Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions

Huan Jia; Naihong Hu; Rongchuan Xiong; Yinhuo Zhang

doi:10.36045/j.bbms.230408

Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions

Huan Jia
, Naihong Hu
, Rongchuan Xiong^*
, Yinhuo Zhang

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

Let k be a field. We study the free bialgebra T generated by the coalgebra C = k{1, g, h} and its quotient bialgebras (or Hopf algebras) over k. We show that the free non-commutative Faà di Bruno bialgebra is a sub-bialgebra of T , and the quotient bialgebra T := T /(E_α | α(g) ≥ 2) is an Ore extension of the well-known Faà di Bruno bialgebra. The image of the free non-commutative Faà di Bruno bialgebra in the quotient T gives a more reasonable non-commutative (non-free) version of the commutative Faà di Bruno bialgebra from the PBW basis point view. If char k = p > 0, we obtain a chain of quotient Hopf algebras of T : T → T _n → T ⁰_n(p) → T _n(p) → T _n(p; d₁) → . . . → T _n(p; d_j, d_j−₁, . . ., d₁) → . . . → T _n(p; d_p−₂, d_p−₃, . . ., d₁) with infinite or finite GK-dimension. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras.

Original language	English
Pages (from-to)	634-667
Number of pages	34
Journal	Bulletin of the Belgian Mathematical Society - Simon Stevin
Volume	30
Issue number	5
DOIs	https://doi.org/10.36045/j.bbms.230408
State	Published - Dec 2023

Keywords

Faà di Bruno Hopf algebra
GK-dimension
Lyndon-Shirshov basis
Pointed Hopf algebras
shuffle type polynomials

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title = "Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions",

abstract = "Let k be a field. We study the free bialgebra T generated by the coalgebra C = k\{1, g, h\} and its quotient bialgebras (or Hopf algebras) over k. We show that the free non-commutative Fa{\`a} di Bruno bialgebra is a sub-bialgebra of T , and the quotient bialgebra T := T /(Eα | α(g) ≥ 2) is an Ore extension of the well-known Fa{\`a} di Bruno bialgebra. The image of the free non-commutative Fa{\`a} di Bruno bialgebra in the quotient T gives a more reasonable non-commutative (non-free) version of the commutative Fa{\`a} di Bruno bialgebra from the PBW basis point view. If char k = p > 0, we obtain a chain of quotient Hopf algebras of T : T → T n → T 0n(p) → T n(p) → T n(p; d1) → . . . → T n(p; dj, dj−1, . . ., d1) → . . . → T n(p; dp−2, dp−3, . . ., d1) with infinite or finite GK-dimension. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras.",

keywords = "Fa{\`a} di Bruno Hopf algebra, GK-dimension, Lyndon-Shirshov basis, Pointed Hopf algebras, shuffle type polynomials",

author = "Huan Jia and Naihong Hu and Rongchuan Xiong and Yinhuo Zhang",

year = "2023",

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doi = "10.36045/j.bbms.230408",

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TY - JOUR

T1 - Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions

AU - Jia, Huan

AU - Hu, Naihong

AU - Xiong, Rongchuan

AU - Zhang, Yinhuo

PY - 2023/12

Y1 - 2023/12

N2 - Let k be a field. We study the free bialgebra T generated by the coalgebra C = k{1, g, h} and its quotient bialgebras (or Hopf algebras) over k. We show that the free non-commutative Faà di Bruno bialgebra is a sub-bialgebra of T , and the quotient bialgebra T := T /(Eα | α(g) ≥ 2) is an Ore extension of the well-known Faà di Bruno bialgebra. The image of the free non-commutative Faà di Bruno bialgebra in the quotient T gives a more reasonable non-commutative (non-free) version of the commutative Faà di Bruno bialgebra from the PBW basis point view. If char k = p > 0, we obtain a chain of quotient Hopf algebras of T : T → T n → T 0n(p) → T n(p) → T n(p; d1) → . . . → T n(p; dj, dj−1, . . ., d1) → . . . → T n(p; dp−2, dp−3, . . ., d1) with infinite or finite GK-dimension. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras.

AB - Let k be a field. We study the free bialgebra T generated by the coalgebra C = k{1, g, h} and its quotient bialgebras (or Hopf algebras) over k. We show that the free non-commutative Faà di Bruno bialgebra is a sub-bialgebra of T , and the quotient bialgebra T := T /(Eα | α(g) ≥ 2) is an Ore extension of the well-known Faà di Bruno bialgebra. The image of the free non-commutative Faà di Bruno bialgebra in the quotient T gives a more reasonable non-commutative (non-free) version of the commutative Faà di Bruno bialgebra from the PBW basis point view. If char k = p > 0, we obtain a chain of quotient Hopf algebras of T : T → T n → T 0n(p) → T n(p) → T n(p; d1) → . . . → T n(p; dj, dj−1, . . ., d1) → . . . → T n(p; dp−2, dp−3, . . ., d1) with infinite or finite GK-dimension. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras.

KW - Faà di Bruno Hopf algebra

KW - GK-dimension

KW - Lyndon-Shirshov basis

KW - Pointed Hopf algebras

KW - shuffle type polynomials

UR - https://www.scopus.com/pages/publications/85186333528

U2 - 10.36045/j.bbms.230408

DO - 10.36045/j.bbms.230408

M3 - 文章

AN - SCOPUS:85186333528

SN - 1370-1444

VL - 30

SP - 634

EP - 667

JO - Bulletin of the Belgian Mathematical Society - Simon Stevin

JF - Bulletin of the Belgian Mathematical Society - Simon Stevin

IS - 5

ER -

Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions

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