# Quantum Groups and Noncommutative Spaces: Perspectives on by Matilde Marcolli, Deepak Parashar

By Matilde Marcolli, Deepak Parashar

This ebook is aimed toward featuring varied equipment and views within the idea of Quantum teams, bridging among the algebraic, illustration theoretic, analytic, and differential-geometric ways. It additionally covers contemporary advancements in Noncommutative Geometry, that have shut relatives to quantization and quantum staff symmetries. the amount collects surveys by way of specialists which originate from an acitvity on the Max-Planck-Institute for arithmetic in Bonn.

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**Example text**

13) (mult ◦ιn )(Bimod(A, n)) = SymS (Z≥0 ). Proof. First, since a unitary equivalence of real A-bimodules of KO-dimension n mod 8 is, in particular, a unitary equivalence of odd A-bimodules, the map ιn is well deﬁned. Next, let (H, J) and (H , J ) be real A-bimodules of KO-dimension n mod 8, and suppose that H and H are unitarily equivalent as bimodules; let U ∈ ULR A (H , H). Now, if m is the multiplicity matrix of H, then H and Hm are unitarily ∗ equivalent, so let V ∈ ULR and V U J U ∗ V ∗ are both real A (H, Hm ).

Real spectral triples of odd KO-dimension. For this section, let (H, J) be a real A-bimodule of odd KO-dimension n mod 8 with multiplicity matrix m. We begin by reducing the study of Dirac operators on (H, J) to that of selfadjoint right A-linear operators on H. 4]). Let (H, J) be a real A-bimodule of odd KO-dimension n mod 8. Then the map Rn : LR A (H)sa → D0 (A, H, J) deﬁned by Rn (M ) := M + ε JM J ∗ is a surjection interwining the action of ULR A (H, J) on LR (H) by conjugation with the action on D (A, H, J) by conjugation, and sa 0 A LR ker(Rn ) ⊆ LA (H)sa .

E. Eλ (T ) ∈ LLA (H1 , H2 ) ρ2 (a)Eρ (T ) = Eρ (T )ρ1 (a) and Eρ (T ) ∈ LR A (H1 , H2 ). FINITE SPECTRAL TRIPLES 37 On the other hand, dμ(u)(T λ1 (u) − λ2 (u)T )λ1 (u−1 ), (id −Eλ )(T ) = U(A) dμ(u)(T ρ1 (u−1 ) − ρ2 (u−1 )T )ρ1 (u), (id −Eρ )(T ) = U(A) so that by the generalised order one condition, (id −Eλ )(T ) ∈ LR A (H1 , H2 ) and (id −Eρ )(T ) ∈ LLA (H1 , H2 ). Finally, the commutation of Eλ and Eρ together with our identiﬁcation of im(Eλ ) and of im(Eρ ) imply the desired result about im(Eλ Eρ ).