We prove that the Hochschild homology (and cohomology) of a symmetric open Frobenius algebra A has a natural coBV and BV structure. The underlying coalgebra and algebra structure may not be resp. counital and unital. Moreover we prove that the product and coproduct satisfy the Frobenius compatibility condition i.e. the coproduct on HH∗(A) is a map of left and right HH∗(A)-modules. If A is commu...

The free loop space of a Lie group is homeomorphic to the product itself and its based space. We show that coproduct on homology was introduced by Goresky Hingston splits into diagonal map so-called This result implies trivial for even-dimensional groups. Using results Bott Samelson, we as well large family simply connected

A product and coproduct in categories are introduced. The concepts included correspond to that presented in 7]. and 6] provide the notation and terminology for this paper.

‎ in the present article‎, ‎we study some categorical properties of the category {$bf‎ cpo_{sep}$-$s$} of all {separately strict $s$-cpo's}; cpo's equipped with‎ a compatible right action of a separately strict cpo-monoid $s$ which is‎ strict continuous in each component‎. ‎in particular‎, we show that this category is reflective and coreflective in the‎ category of $s$-cpo's‎, ‎find the free a...

Increase in demand for ethanol as a fuel additive has resulted in dramatic growth in ethanol production. Ethanol is produced from corn by wet milling or dry grind processing. Wet mill plants are capital intensive due to equipment requirements; they produce large volumes of ethanol and are corporate owned. In dry grind processing, the kernel is not fractionated and only one coproduct, distillers...

Based on the structure of Casimir elements associated with general Hopf algebras there are constructed Liouville-Arnold integrable flows related with naturally induced Poisson structures on arbitrary co-algebra and their deformations. Some interesting special cases including the oscillatory Heisenberg-Weil algebra related co-algebra structures and adjoint with them integrable Hamiltonian system...

With each coassociative coalgebra, we associate an oriented graph. The coproduct ∆, obeying the coassociativity equation (∆ ⊗ id)∆ = (id ⊗ ∆)∆ is then viewed as a physical propagator which can convey information. We notice that such a coproduct is non local. To recover locality we have to break the coassociativity of the coproduct and restore it, in some sense by introducing another coproduct ∆̃...