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).

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 Dirac Structures on Banach Lie Algebroids

2014 ◽  
Vol 22 (3) ◽  
pp. 219-228
Author(s):  
Vlad-Augustin Vulcu
Keyword(s):  
Vector Bundle ◽  
Tangent Bundle ◽  
Differential Calculus ◽  
Vector Bundles ◽  
Dirac Structure ◽  
Lie Algebroid ◽  
Symmetric Form ◽  
Courant Bracket ◽  
Dirac Structures ◽  
Original Definition

Abstract In the original definition due to A. Weinstein and T. Courant a Dirac structure is a subbundle of the big tangent bundle T M ⊕ T* M that is equal to its ortho-complement with respect to the so-called neutral metric on the big tangent bundle. In this paper, instead of the big tangent bundle we consider the vector bundle E ⊕ E*, where E is a Banach Lie algebroid and E* its dual. Recall that E* is not in general a Lie algebroid. We define a bilinear and symmetric form on the vector bundle E ⊕ E* and say that a subbundle of it is a Dirac structure if it is equal with its orthocomplement. Our main result is that any Dirac structure that is closed with respect to a type of Courant bracket, endowed with a natural anchor is a Lie algebroid. In the proof the differential calculus on a Lie algebroid is essentially involved. We work in the category of Banach vector bundles.

scholarly journals Gradings of non-graded Hamiltonian Lie algebras

2005 ◽  
Vol 79 (3) ◽  
pp. 399-440 ◽  
Author(s):  
A. Caranti ◽  
S. Mattarei
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Prime Characteristic ◽  
Infinite Dimensional ◽  
Derivation Algebra ◽  
Dimension One ◽  
Positive Integers ◽  
Block Algebra

AbstractA thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2: n; ω2) (of dimension one less than a power of p) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2: n; ω2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).

DERIVATIONS OF THE EVEN PARTS FOR MODULAR LIE SUPERALGEBRAS OF CARTAN TYPE W AND S

2007 ◽  
Vol 17 (04) ◽  
pp. 661-714 ◽  
Author(s):  
WENDE LIU ◽  
YONGZHENG ZHANG
Keyword(s):  
Lie Algebras ◽  
Lie Superalgebra ◽  
Adjoint Representation ◽  
Lie Superalgebras ◽  
Base Field ◽  
Characteristic P ◽  
Derivation Algebra ◽  
Finite Dimensional ◽  
Outer Derivation ◽  
Generator Sets

Let 𝔽 be the underlying base field of characteristic p < 3 and denote by [Formula: see text] and [Formula: see text] the even parts of the finite-dimensional generalized Witt Lie superalgebra W and the special Lie superalgebra S, respectively. We first give the generator sets of the Lie algebras [Formula: see text] and [Formula: see text]. Using certain properties of the canonical tori of [Formula: see text] and [Formula: see text], we then determine the derivation algebra of [Formula: see text] and the derivation space of [Formula: see text] to [Formula: see text], where [Formula: see text] is viewed as a [Formula: see text]-module by means of the adjoint representation. As a result, we describe explicitly the derivation algebra of [Formula: see text]. Furthermore, we prove that the outer derivation algebras of [Formula: see text] and [Formula: see text] are abelian Lie algebras or metabelian Lie algebras with explicit structure. In particular, we give the dimension formulas of the derivation algebras and outer derivation algebras of [Formula: see text] and [Formula: see text]. Thus, we may make a comparison between the even parts of the (outer) superderivation algebras of W and S and the (outer) derivation algebras of the even parts of W and S, respectively.

Derivation algebra of semi-direct sum of Lie algebras

2021 ◽  
pp. 2250063
Author(s):  
Mohammad Reza Alemi ◽  
Farshid Saeedi
Keyword(s):  
Lie Algebras ◽  
Arbitrary Field ◽  
Direct Sum ◽  
Derivation Algebra

Let [Formula: see text] and [Formula: see text] be two Lie algebras over an arbitrary field [Formula: see text], and let [Formula: see text] be the semidirect sum of [Formula: see text] by [Formula: see text]. In this paper, we give the structure of derivation algebra of [Formula: see text]; then as a consequence we illustrate the structure and dimension derivation algebra of Heisenberg Lie algebras.

Derivations and Automorphism Group of Completed Witt Lie Algebra

2012 ◽  
Vol 19 (03) ◽  
pp. 581-590 ◽  
Author(s):  
Yongping Wu ◽  
Ying Xu ◽  
Lamei Yuan
Keyword(s):  
Finite Element ◽  
Automorphism Group ◽  
Lie Algebra ◽  
Cohomology Group ◽  
Locally Finite ◽  
Simple Lie Algebra ◽  
Derivation Algebra ◽  
First Cohomology Group

In this paper, a simple Lie algebra, referred to as the completed Witt Lie algebra, is introduced. Its derivation algebra and automorphism group are completely described. As a by-product, it is obtained that the first cohomology group of this Lie algebra with coefficients in its adjoint module is trivial. Furthermore, we completely determine the conjugate classes of this Lie algebra under its automorphism group, and also obtain that this Lie algebra does not contain any nonzero ad -locally finite element.

scholarly journals Dirac structures on Hilbert spaces

1999 ◽  
Vol 22 (1) ◽  
pp. 97-108 ◽  
Author(s):  
A. Parsian ◽  
A. Shafei Deh Abad
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Smooth Manifold ◽  
Real Hilbert Space ◽  
Dirac Structure ◽  
Dirac Structures ◽  
Basic Properties ◽  
Hilbert Subspace ◽  
Densely Defined

For a real Hilbert space(H,〈,〉), a subspaceL⊂H⊕His said to be a Dirac structure onHif it is maximally isotropic with respect to the pairing〈(x,y),(x′,y′)〉+=(1/2)(〈x,y′〉+〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures onHare in one-to-one correspondence with isometries onH, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structureLonH, everyz∈His uniquely decomposed asz=p1(l)+p2(l)for somel∈L, wherep1andp2are projections. Whenp1(L)is closed, for any Hilbert subspaceW⊂H, an induced Dirac structure onWis introduced. The latter concept has also been generalized.

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 Derivation algebra of direct sum of lie algebras

2019 ◽  
Vol 6 (1) ◽  
pp. 1624244
Author(s):  
Mohammad Reza Alemi ◽  
Farshid Saeedi ◽  
Hari M. Srivastava
Keyword(s):  
Lie Algebras ◽  
Direct Sum ◽  
Derivation Algebra

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.

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