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
N1 - Publisher Copyright:
© 2023 Belgian Mathematical Society. All rights reserved.
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 -