Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras
- PDF / 487,176 Bytes
- 33 Pages / 439.37 x 666.142 pts Page_size
- 7 Downloads / 136 Views
Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras Yi Zhang1
· Xing Gao2 · Yanfeng Luo2
Received: 18 August 2019 / Accepted: 24 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The concept of weighted infinitesimal unitary bialgebra is an algebraic meaning of the nonhomogenous associative Yang–Baxter equation. In this paper, we equip the space of decorated planar rooted forests with a coproduct which makes it a weighted infinitesimal unitary bialgebra. Further, we construct an infinitesimal unitary Hopf algebra on decorated planar rooted forests in the sense of Loday and Ronco. We then introduce the concept of symmetric 1-cocycle condition, which is derived from the dual of the Hochschild cohomology. We study the universal properties of the space of decorated planar rooted forests with the symmetric 1-cocycle, leading to the notation of a weighted -cocycle infinitesimal unitary bialgebra. As an application, we obtain the initial object in the category of free cocycle infinitesimal unitary bialgebras on the undecorated planar rooted forests, which is the object studied in the wellknown noncommutative Connes–Kreimer Hopf algebra. Finally, we construct a preLie algebra on decorated planar rooted forests. Keywords Rooted forest · Infinitesimal bialgebra · Cocycle condition · Pre-Lie algebra Mathematics Subject Classification 16W99 · 05C05 · 16S10 · 16T10 · 16T30 · 17B60
B
Yanfeng Luo [email protected] Yi Zhang [email protected] Xing Gao [email protected]
1
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, People’s Republic of China
2
School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, Gansu, People’s Republic of China
123
Journal of Algebraic Combinatorics
1 Introduction Weighted infinitesimal unitary bialgebras first appeared in [39] and were further studied in [21,44,46,47], in order to give an algebraic meaning of nonhomogenous associative classical Yang–Baxter equations [39]. More precisely, a weighted infinitesimal unitary bialgebra is a module A, which is simultaneously an algebra (possibly without a unit) and a coalgebra (possibly without a counit) such that the coproduct is a weighted derivation of A in the sense that (ab) = a · (b) + (a) · b + λ(a ⊗ b) for a, b ∈ A, where λ ∈ k is a fixed constant. Parallel to the well-known fact that the solutions of a classical Yang–Baxter equation give rise to Lie bialgebras and quantum groups [11], Aguiar [1] introduced the associative Yang–Baxter equation (AYBE) r13r12 − r12 r23 + r23r13 = 0, and showed that any solution r of AYBE in an algebra A endows A with an infinitesimal unitary bialgebra of weight zero, involving a principle derivation. This result was generalized by Ogievetsky and Popov in [39], by the concept of nonhomogenous associative classical Yang–Baxter equation r13r12 − r12 r23 + r23r13 = λr13 . In [39], Ogievetsky and Popov clarifie
Data Loading...
