381 research outputs found
Albert algebras over curves of genus zero and one
Albert algebras and other Jordan algebras are constructed over curves of
genus zero and one, using a generalization of the Tits process and the first
Tits construction due to Achhammer.Comment: 37 page
Conjugacy classes of trialitarian automorphisms and symmetric compositions
The trialitarian automorphisms considered in this paper are the outer
automorphisms of order 3 of adjoint classical groups of type D_4 over arbitrary
fields. A one-to-one correspondence is established between their conjugacy
classes and similarity classes of symmetric compositions on 8-dimensional
quadratic spaces. Using the known classification of symmetric compositions, we
distinguish two conjugacy classes of trialitarian automorphisms over
algebraically closed fields. For type I, the group of fixed points is of type
G_2, whereas it is of type A_2 for trialitarian automorphisms of type II
Trialitarian automorphisms of lie algebras
Over an algebraically closed field of characteristic zero simple Lie algebras admit outer automorphisms of order 3 if and only if they are of type D4. Moreover, thereare two conjugacy classes of such automorphisms. Among orthogonal Lie algebras over arbitrary fields of characteristic zero, only orthogonal Lie algebras relative to quadratic norm forms of Cayley algebras admit outer automorphisms of order 3. We give a complete list of conjugacy classes of outer automorphisms of order 3 for orthogonal Lie algebras over arbitrary fields of characteristic zero. For the norm form of a given Cayley algebra, one class is associated with the Cayley algebra and the others with central simple algebras of degree 3 with involution of the second kind such that the cohomological invariant of the involution is the norm for
Surjectivity of the total Clifford invariant and Brauer dimension
Merkurjev's theorem--the statement that the 2-torsion of the Brauer group is
represented by Clifford algebras of quadratic forms--is in general false when
the base is no longer a field. The work of Parimala, Scharlau, and Sridharan
proves the existence of smooth complete curves over local fields, over which
Merkurjev's theorem is equivalent to the existence of a rational theta
characteristic. Here, we prove that for smooth curves over a local field or
surfaces over a finite field, replacing the Witt group by the total Witt group
of all line bundle-valued quadratic forms recovers Merkurjev's theorem: the
2-torsion of the Brauer group is always represented by even Clifford algebras
of line bundle-valued quadratic forms.Comment: Final published versio
Vector bundles of rank four and A_3 = D_3
Over a scheme with 2 invertible, we show that a vector bundle of rank four
has a sub or quotient line bundle if and only if the canonical symmetric
bilinear form on its exterior square has a lagrangian subspace. For this, we
exploit a version of "Pascal's rule" for vector bundles that provides an
explicit isomorphism between the moduli functors represented by projective
homogeneous bundles for reductive group schemes of type A_3 and D_3. Under
additional hypotheses on the scheme (e.g. proper over a field), we show that
the existence of sub or quotient line bundles of a rank four vector bundle is
equivalent to the vanishing of its Witt-theoretic Euler class.Comment: 16 pages, final version; IMRN 2012 rns14
Beno Eckmann 1917-2008
Am 25. November 2008 verstarb Beno Eckmann in seinem 92. Lebensjahr. Dieser Nachruf beleuchtet Leben und Werk dieses bedeutenden Vertreters der Algebra und der Topologi
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