quantum groups and noncommutative spaces - persps. on quantum geometry - m. marcolli, d. parashar (teubner, 2011) ww

[...]... Section 3.9] contramodules of a coring with a grouplike element correspond to flat hom-connections Thus, in particular, anti-YetterDrinfeld contramodules are flat hom-connections We illustrate this discussion by the example of right-right anti-Yetter-Drinfeld contramodules First recall the definition of hom-connections from [3] Fix a differential graded algebra ΩA over an algebra A A hom-connection is... the second equality follows by the associative law for left contramodules and n+1 the third one by the definition of a left H-module algebra The equality τn = id follows by the associative law of contramodules, the definition of left H-action on A⊗n+1 , and by the stability of anti-Yetter-Drinfeld contramodules In the case of a contramodule M constructed on the dual vector space of a stable right-right... (Eds.), Quantum Groups and Noncommutative Spaces, DOI: 10.1007/97 8-3 -8 34 8-9 83 1-9 _2, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011 10 ´ ´ ´ BRANIMIR CACIC Connes and Marcolli The finite spectral triple of the current version has KOdimension 6 mod 8 instead of 0 mod 8, fails to be orientable, and only satisfies a certain modified version of Poincar´ duality It also no longer satisfies S 0 -reality,... Definition 2.8 A bimodule structure P consists of the following data: • A set P = Pγ PJ P , where each set PX is either empty or the singleton {X}, and where P is non-empty only if PJ is non-empty; • If PJ is non-empty, a choice of KO-dimension n mod 8, where n is even if and only if Pγ is non-empty In particular, we call a structure P : • odd if P is empty; • even if P = Pγ = {γ}; • real if PJ is non-empty... real C ∗ -algebras with fixed Wedderburn decomposition 2.2 Representation theory In keeping with the conventions of noncommutative differential geometry, we shall consider ∗-representations of real C ∗ -algebras on complex Hilbert spaces Recall that such a (left) representation of a real C ∗ algebra A consists of a complex Hilbert space H together with a ∗-homomorphism λ : A → L(H) between real C ∗ -algebras... M , h → h·m (2) A left-right anti-Yetter-Drinfeld contramodule is a left H-module and a right H-contramodule, such that, for all h ∈ H and f ∈ Hom(H, M ), h·α(f ) = α h(2) ·f S(h(3) )(−)h(1) M is said to be stable, provided that, for all m ∈ M , α(rm ) = m (3) A right-left anti-Yetter-Drinfeld contramodule is a right H-module and a left H-contramodule, such that, for all h ∈ H and f ∈ Hom(H, M ), α(f... if PJ is non-empty and P is empty • S 0 -real if P is non-empty Finally, if P is a graded structure, we call γ the grading, and if P is real or S 0 -real, we call J the charge conjugation Since this notion of KO-dimension is meant to correspond with the usual KOdimension of a real spectral triple, we assign to each real or S 0 -real structure P of KO-dimension n mod 8 constants ε, ε and, in the case... and a left H-contramodule M , Hom(A, M ) is an A-bimodule with the left and right A-actions defined by (f ·a)(b) = α (f (((−)·a) b)) , (a·f )(b) = f (ba), for all a, b ∈ A and f ∈ Hom(A, M ) Proof The definition of left A-action is standard, compatibility between left and right actions is immediate To prove the associativity of the right A-action, take any a, a , b ∈ A and f ∈ Hom(A, M ), and compute... composite F = ∇0 ◦ ∇1 is called the curvature of (M, ∇0 ) The hom-connection (M, ∇0 ) is said to be flat provided its curvature is equal to zero Hom-connections are non-commutative versions of right connections or co-connections studied in [13, Chapter 4 § 5], [16], [17] Consider a Hopf algebra H with a bijective antipode, and define an H-coring C = H⊗H as follows The H bimodule structure of C is given... anti-Yetter-Drinfeld module N , the complex described in Theorem 2 is the right-right version of Hopf-cyclic complex of a left module algebra with coefficients in N discussed in [8, Theorem 2.2] 6 Anti-Yetter-Drinfeld contramodules and hom-connections Anti-Yetter-Drinfeld modules over a Hopf algebra H can be understood as comodules of an H-coring; see [2] for explicit formulae and [4] for more information .