Skip Garibaldi’s research while affiliated with Center for Communications Research - Princeton and other places

Albert algebras provide key tools for understanding exceptional groups and related structures such as symmetric spaces. This self-contained book provides the first comprehensive reference on Albert algebras over fields without any restrictions on the characteristic of the field. As well as covering results in characteristic 2 and 3, many results are proven for Albert algebras over an arbitrary commutative ring, showing that they hold in this greater generality. The book extensively covers requisite knowledge, such as non-associative algebras over commutative rings, scalar extensions, projective modules, alternative algebras, and composition algebras over commutative rings, with a special focus on octonion algebras. It then goes into Jordan algebras, Lie algebras, and group schemes, providing exercises so readers can apply concepts. This centralized resource illuminates the interplay between results that use only the structure of Albert algebras and those that employ theorems about group schemes, and is ideal for mathematics and physics researchers.

This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024.

Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative nonassociative algebras and also arise naturally in the context of simple affine group schemes of type $\mathsf {F}_4$ , $\mathsf {E}_6$ , or $\mathsf {E}_7$ . We study these objects over an arbitrary base ring R , with particular attention to the case $R = \mathbb {Z}$ . We prove in this generality results previously in the literature in the special case where R is a field of characteristic different from 2 and 3.

Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative non-associative algebras and also arise naturally in the context of simple affine group schemes of type $F_4$, $E_6$, or $E_7$. We study these objects over an arbitrary base ring R, with particular attention to the case of the integers. We prove in this generality results previously in the literature in the special case where R is a field of characteristic different from 2 and 3.

We prove a myriad of results related to the stabilizer in an algebraic group G of a generic vector in a representation V of G over an algebraically closed field k. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of G and the group G(k) of k-points. For G simple and V faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those G and V for which the stabilizer in general position is smooth, or $\dim V/G < \dim G$, or there is a $v \in V$ whose stabilizer in G is trivial.

We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type $E_8$, the algebra A is obtained by adjoining a unit to the 3875-dimensional representation; and (2) it is effective, in that the product operation on A can be implemented on a computer. A description of the algebra in the $E_8$ case has been requested for some time, and interest has been increased by the recent proof that $E_8$ is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum.

We study embeddings J→G of simple linear algebraic groups with the following property: the simple components of the J module Lie(G)∕Lie(J) are all minuscule representations of J. One family of examples occurs when the group G has roots of two different lengths and J is the subgroup generated by the long roots. We classify all such embeddings when J=SL2 and J=SL3, show how each embedding implies the existence of exceptional algebraic structures on the graded components of Lie(G), and relate properties of those structures to the existence of various twisted forms of G with certain relative root systems.

We study embeddings $J \rightarrow G$ of simple linear algebraic groups with the following property: the simple components of the J module Lie(G)/Lie(J) are all minuscule representations of J. One family of examples occurs when the group G has roots of two different lengths and J is the subgroup generated by the long roots. We classify all such embeddings when $J = SL_2$ and $J = SL_3$, show how each embedding implies the existence of exceptional algebraic structures on the graded components of Lie(G), and relate properties of those structures to the existence of various twisted forms of G with certain relative root systems.

In parts I and II, we determined which irreducible representations V of a simple linear algebraic group G are generically free for Lie(G), i.e., which V have an open subset consisting of vectors whose stabilizer in Lie(G) is zero, with some assumptions on the characteristic of the field. This paper settles the remaining cases, which are of a different nature because Lie(G) has a more complicated structure and there need not exist general dimension bounds of the sort that exist in good characteristic. In a later work, we combine these results with those of Guralnick--Lawther--Liebeck to show that for any irreducible module for a simple algebraic group, there is a generic stabilizer (as a group scheme) and gives a classification of the generic stabilizers in all cases.