arXiv subjects
Publications and source records attributed to Michiel Smet.
The Frank Lie algebras are simple Lie algebras that only occur over fields of characteristic 3. These come equipped with distinguished inner derivations that make them algebras in the category $\textbf{Rep}(\alpha_3)$. We apply the semisimplification functor to these Frank Lie algebras and obtain modular contact Lie superalgebras. We also obtain a class of simple $J$-ternary algebras whose associated Jordan algebras are not simple.
We describe a class of Lie superalgebras in characteristic $3$, containing the Elduque-Cunha superalgebras $\mathfrak{g}(3,3), \mathfrak{g}(6,6)$ and the Elduque superalgebra $\mathfrak{el}(5,3)$, using the tensor product of composition algebras. For the Lie superalgebra $\mathfrak{el}(5,3)$, this allows us to move beyond the contragredient construction and it also allows us to construct more general forms. We also describe how one obtains these Lie superalgebras using the semisimplification functor on the representation category $\mathbf{Rep}(\alpha_3)$ to Lie algebras of type $E_6, E_7$ and $E_8$, in line with how Arun Kannan applied this functor to the split algebras. We further apply this functor more broadly to the class of Lie algebras coming from $J$-ternary algebras over fields of characteristic $3$.
Hermitian cubic norm structures were recently introduced in order to study the class of skew-dimension one structurable algebras (which are typically only defined over fields of characteristic different from $2$ and $3$) over arbitrary rings and fields. Here, we generalize the quartic norm for these algebras and show that elements for which the quartic norm is invertible are conjugate invertible. This leads to the notion of a division hermitian cubic norm structure, defined as one in which the quartic norm is anisotropic. We classify such structures in terms of the Tits index of an associated (rank one) adjoint simple linear algebraic group and show that any adjoint linear algebraic group with such a Tits index defines a corresponding hermitian cubic norm structure.
We generalize cubic norm structures to cubic norm pairs and extend hermitian cubic norm structures to arbitrary commutative unital rings. For the associated ``skew dimension one structurable algebra" of these pairs, we construct a corresponding Lie algebra and a group of automorphisms of the Lie algebra.
Kantor pairs, (quadratic) Jordan pairs, and similar structures have been instrumental in the study of $\mathbb{Z}$-graded Lie algebras and algebraic groups. We introduce the notion of an operator Kantor pair, a generalization of Kantor pairs to arbitrary (commutative, unital) rings, similar in spirit as to how quadratic Jordan pairs and algebras generalize linear Jordan pairs and algebras. Such an operator Kantor pair is formed by a pair of $\Phi$-groups $(G^+,G^-)$ of a specific kind, equipped with certain homogeneous operators. For each such a pair $(G^+,G^-)$, we construct a $5$-graded Lie algebra $L$ together with actions of $G^\pm$ on $L$ as automorphisms. Moreover, we can associate a group $G(G^+,G^-) \subset \operatorname{Aut}(L)$ to this pair generalizing the projective elementary group of Jordan pairs. If the non-$0$-graded part of $L$ is projective, we can uniquely recover $G^+,G^-$ from $G(G^+,G^-)$ and the grading on $L$ alone. We establish, over rings $\Phi$ with $1/30 \in \Phi$, a one to one correspondence between Kantor pairs and operator Kantor pairs. Finally, we construct operator Kantor pairs for the different families of central simple structurable algebras.