Tiling the (n², 1)-De Bruijn Graph with n Coassociative Coalgebras

Abstract

We construct, via usual graph theory a class of associative dialgebras as well as a class of coassociative L-coalgebras, the two classes being related by a tool from graph theory called the line-extension. As a corollary, a tiling of the n ²-De Bruijn graph with n (geometric supports of) coassociative coalgebras is obtained. Via the tiling of the (3, 1)-De Bruijn graph, we also get an example of cubical trialgebra, notion introduced by Loday and Ronco. Other examples are obtained by letting M _n (k) act on axioms defining such tilings. Examples of associative products which split into several associative ones are also given.

Key Words:

• Directed graphs
• Hopf algebras
• (Markov) L-coalgebras
• Coassociative co-dialgebras
• Cubical trialgebras
• Achirality

Mathematics Subject Classification (2000):

• 16W30
• 05C20
• 05C90

Acknowledgments

The author wishes to thank Dimitri Petritis for fruitful advice for the redaction of this paper and to S. Severini for pointing him the precise definition of the De Bruijn graphs.

Notes

^aA graph isomorphism f : G → H between two graphs G and H is a pair of bijection f ₀ : G ₀ → H ₀ and f ₁ : G ₁ → H ₁ such that f ₀(s _G (a)) = s _H (f ₁(a)) and f ₀(t _G (a)) = t _H (f ₁(a)) for all a ∈ G ₁. All the directed graphs in this formalism will be considered up to a graph isomorphism.

^bLet A be an associative algebra with unit e, M be a A-bimodule and f : A → M be a linear map. The map f is said to be a Leibniz–Ito derivative if f(e) = 0 and f(xy) = f(x)f(y) + xf(y) + f(x)y, for all x, y ∈ A.

^cFix n > 0. A L-Hopf algebra of degree n, (H, Δ _H , Δ˜ _H ), is by definition a L-bialgebra of degree n, equipped with right and left counits, ε˜ _H , ε _H of degree n, such that its antipodes S, [Stilde] : H → H verify (i d _n−1 ⊗ m)(i d _n  ⊗ S)Δ _H  = η _n ε _H and (m ⊗ id)([Stilde] ⊗ i d _n )Δ˜ _H  = η˜ _n ε˜ _H , with η _n , η˜ _n  : H ^⊗(n−1) → H ^⊗n such that η _n (h) ≔ h ⊗ 1 _H and η˜ _n (h) ≔ 1 _H  ⊗ h, h ∈ H ^⊗(n−1) (Leroux, 2003).

^dIn Leroux (eprint arXiv:math.QA/0301080), such algebraic objects will be called coassociative manifolds.

^eEntangled by the achiral entanglement equation.

^fThese notions have been discovered by Loday and Ronco (2003) and rediscovered independently, via graph theory by the author (Private Communication).

^#Communicated by J. Alev.