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In the case of 3-dimensional commutative algebras a new proof of a recent theorem of Katsylo and Mikhailov about the 28 bitangents to the associated plane quartic is given.
      
If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1).
      
If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (Theorem 3).
      
On the other hand, there is a locally trivialGa-action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2).
      
As an application of the results we prove a generalization of Chevalley's restriction theorem for the classical Lie algebras.
      
As a corollary we obtain an easy proof of a theorem of Borel and Serre: AnS-arithmetic subgroup of a semisimple group has all the finiteness propertiesFn.
      
Andersen's theorem about the ideal of negligible modules which in our notations is nothing else then.
      
As an immediate application we obtain a new proof of the main theorem of standard monomial theory for classical groups.
      
A convexity theorem for noncommutative gradient flows
      
In this survey we shall prove a convexity theorem for gradient actions of reductive Lie groups on Riemannian symmetric spaces.
      
In the final section the theorem is applied to gradient actions on other homogeneous spaces and we show, that Hilgert's Convexity Theorem for moment maps can be derived from the results.
      
Another proof of Joseph and Letzter's separation of variables theorem for quantum groups
      
A theorem of Kostant states that the universal enveloping algebra of g is a free module over its center.
      
A theorem of Richardson states that the algebra of regular functions ofG is a free module over the subalgebra of regular class functions.
      
Joseph and Letzter extended Kostant's theorem to the case of the quantized enveloping algebra of g.
      
Using the theory of crystal bases as the main tool, we prove a quantum analogue of Richardson's theorem.
      
In particular, we obtain an elementary proof of the celebrated theorem of Ellingsrud and Skjelbred [6].
      
The theorem of Hochster and Roberts says that, for every moduleV of a linearly reductive groupG over a fieldK, the invariant ringK[V]G is Cohen-Macaulay.
      
Our lower bound comes from applying a decomposition theorem for "hyperbolic localization" [Br] to this torus action.
      
We show the following: if there exists a k-variety which is birational to X and which has a smooth k-rational point, then X also has a k-rational point (Theorem 5.7).
      
 

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