Nonlinear flight physics of the Lie Bracket roll mechanism

In this paper, we review the concept of Lie brackets and how it can be exploited in generating motion in unactuated directions through nonlinear interactions between two or more control inputs. Applying this technique to the airplane flight dynamics near stall, a new rolling mechanism is discovered through nonlinear interactions between the elevator and the aileron control inputs. This mechanism, referred to as the Lie Bracket Roll Augmentation (LIBRA) mechanism, possesses a significantly higher roll control authority near stall compared to the conventional roll mechanism using ailerons only; it produces more than an order-of-magnitude stronger roll motion over the first second. The main contribution of this paper is to study the nonlinear flight physics that lead to this superior performance of the LIBRA mechanism. In fact, the LIBRA performance in free flight (six DOF) is double that in a confined environment of two-DOF roll-pitch dynamics. The natural feedback from the airplane motion (roll, yaw, and sideslip) into the LIBRA mechanism boosts its performance through interesting nonlinear interplay between roll and yaw, while exploiting some of the changes in the airplane characteristics near stall.

Generalized Ricci flow on nilpotent Lie groups

We define solitons for the generalized Ricci flow on an exact Courant algebroid. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.

Graded Medial n-Ary Algebras and Polyadic Tensor Categories

Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or ε-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with n−1 associators of the arity 2n−1 satisfying a n2+1-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.

Local and global integrability of Lie brackets

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

From Schouten to Mackenzie: Notes on brackets

<p style='text-indent:20px;'>In this paper, dedicated to the memory of Kirill Mackenzie, I relate the origins and early development of the theory of graded Lie brackets, first in the publications on differential geometry of Schouten, Nijenhuis, and Frölicher–Nijenhuis, then in the work of Gerstenhaber and Nijenhuis–Richardson in cohomology theory.</p>

Periodic Staircase Matrices and Generalized Cluster Structures

As is well known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plücker relations, Desnanot–Jacobi identities, and their generalizations. We present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in $GL_n$ compatible with a certain subclass of Belavin–Drinfeld Poisson–Lie brackets, in the Drinfeld double of $GL_n$, and in spaces of periodic difference operators.