2,107 research outputs found
A quadratic Poisson Gel'fand-Kirillov problem in prime characteristic
The quadratic Poisson Gel’fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is Poisson birationally equivalent to a Poisson affine space, i.e. to a polyno-mial algebra K[X1,..., Xn] with Poisson bracket defined by {Xi, Xj} = λijXiXj for some skew-symmetric matrix (λij) ∈Mn(K). This problem was studied in [9] over a field of charac-teristic 0 by using a Poisson version of the deleting derivation homomorphism of Cauchon. In this paper, we study the quadratic Poisson Gel’fand-Kirillov problem over a field of arbitrary characteristic. In particular, we prove that the quadratic Poisson Gel’fand-Kirillov problem is satisfied for a large class of Poisson algebras arising as semiclassical limits of quantised co-ordinate rings. For, we introduce the concept of higher Poisson derivation which allows us to extend the Poisson version of the deleting derivation homomorphism from the characteristic 0 case to the case of arbitrary characteristic. When a torus is acting rationally by Poisson automorphisms on a Poisson polynomial algebra arising as the semiclassical limit of a quantised coordinate ring, we prove (under some technical assumptions) that quotients by Poisson prime torus-invariant ideals also satisfy the quadratic Poisson Gel’fand-Kirillov problem. In particular, we show that coordinate rings of determinantal varieties satisfy the quadratic Poisson Gel’fand-Kirillov problem
On the dimension of H-strata in quantum matrices
We study the topology of the prime spectrum of an algebra supporting a
rational torus action. More precisely, we study inclusions between prime ideals
that are torus-invariant using the -stratification theory of Goodearl and
Letzter on one hand and the theory of deleting derivations of Cauchon on the
other. We apply the results obtained to the algebra of generic
quantum matrices to show that the dimensions of the -strata described by
Goodearl and Letzter are bounded above by the minimum of and , and that
moreover all the values between 0 and this bound are achieved.Comment: New introduction; results improve
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