In this paper we show that if A is a Poisson algebra equipped with a set of maps ∆ (i) λ : A → A ⊗ N satisfying suitable conditions, then the images of the Casimir functions of A under the maps ∆ (i) λ (that we call " loop coproducts ") are in involution. Rational, trigonometric and elliptic Gaudin models can be recovered as particular cases of this result, and we show that the same happens for...

We study the action of the symmetric group Σn on a tensor product of n − 1 copies of a commutative Hopf algebra A, defined by the second author [8]. We show that for ‘nice’ Hopf algebras, the cohomology algebra H∗(Σn;A⊗n−1) is independent of the coproduct if n.(n − 2)! is invertible in the ground ring. Let A be a graded, connected, unital, counital, associative, coassociative Hopf algebra. In s...

It is well known that a sum (coproduct) of a family {Xi : i ∈ I} of Priestley spaces is a compactification of their disjoint union, and that this compactification in turn can be organized into a union of pairwise disjoint order independent closed subspaces Xu, indexed by the ultrafilters u on the index set I. The nature of those subspaces Xu indexed by the free ultrafilters u is not yet fully u...

In this paper, we studied the category of EQ-algebras and showed that it is complete, but not cocomplete, in general. We proved multiplicatively relative have coequlizers calculated coproduct pushout a special case. Also, constructed free EQ-algebra on singleton.