Homogeneous algebras

Rational Homogeneous Algebras

An algebra A is homogeneous if the automorphism group of A acts transitively on the one dimensional subspaces of A. The existence of homogeneous algebras depends critically on the choice of the scalar field. We examine the case where the scalar field is the rationals. We prove that if A is a rational homogeneous algebra with dimA > 1 then A = 0.

Inductive Algebras and Homogeneous Shifts

Inductive algebras for the irreducible unitary representations of the universal cover of the group of unimodular two by two matrices are classified. The classification of homogeneous shift operators is obtained as a direct consequence. This gives a new approach to the results of Bagchi and Misra.

Homogeneous orthocomplete effect algebras are covered by MV-algebras

Algebras of Curvature Forms on Homogeneous Manifolds

Let C(X) be the algebra generated by the curvature two-forms of standard holomorphic hermitian line bundles over the complex homogeneous manifold X = G/B. The cohomology ring of X is a quotient of C(X). We calculate the Hilbert polynomial of this algebra. In particular, we show that the dimension of C(X) is equal to the number of independent subsets of roots in the corresponding root system. We...

Koszul and Gorenstein properties for homogeneous algebras

Koszul property was generalized to homogeneous algebras of degree N > 2 in [5], and related to N -complexes in [7]. We show that if the N -homogeneous algebra A is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem [23] to A, i.e., there is a Poincaré duality between Hochschild homology and cohomology of A, as for N = 2. Mathem...