In this paper, Jordan-Kronecker invariants are calculated for all nilpotent 6-and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space g proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A − λB)(x), we consider Jordan-Kronecker invariants for this pencil and how they change with x (the latter is done for 6-dimensional Lie algebras).

No abstract

An ultrafilter p on ω is said to be discrete if, given any function f : ω → X to any completely regular Hausdorff space, there is an A ∈ p such that f (A) is discrete. Basic properties of discrete ultrafilters are studied. Three intermediate classes of spaces R1 ⊂ R2 ⊂ R3 between the class of F -spaces and the class of van Douwen's βωspaces are introduced. It is proved that no product of infinite compact R2-spaces is homogeneous; moreover, under the assumption d = c, no product of βω-spaces is homogeneous.In [1] Frolík proved the nonhomogeneity of the Stone-Čech remainder ω * = βω \ ω of ω by noticing that if two discrete sequences in ω * converge to the same point x ∈ ω * along ultrafilters p and q, then p and q are compatible in the Rudin-Frolík order. The idea is quite natural: if p, q ∈ ω * are incompatible and D = {d n : n ∈ ω} is a countable discrete subset of ω * , then there cannot exist a homeomorphism h : ω * → ω * taking q-lim n d n to p-lim n d n , because if it existed, then (d n ) n∈ω and (h(d n )) n∈ω would be discrete sequences converging to the same point p-lim n d n along p and q, respectively.Frolík's idea of proving nonhomogeneity by considering orderings of ultrafilters was developed by Kunen. One of the key ingredients in his proof of the inhomogeneity of any product of infinite compact F -spaces is the following lemma on the Rudin-Keisler comparability of ultrafilters.Kunen's lemma ([2, Lemma 4]). Let p, q ∈ ω * be Rudin-Keisler incomparable weak P -points, and let X be any compact F -space. Suppose that x ∈ X, (d m ) m∈ω is a discrete sequence of distinct points in X, (e n ) n∈ω is any sequence of points in X, and x = p-lim m d m = q-lim n e n . Then {n : e n = x} ∈ q.The present paper arose from an attempt to extend Kunen's lemma and, thereby, his result on the inhomogeneity of product spaces to other classes of spaces. This can be done by looking for larger classes for which Kunen's argument still works or by strengthening the assumptions on the ultrafilters p and q. In [3] we introduced new classes R 1 , R 2 , and R 3 of topological spaces, which lie strictly between the classes of F -and βω-spaces, and proved that

No abstract

No abstract