRAC STRUCTURES OF OMNI-LIE ALGEBROIDS

2011 ◽  
Vol 22 (08) ◽  
pp. 1163-1185 ◽  
Author(s):  
ZHUO CHEN ◽  
ZHANG JU LIU ◽  
YUNHE SHENG
Keyword(s):  
Lie Algebras ◽  
Cohomology Group ◽  
Adjoint Representation ◽  
Lie Algebroids ◽  
Reduction Theory ◽  
Dirac Structure ◽  
Lie Algebroid ◽  
Dirac Structures ◽  
Derivation Algebra ◽  
One To One

Omni-Lie algebroids are generalizations of Alan Weinstein's omni-Lie algebras. A Dirac structure in an omni-Lie algebroid 𝔇E ⊕ 𝔍E is necessarily a Lie algebroid together with a representation on E. We study the geometry underlying these Dirac structures in the light of reduction theory. In particular, we prove that there is a one-to-one correspondence between reducible Dirac structures and projective Lie algebroids in [Formula: see text]; we establish the relation between the normalizer NL of a reducible Dirac structure L and the derivation algebra Der (b (L)) of the projective Lie algebroid b(L); we study the cohomology group H •(L, ρL) and the relation between NL and H 1(L, ρL); we describe Lie bialgebroids using the adjoint representation; we study the deformation of a Dirac structure L, which is related with H 2(L, ρL).

EXT ALGEBRA OF NICHOLS ALGEBRAS OF CARTAN TYPE A2

2013 ◽  
Vol 12 (04) ◽  
pp. 1250191
Author(s):  
XIAOLAN YU ◽  
YINHUO ZHANG
Keyword(s):  
Spectral Sequence ◽  
Hopf Algebras ◽  
Type A ◽  
Serre Spectral Sequence ◽  
Nichols Algebra ◽  
Cartan Type ◽  
Dynkin Diagrams ◽  
Full Structure ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras

We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A2 by using the Hochschild–Serre spectral sequence. As an application, we show that the pointed Hopf algebras [Formula: see text] with Dynkin diagrams of type A, D, or E, except for A1 and A1 × A1 with the order NJ > 2 for at least one component J, are wild.

scholarly journals On local integration of Lie brackets

2020 ◽  
Vol 2020 (760) ◽  
pp. 267-293 ◽  
Author(s):  
Alejandro Cabrera ◽  
Ioan Mărcuţ ◽  
María Amelia Salazar
Keyword(s):  
Vector Field ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Lie Theory ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Local Integration ◽  
Lie Brackets ◽  
Complete Account

AbstractWe give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map. This construction leads to a complete account of local Lie theory and, in particular, to a finite-dimensional proof of the fact that the category of germs of local Lie groupoids is equivalent to that of Lie algebroids.

scholarly journals A LERAY SPECTRAL SEQUENCE FOR NONCOMMUTATIVE DIFFERENTIAL FIBRATIONS

2013 ◽  
Vol 10 (05) ◽  
pp. 1350015 ◽  
Author(s):  
EDWIN BEGGS ◽  
IBTISAM MASMALI
Keyword(s):  
Spectral Sequence ◽  
Sheaf Cohomology ◽  
Left Module ◽  
Graded Algebras ◽  
Serre Spectral Sequence ◽  
Differential Graded Algebras ◽  
Zero Curvature ◽  
Total Differential ◽  
Leray Spectral Sequence ◽  
Special Case

This paper describes the Leray spectral sequence associated to a differential fibration. The differential fibration is described by base and total differential graded algebras. The cohomology used is noncommutative differential sheaf cohomology. For this purpose, a sheaf over an algebra is a left module with zero curvature covariant derivative. As a special case, we can recover the Serre spectral sequence for a noncommutative fibration.

scholarly journals The Serre spectral sequence of a multiplicative fibration

2001 ◽  
Vol 353 (9) ◽  
pp. 3803-3831 ◽  
Author(s):  
Yves Félix ◽  
Stephen Halperin ◽  
Jean-Claude Thomas
Keyword(s):  
Spectral Sequence ◽  
Serre Spectral Sequence

scholarly journals LIE ALGEBROIDS AS SPACES

2018 ◽  
Vol 19 (2) ◽  
pp. 487-535 ◽  
Author(s):  
Ryan Grady ◽  
Owen Gwilliam
Keyword(s):  
Tangent Bundle ◽  
Symplectic Structure ◽  
Vector Bundles ◽  
Natural Generalization ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Faithful Functor ◽  
Derived Geometry

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

scholarly journals GEOMETRICAL MECHANICS ON ALGEBROIDS

2006 ◽  
Vol 03 (03) ◽  
pp. 559-575 ◽  
Author(s):  
KATARZYNA GRABOWSKA ◽  
PAWEŁ URBAŃSKI ◽  
JANUSZ GRABOWSKI
Keyword(s):  
Vector Bundle ◽  
Lie Algebroids ◽  
Legendre Transform ◽  
Analytical Mechanics ◽  
Lie Algebroid ◽  
Noether Theorem ◽  
Geometric Framework ◽  
Hamiltonian Functions

A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler–Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem.

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