The authors prove that a local n-quasigroup defined by the equation $x_{n+1}=F(x₁,...,xₙ)=(f₁\left(x₁\right)+...+fₙ\left(xₙ\right))/(x₁+...+xₙ)$, where $f_i\left(x_i\right)$, i,j = 1,...,n, are arbitrary functions, is irreducible if and only if any two functions $f_i\left(x_i\right)$ and $f_j\left(x_j\right)$, i ≠ j, are not both linear homogeneous, or these functions are linear homogeneous but $f_i\left(x_i\right)/x_i\ne f_j\left(x_j\right)/x_j$. This gives a solution of Belousov’s problem to construct examples of irreducible n-quasigroups for any n ≥ 3.

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

We introduce the modular class of a Poisson map. We look at several examples and we use the modular classes of Poisson maps to study the behavior of the modular class of a Poisson manifold under different kinds of reduction. We also discuss their symplectic groupoid version, which lives in groupoid cohomology.

It is shown that the Lagrange's equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of Lagrangian reduction and the relation with the method of Lagrange multipliers are also studied.