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The paper begins with a review of the well known KdV hierarchy, N-th Novikov equations and its finite hierarchiey in the classical commutative case. Its finite hierarchy consists of N compatible integrable polynomial dynamical systems in C2N. Then we discuss a non-commutative version of the N-th Novikov hierarchy defined on a finitely generated free associative algebra BN with 2N generators. Using the quantisation ideals method in BN, for N = 1, 2, 3, 4, we have found two-sided homogeneous ideals QN / BN (quantisation ideals) which are invariant with respect to the N-th Novikov equation and such that the quotient algebra CN = BN/QN has a well defined Poincare--Birkhoff--Witt basis. It enables us to define the quantum N-th Novikov equation and its hierarchy on the CN. We have found N commuting quantum first integrals (Hamiltonians) and represented equations of the hierarchy in the Heisenberg form. In this paper we introduce the concept of Frobenius-Hochschild algebras and in its terms we express explicitly first integrals of the N-th Novikov hierarchy in the commutative, free and quantum cases. [ABSTRACT FROM AUTHOR]

The aim of this paper is to further study the universal toric genus of compact homogeneous spaces and their homogeneous fibrations. We consider the homogeneous spaces with positive Euler characteristic. It is well known that such spaces carry many stable complex structures equivariant under the canonical action of the maximal torus Tk. As the torus action in this case only has isolated fixed points it is possible to effectively apply localization formula for the universal toric genus. Using this, we prove that the famous topological results related to rigidity and multiplicativity of a Hirzebruch genus can be obtained on homogeneous spaces just using representation theory. In this context, for homogeneous SU-spaces, we prove the well-known result about rigidity of the Krichever genus. We also prove that for a large class of stable complex homogeneous spaces any Tk-equivariant Hirzebruch genus given by an odd-power series vanishes. With regard to the problem of multiplicativity, we provide construction of stable complex Tk-fibrations for which the universal toric genus is twistedly multiplicative. We prove that it is always twistedly multiplicative for almost complex homogeneous fibrations and describe those fibrations for which it is multiplicative. As a consequence for such fibrations the strong relations between rigidity and multiplicativity for an equivariant Hirzebruch genus is established. The universal toric genus of the fibrations for which the base does not admit any stable complex structure is also considered. The main examples here for which we compute the universal toric genus are the homogeneous fibrations over quaternionic projective spaces. [ABSTRACT FROM AUTHOR]

We extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples Bi,j, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the Bi,j allows us to deduce that every complex cobordism class of dimension > 2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum # for simple n-dimensional polytopes; when Pn is a product of simplices, we describe Pn # Qn by applying an appropriate sequence of pruning operators, or hyperplane cuts, to Qn. [ABSTRACT FROM PUBLISHER]