We investigate the quantization problem of $(-1)$-shifted derived Poisson manifolds in terms $\BV_\infty$-operators on space Berezinian half-densities. prove that quantizing such a manifold is equivalent to lifting consecutive sequences Maurer-Cartan elements short exact differential graded Lie algebras, where obstruction certain class second cohomology. Consequently, quantizable if cohomology ...

We show that we can skip the skew-symmetry assumption in the definition of Nambu-Poisson brackets. In other words, a n-ary bracket on the algebra of smooth functions which satisfies the Leibniz rule and a n-ary version of the Jacobi identity must be skew-symmetric. A similar result holds for a non-antisymmetric version of Lie algebroids.

<p style='text-indent:20px;'>A theory of local convexity for a second order differential equation (${\text{sode}}$) on Lie algebroid is developed. The particular case when the ${\text{sode}}$ homogeneous quadratic extensively discussed.</p>

A symplectic Lie group is a with left-invariant form. Its algebra structure that of quasi-Frobenius algebra. In this note, we identify the groupoid analogue group. We call aforementioned \textit{$t$-symplectic groupoid}; $t$ motivated by fact each target fiber $t$-symplectic manifold. For $\mathcal{G}\rightrightarrows M$, show there one-to-one correspondence between algebroid structures on $A\m...

A number of issues in heterotic double field theory are studied. This includes the analysis of the T-dual configurations of a flat constant gauge flux background, which turn out to be non-geometric. Performing a field redefinition to a nongeometric frame, these T-duals take a very simple form reminiscent of the constant Qand R-flux backgrounds. In addition, it is shown how the analysis of arXiv...

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 i...

Associated with the canonical symplectic structure on a cotangent bundle T M is the diffeomorphism #: T (T M) −→ T (T M). This and the Tulczyjew diffeomorphism T (T M) −→ T (TM) may be derived from the canonical involution T (TM) −→ T (TM) by suitable dualizations. We show that the constructions which yield these maps extend very generally to the double Lie algebroids of double Lie groupoids, w...

We discuss actions of Lie n-groups and the corresponding action Lie n-groupoids; discuss actions of Lie n-algebras (L∞-algebras) and the corresponding action Lie n-algebroids; and discuss the relation between the two by integration and differentiation. As an example of interest, we discuss the BRST complex that appears in quantum field theory. We describe it as the Chevalley-Eilenberg algebra o...