Abstract. Let G be an algebraic group and let X be a generically free G-variety. We show that X can be transformed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety X ′ with the following property: the stabilizer of every point of X ′ is isomorphic to a semidirect product U> ⊳ A of a unipotent group U and a diagonalizable group A. As an application of this and related results, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus

This paper surveys the classi cation of integrable evolution equations whose eld variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to associative algebra-valued version of the Painleve transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the bi-Hamiltonian structures for several examples are found.

We discuss a possibility of the extension of a primal-dual interior-point algorithm suggested recently in [1]. We consider optimization problems defined on the intersection of a symmetric cone and an affine subspace. The question of solvability of a linear system arising in the implementation of the primal-dual algorithm is analyzed. A nondegeneracy theory for the considered class of problems is developed. The Jordan algebra technique suggested in [5] plays major role in the present paper. 1 Introduction Recently F.Alizadeh, J.-P. Haeberly and M. Overton suggested a primal-dual interior-point algorithm for solving semidefinite problems [1] that shows extremely good convergence properties and a high degree of accuracy [2]. In the present paper we discuss a possibility of an extension of this algorithm to a broader class of optimization problems defined on the intersection of an affine subspace with a symmetric (i.e. self-dual, homogeneous ) cone. This class of problems includes linear ...

We study the “conformal groups ” of Jordan algebras along the lines suggested by Kantor. They provide a natural generalization of the concept of conformal transformations that leave 2-angles invariant to spaces where “p-angles ” (p ≥ 2) can be defined. We give an oscillator realization of the generalized conformal groups of Jordan algebras and Jordan triple systems. A complete list of the generalized conformal algebras of simple Jordan algebras and hermitian Jordan triple systems is given. These results are then extended to Jordan superalgebras and super Jordan triple systems. By going to a coordinate representation of the (super)oscillators one then obtains the differential operators representing the action of these generalized (super) conformal groups on the corresponding (super) spaces. The superconformal algebras of the Jordan superalgebras in Kac’s classification is also presented.

Abstract. For G an almost simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map RG: 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 RG has trivial kernel if G is quasi-split of type E6 or E7. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank. For G an almost simple simply connected algebraic group over a field F, the set of all natural transformations of functors H 1 (?, G) − → H 3 (?, Q/Z(2)) is a finite cyclic group [KMRT98, §31] with a canonical generator. (Here Hi (?, M) is the Galois cohomology functor which takes a field extension of your base field F and returns a group if M is abelian and a pointed set otherwise. When F has characteristic zero, Q/Z(2) is defined to be lim → µ ⊗2 n for µ n the algebraic groups of nth roots of unity; see [KMRT98, p. 431] or [Gilb, I.1(b)] for a more complete definition.) This generator is called the Rost

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.

Let F be a local field, ψ a nontrivial unitary additive character of F, and V a finite dimensional vector space over F. Let us say that a complex function on V is elementary if it has the form k∏ g(x) = Cψ(Q(x)) χj(Pj(x)), x ∈ V, j=1 where C ∈ C, Q a rational function (the phase function), Pj are polynomials, and χj multiplicative characters of F. For generic χj, this function canonically extends to a distribution on V (if char(F)=0). Occasionally, the Fourier transform of an elementary function is also an elementary function (the basic example is the Gaussian integral: k = 0, Q is a nondegenerate quadratic form). It is interesting to determine when exactly this happens. This question is the main subject of our study. In the first part of this paper, we show that for F = R or C, if the Fourier transform of an elementary function g ̸ = 0 with phase function −Q such that detd 2 Q ̸ = 0 is another elementary function g ∗ with phase function Q ∗ , then Q ∗ is the Legendre transform of Q (the “semiclassical condition”). We study properties and examples of phase functions satisfying this condition, and give a classification of phase functions such that both Q and Q ∗ are of the form f(x)/t, where f is a homogeneous cubic polynomial and t is an additional variable (this is one of the simplest possible situations). Unexpectedly, the proof uses Zak’s classification theorem for Severi varieties. In the second part of the paper, we give a necessary and sufficient condition for an elementary function to have an elementary Fourier transform (in an appropriate “weak ” sense) and explicit formulas for such Fourier transforms in the case when Q and Pj are monomials, over any local field F. We also describe a generalization of these results to the case of monomials of norms of finite extensions of F. Finally, we generalize some of the above results (including Fourier integration formulas) to the case when F = C and Q comes from a prehomogeneous vector space. 1.

Abstract. It is well-known that every group of type F4 is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type 1 E6 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 E6 and groups of type E7 and use it to give explicit descriptions of the homogeneous projective varieties associated to groups of type E7 with trivial Tits algebras. It is well-known that over an arbitrary field F (which for our purposes we will assume has characteristic ̸ = 2, 3) every algebraic group of type F4 is obtained as the automorphism group of some 27-dimensional exceptional Jordan algebra and that some groups of type E6 can be obtained as automorphism groups of norm forms of such algebras. In [Bro63], R.B. Brown introduced a new kind of F-algebra, which we will call a Brown algebra. The automorphism groups of Brown algebras provide a somewhat wider class of groups of type E6, specifically all of those with trivial Tits algebras. Allison and Faulkner [AF84] showed that there is a Freudenthal triple system (i.e., a quartic form and a skewsymmetric bilinear form satisfying certain relations) determined up to similarity by every

5.3 Divisible designs vs. finite chain geometries........... 64

We describe canonical forms for elements of a classical Lie group of matrices under similarity transformations in the group. Matrices in the associated Lie algebra and Jordan algebra of matrices inherit related forms under these similarity transformations. In general, one cannot achieve diagonal or Schur form, but the form that can be achieved displays the eigenvalues of the matrix. We also discuss matrices in intersections of these classes and their Schur-like forms. Such multistructured matrices arise in applications from quantum physics and quantum chemistry.