Symmetry groups of Lie algebras and superalgebras constructed from (∈, δ)-Freudenthal-Kantor triple systems have been studied. In particular, for a special (ε, ε)-Freudenthal–Kantor triple, it is the SL(2) group. Also, the relationship between two constructions of Lie algebras from structurable algebras has been investigated.

We construct, for all $d\,\ge \,4$, a cellulation of ${{\mathbb{S}}^{d-1}}$. We prove that these cellulations cannot be polytopal with maximal combinatorial symmetry. Such non-realizability phenomenon was first described in dimension 4 by Bokowski, Ewald and Kleinschmidt, and, to the knowledge of the author, until now there have not been any known examples in higher dimensions. As a starting point for the construction, we introduce a new class of (Wythoffian) uniform polytopes, which we call duplexes. In proving our main result, we use some tools that we developed earlier while studying perfect polytopes. In particular, we prove perfectness of the duplexes; furthermore, we prove and make use of the perfectness of another new class of polytopes which we obtain by a variant of the so-called $E$-construction introduced by Eppstein, Kuperberg and Ziegler.

Let G be a real reductive Lie group, K its compact subgroup. Let A be the algebra of G-invariant real-analytic functions on T*(G/K) (with respect to the Poisson bracket) and let C be the center of A. Denote by 2ε(G,K) the maximal number of functionally independent functions from A\C. We prove that ε(G,K) is equal to the codimension δ(G,K) of maximal dimension orbits of the Borel subgroup B⊂G$^{\Bbb C}$ in the complex algebraic variety G$^{\Bbb C}$/K$^{\Bbb C}$. Moreover, if δ(G,K)=1, then all G-invariant Hamiltonian systems on T*(G/K) are integrable in the class of the integrals generated by the symmetry group G. We also discuss related questions in the geometry of the Borel group action.

This paper is devoted to the complete classification of integrable one-component evolution equations whose field variable takes its values in an associative algebra. The proof that the list of non-commutative integrable homogeneous evolution equations is complete relies on the symbolic method. Each equation in the list has infinitely many local symmetries and these can be generated by its recursion (recursive) operator or master symmetry. 1991 Mathematics Subject Classification: 13A50, 16-XX, 22E70, 35A30, 35Q53, 58F07.