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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 4-dimensional 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 ann-Hopf algebra and show that the ring of functions on ann-valued group and, in the topological case, the cohomology has ann-Hopf algebra structure.
      
The cohomology algebra of the classifying space of a compact Lie group admits the structure of ann-Hopf algebra, wheren is the order of the Weyl group; the homology with dual structure is also ann-Hopf algebra.
      
Using the properties ofn-Hopf algebras we show that certain spaces do not admit the structure of ann-valued group and that certain commutativen-valued groups do not arise by applying then-coset construction to any commutative group.
      
Cayley transforms and orbit structure in complex flag manifolds
      
Then theG0-orbit structure ofZ is described explicitly by the partial Cayley transforms of a certain hermitian symmetric sub-flagF?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 level-1 highest weight-modules.
      
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 semi-simple 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 aG-variety.
      
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.
      
 

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