arXiv subjects
Publications and source records attributed to R. Skip Garibaldi.
The Rost invariant associated with a simple simply connected algebraic group G is used to define an invariant of strongly inner forms of G. This invariant takes values in a quotient of H^3(k, Q/Z(2)). It is used to prove a generalization of Gille's splitting criterion for groups of type E_6 and E_7.
Let BG be a classifying variety for an exceptional simple simply connected algebraic group G. We compute the degree 3 unramified Galois cohomology of BG with values in Q/Z(2) over an arbitrary field F. Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for G semisimple simply connected over all fields. These computations provide another example of a simple simply connected group G such that BG is not stably rational.
The typical definition of the characteristic polynomial seems totally ad hoc to me. This note gives a canonical construction of the characteristic polynomial as the minimal polynomial of a "generic" matrix. This approach works not just for matrices but also for a very broad class of algebras including the quaternions, all central simple algebras, and Jordan algebras. The main idea of this paper dates back to the late 1800s. (In particular, it is not due to the author.) This note is intended for a broad audience; the only background required is one year of graduate algebra.
For G an almost simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map R_G: H^1(F, G) --> H^3(F, Q/Z(2)). This includes the Arason invariant for quadratic forms and Rost's mod 3 invariant for Albert algebras as special cases. We show that R_G has trivial kernel if G is quasi-split of type E_6 or E_7. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank.
Freudenthal triple systems come in two flavors, degenerate and nondegenerate. The best criterion for distinguishing between the two which is available in the literature is by descent. We provide an identity which is satisfied only by nondegenerate triple systems. We then use this to define algebraic structures whose automorphism groups produce all adjoint algebraic groups of type E_7 over an arbitrary field of characteristic not 2 or 3. The main advantage of these new structures is that they incorporate a previously unconsidered invariant (a symplectic involution) of these groups in a fundamental way. As an application, we give a construction of adjoint groups with Tits algebras of index 2 which provides a complete description of this involution and apply this to groups of type E_7 over a real-closed field.
It is well-known that every algebraic group of type F_4 is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type ^1E_6 with trivial Tits algebras arise as the isometry groups of norm forms of such Jordan algebras. We describe a similar relationship between groups of type E_6 and groups of type E_7 and use it to give explicit descriptions of the homogeneous projective varieties associated to groups of type E_7 with trivial Tits algebras. The underlying algebraic structure for the relationship considered here are a sort of 56-dimensional structurable algebra which are forms of an algebra constructed from an exceptional Jordan algebra.