structure 
The paper studies generic commutative and anticommutative algebras of a fixed dimension, their invariants, covariants and algebraic properties (e.g., the structure of subalgebras).


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.


We study the multiplicative structure of rings of coinvariants for finite groups.


Fibonacci manifolds have a hyperbolic structure which may be defined via Fibonacci numbers.


In this paper we prove that the homogeneous spaceG/K has a structure of a globally symmetric space for every choice ofG andK, especially forG being compact.


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.


The cohomology algebra of the classifying space of a compact Lie group admits the structure of annHopf algebra, wheren is the order of the Weyl group; the homology with dual structure is also annHopf algebra.


Using the properties ofnHopf algebras we show that certain spaces do not admit the structure of annvalued group and that certain commutativenvalued groups do not arise by applying thencoset construction to any commutative group.


Cayley transforms and orbit structure in complex flag manifolds


Then theG0orbit structure ofZ is described explicitly by the partial Cayley transforms of a certain hermitian symmetric subflagF?Z.


This extends the results and simplifies the proof for the classical orbit structure description of [10] and [11], which applies whenF=Z.


Similarly, by a specific choice of the parameter, the level (1,0) vertex representation of the quantum totoidal algebra gives rise to a structure on irreducible level1 highest weightmodules.


Structure of some ?graded lie superalgebras of vector fields


We also study the structure of the exceptional?graded transitive Lie superalgebras and give their geometric realization.


Coordinates on Schubert cells, Kostant's harmonic forms, and the Bruhat Poisson structure onG/B


For the flag manifoldX=G/B of a complex semisimple Lie groupG, we make connections between the Kostant harmonic forms onG/B and the geometry of the Bruhat Poisson structure.


We show that on each Schubert cell, the corresponding Kostant harmonic form can be described using only data coming from the Bruhat Poisson structure.


We give criteria for a divisor to be Cartier, globally generated and ample, and apply them to a study of the local structure and the intersection theory of aGvariety.


LetM=G/Γ be a compact nilmanifold endowed with an invariant complex structure.


to obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure.

