In the theory of noncommutative geometry and quantum groups, we have the useful notions of “twisted” spectral triples introduced by Connes and Moscovici, and covariant differential calculi introduced by Woronowicz.
Dropping the analytic requirements and formulating in a purely algebraic sense, the authors show that any finite-dimensional covariant differential calculus over a Hopf algebra with invertible antipode can be realized by a twistedspectral triple. Roughly speaking, a twisted spectral triple \(\left( A,H,\sigma,D\right) \) is said to define a realization of the differential calculus \(\Omega^{1}:=\text{Span}\left\{ a\vartriangleright db:a,b\in A\right\} \) over \(A\), where \(\text{End}_{k}\left( H\right) \) for a \(k\)-vector space \(H\) is endowed with the \(A\)-bimodule structure \(a\vartriangleright\omega\vartriangleleft b:=\sigma\left( a\right) \circ\omega\circ b\) for \(\omega\in\text{End}_{k}\left( H\right) \) and \(a,b\in A\) associated with an associative subalgebra \(A\subset \text{End}_{k}\left( H\right) \) and a representation \(\sigma\) of \(A\) on \(H\), and \(D\in\text{End}_{k}\left( H\right) \) defines a derivation \(d:A\rightarrow\text{End}_{k}\left( H\right) \) by \(da:=D\circ a-\sigma\left( a\right) \circ D\) (which replaces the commutator bracket \(\left[ D,a\right] =D\circ a-a\circ D\) included in the definition of an ordinary “untwisted” spectral triple).
When the algebra \(A\) is a compact quantum group, it is shown that any covariant differential calculus of finite rank over \(A\) can be realized by a twistedspectral triple on a Hilbert space \(H\) with all elements of the calculus being bounded operators. Finally a particular calculus for the quantum group \(SU_{q}\left( 2\right) \) found by Heckenberger is analyzed as an example with \(\sigma\) an automorphism, in which however the operator \(D\) does not have a compact resolvent.
For the entire collection see [Zbl 1230.53004].