We study the family of vertex algebras associated with vertex algebroids, constructed by Gorbounov, Malikov, and Schechtman. As the main result, we classify all the (graded) simple modules for such vertex algebras and we show that the equivalence classes of graded simple modules one-to-one correspond to the equivalence classes of simple modules for the Lie algebroids associated with the vertex ...

a unital $c^*$ -- algebra $mathcal a,$ endowed withthe lie product $[x,y]=xy- yx$ on $mathcal a,$ is called a lie$c^*$ -- algebra. let $mathcal a$ be a lie $c^*$ -- algebra and$g,h:mathcal a to mathcal a$ be $bbb c$ -- linear mappings. a$bbb c$ -- linear mapping $f:mathcal a to mathcal a$ is calleda lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all ...

<p style='text-indent:20px;'>We survey recent results on the local and global integrability of a Lie algebroid, as well infinitesimal multiplicative geometric structures it.</p>

We introduce the notion of -algebroid, generalising both Lie and Courant algebroids, as well algebroids used in exceptional generalised geometry for . Focusing on case, we prove a classification “exact” translate related Leibniz parallelisable spaces into tractable algebraic problem. After discussing general Poisson–Lie duality, show that U-duality is compatible with equations motion supergravi...

We give an integrability criterion for a projective limit of Banach distributions on Fr\'echet manifold which is manifolds. This leads to result involutive bundles sequence can be seen as version Frobenius Theorem in setting. As consequence, we obtain the third Lie theorem Fr\'echet-Lie group submersive groups. also application prolongations algebroid.

In order to obtain a framework in which both non-holonomic mechanical systems and non-holonomic mechanical systems with symmetry can be described, we introduce in this paper the notion of a Lagrangian system on a subbundle of a Lie algebroid.

A double field theory algebroid (DFT algebroid) is a special case of the metric (or Vaisman) algebroid, shown to be relevant in understanding symmetries theory. In particular, DFT structure defined on vector bundle over doubled spacetime equipped with C-bracket this paper, we give definition as curved L∞-algebra and show how implementation strong constraint can formulated an morphism. Our resul...

Main ideas of the differential geometry on affine bundles are presented. Affine counterparts of Lie algebroid and Poisson structures are introduced and discussed. The developed concepts are applied in a frame-independent formulation of the time-dependent and the Newtonian mechanics. MSC (2000): 37J05, 53A40, 53D99, 55R10, 70H99.