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Given a bicovariant differential calculus $(mathcal{E}, d)$ such that the braiding map is diagonalisable in a certain sense, the bimodule of two-tensors admits a direct sum decomposition into symmetric and anti-symmetric tensors. This is used to prove the existence of a bicovariant torsionless connection on $mathcal{E}$. Following Heckenberger and Schm{u}dgen, we study invariant metrics and the compatibility of covariant connections with such metrics. A sufficient condition for the existence and uniqueness of bicovariant Levi-Civita connections is derived. This condition is shown to hold for cocycle deformations of classical Lie groups.
We study pseudo-Riemannian invariant metrics on bicovariant bimodules over Hopf algebras. We clarify some properties of such metrics and prove that pseudo-Riemannian invariant metrics on a bicovariant bimodule and its cocycle deformations are in one to one correspondence.
We study covariant derivatives on a class of centered bimodules $mathcal{E}$ over an algebra A. We begin by identifying a $mathbb{Z} ( A ) $-submodule $ mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ mathca
Using the fact that the algebra M(3,C) of 3 x 3 complex matrices can be taken as a reduced quantum plane, we build a differential calculus Omega(S) on the quantum space S defined by the algebra C^infty(M) otimes M(3,C), where M is a space-time manifo
Given a tame differential calculus over a noncommutative algebra $mathcal{A}$ and an $mathcal{A}$-bilinear pseudo-Riemannian metric $g_0,$ consider the conformal deformation $ g = k. g_0, $ $k$ being an invertible element of $mathcal{A}.$We prove tha
We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation gl_N[t]-modules and the scheme-theoretic intersection of suitable Schubert varieties. Moreover, we prove that the multipli