Abstract This paper presents several properties and relations that satisfy the components of a bicomplex holomorphic function. It also exhibits analogies differences with case analytic functions.

For any Lie algebra g we introduce the notion of gsymplectic structures and show that every orbit of a principal G-bundle carries a natural g-symplectic form and an associated momentum map induced by the Maurer–Cartan form on G. We apply this to the BRST bicomplex and show that the associated momentum map is a solution of the Wess–Zumino consistency condition for the anomaly.

Some enumerative aspects of the fans called generalized associahedra, introduced by S. Fomin and A. Zelevinsky in their theory of cluster algebras, are considered in relation with a bicomplex and its two spectral sequences. A precise enumerative relation with the lattices of generalized noncrossing partitions is conjectured and some evidence is given.

This paper reviews the moving frame approach to the construction of the invariant variational bicomplex. Applications include explicit formulae for the Euler-Lagrange equations of an invariant variational problem, and for the equations governing the evolution of differential invariants under invariant submanifold flows.

Some enumerative aspects of the fans, called generalized associahedra, introduced by S. Fomin and A. Zelevinsky in their theory of cluster algebras are considered, in relation with a bicomplex and its two spectral sequences. A precise enumerative relation with the lattices of generalized noncrossing partitions is conjectured and some evidence is given.

We give a relatively simple algorithm to do Hodge decomposition on the variational bicomplex based on the imbedding of the total differentiation operator in a Heisenberg algebra. The method is illustrated with several examples taken from the literature. It has been implemented by the authors in the computer algebra system Form [Ver91].

Abstract We define a bicomplex whose Euler characteristic is the idempotented version of ribbon element quantum s l (2). show that properties this descend to centrality, invertibility, and symmetries after decategorification.