Cocommutative Hopf Dialgebras and Rack Combinatorics

TL;DR

This paper explores cocommutative Hopf dialgebras using rack combinatorics, establishing connections with generalized digroups and providing explicit formulas for their algebraic structures.

Contribution

It introduces a new framework linking cocommutative Hopf dialgebras with rack theory and generalized digroups, including explicit formulas and algebraic constructions.

Findings

Rack of set-like elements is isomorphic to conjugation rack of the digroup.

Explicit formulas for conjugation rack, inner group, and orbit structures are derived.

Constructs the digroup algebra as a cocommutative Hopf dialgebra with specific properties.

Abstract

We study cocommutative Hopf dialgebras through generalized digroups and rack combinatorics. We prove that the rack functor obtained from the adjoint rack bialgebra factorizes through the digroup of group-like elements. More precisely, for every cocommutative Hopf dialgebra $A$, the rack of set-like elements of its adjoint rack bialgebra is naturally isomorphic to the conjugation rack of the digroup $\Glike(A)$. For finite generalized digroups $D\simeq G\times E$, with $G$ acting on the halo $E$, we derive explicit formulas for the conjugation rack, its inner group, left-translation cycle index, fixed-point polynomial, orbit count and subrack structure. Finally, we construct the digroup algebra $K[D]$, prove that it is a cocommutative Hopf dialgebra, and show that $\Glike(K[D])=D\$.