222 research outputs found
Octonion algebras over rings are not determined by their norms
Answering a question of H. Petersson, we provide a class of examples of pair
of octonion algebras over a ring having isometric norms.Comment: 8 page
A lower bound on the essential dimension of a connected linear group
Let G be a connected linear algebraic group defined over an algebraically
closed field k and H be a finite abelian subgroup of G whose order is prime to
char(k). We show that the essential dimension of G is bounded from below by
rank(H) - rank C_G(H)^0, where rank C_G(H)^0 denotes the rank of the maximal
torus in the centralizer C_G(H). This inequality, conjectured by J.-P. Serre,
generalizes previous results of Reichstein -- Youssin (where char(k) is assumed
to be 0 and C_G(H) to be finite) and Chernousov -- Serre (where H is assumed to
be a 2-group).Comment: 21 page
Three-point Lie algebras and Grothendieck's dessins d'enfants
We define and classify the analogues of the affine Kac-Moody Lie algebras for
the ring corresponding to the complex projective line minus three points. The
classification is given in terms of Grothendieck's dessins d'enfants. We also
study the question of conjugacy of Cartan subalgebras for these algebras.Comment: 16 page
RĂ©sidus sur les grassmanniennes affines
Given a linear group G over a field k, we define a notion of index and residue of an element g of G(k((t)). This provides an alternative proof of Gabber's theorem stating that G has no subgroups isomorphic to the additive or the commutative group iff G(k[[t]])= G(k((t))). In the case of a reductive group, we offer an explicit connection with the theory of affine grassmannians
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