TY - JOUR
T1 - Metric compatibility and Levi-Civita connections on quantum groups
AU - ASCHIERI, Paolo Maria
AU - Weber, T.
N1 - Publisher Copyright:
© 2024
PY - 2025
Y1 - 2025
N2 - Arbitrary connections on a generic Hopf algebra H are studied and shown to extend to connections on tensor fields. On this ground a general definition of metric compatible connection is proposed. This leads to a sufficient criterion for the existence and uniqueness of the Levi-Civita connection, that of invertibility of an H-valued matrix. Provided invertibility for one metric, existence and uniqueness of the Levi-Civita connection for all metrics conformal to the initial one is proven. This class consists of metrics which are neither central (bimodule maps) nor equivariant, in general. For central and bicoinvariant metrics the invertibility condition is further simplified to a metric independent one. Examples include metrics on SLq(2).
AB - Arbitrary connections on a generic Hopf algebra H are studied and shown to extend to connections on tensor fields. On this ground a general definition of metric compatible connection is proposed. This leads to a sufficient criterion for the existence and uniqueness of the Levi-Civita connection, that of invertibility of an H-valued matrix. Provided invertibility for one metric, existence and uniqueness of the Levi-Civita connection for all metrics conformal to the initial one is proven. This class consists of metrics which are neither central (bimodule maps) nor equivariant, in general. For central and bicoinvariant metrics the invertibility condition is further simplified to a metric independent one. Examples include metrics on SLq(2).
KW - Bicovariant bimodules
KW - Non-equivariant connections
KW - Non-equivariant non-central metrics
KW - Noncommutative Riemannian geometry
KW - Rational morphisms
KW - Tensor products of noncommutative connections
KW - Bicovariant bimodules
KW - Non-equivariant connections
KW - Non-equivariant non-central metrics
KW - Noncommutative Riemannian geometry
KW - Rational morphisms
KW - Tensor products of noncommutative connections
UR - https://iris.uniupo.it/handle/11579/193344
U2 - 10.1016/j.jalgebra.2024.07.039
DO - 10.1016/j.jalgebra.2024.07.039
M3 - Article
SN - 0021-8693
VL - 661
SP - 479
EP - 544
JO - Journal of Algebra
JF - Journal of Algebra
ER -