case 
In the present paper we study the remaing nontrivial case, that of a negative central chargeN.


The method applies to the standard arithmetic subgroups ofSO(n,1) (a case which was proved previously by Millson [Mi]), to the nonarithmetic lattices inSO(n,1) constructed by Gromov and PiatetskiShapiro [GPS] and to groups generated by reflections.


In the case of 4dimensional anticommutative algebras a construction is given that links the associated cubic surface and the 27 lines on it with the structure of subalgebras of the algebra.


In the case of 3dimensional commutative algebras a new proof of a recent theorem of Katsylo and Mikhailov about the 28 bitangents to the associated plane quartic is given.


As in the case of Mumford's geometric invariant theory (which concerns projective good quotients) the problem can be reduced to the case of an action of a torus.


As in the case of affine Weyl groups, they can be obtained by adding a further node to the diagram for the linear part.


The theory is applied to the case of cubic hypersurfaces, which is the one most relevant to special geometry, obtaining the solution of the two classification problems and the description of the corresponding homogeneous special K?hler manifolds.


For the case of positive characteristic we use the classification of finite irreducible groups generated by pseudoreflections due to Kantor, Wagner, Zalesski? and Sere?kin.


We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety.


In the interesting case when the group is of Coxeter typeDn (n≥4) we use higher polarization operators introduced by Wallach.


We study Edidin and Graham's equivariant Chow groups in the case of torus actions.


We introduce the (purely algebraic) notion of annHopf algebra and show that the ring of functions on annvalued group and, in the topological case, the cohomology has annHopf algebra structure.


[M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rankone case.


We also use known results about canonical bases forUq2 to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmannians), due to LascouxSchützenberger and Zelevinsky.


Joseph and Letzter extended Kostant's theorem to the case of the quantized enveloping algebra of g.


Of course, our proof would work also in the finite type case.


In the course of the proof we show that one can reduce the study of generating semiinvariants to the case when the quiver has no oriented paths of length greater than one.


In the special case whenFn is the projective spaceRPn, one also obtains the upper bound.


Thus we confirm a conjecture of Brundan for one more case.


We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in ?(G) are diagonalizable.

