By Matilde Marcolli, Deepak Parashar
This e-book is aimed toward providing varied equipment and views within the idea of Quantum teams, bridging among the algebraic, illustration theoretic, analytic, and differential-geometric techniques. It additionally covers fresh advancements in Noncommutative Geometry, that have shut family members to quantization and quantum staff symmetries. the amount collects surveys by means of specialists which originate from an acitvity on the Max-Planck-Institute for arithmetic in Bonn.
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Additional info for Quantum Groups and Noncommutative Spaces: Perspectives on Quantum Geometry
Example text
Let T ∈ LR A (H1 , H2 ). 4, it suffices to consider γβ for α, β, γ ∈ A, which take the form components Tαβ γβ γ Tαβ = Mαβ ⊗ 1nβ γ for Mαβ ∈ Mnγ ×nα (C) ⊗ M(m2 )γβ ×(m1 )αβ (C). Now fix α, β, γ ∈ A, and write γ Mαβ = k Ai ⊗ Bi i=1 for Ai ∈ Mnγ ×nα (C) and for Bi ∈ M(m2 )γβ ×(m1 )αβ (C) linearly independent. 5 that γβ Eλ (Tαβ )= 1 nα 0 k i=1 tr(Ai )1nα ⊗ Bi ⊗ 1nβ if α = γ, otherwise, γβ ) vanishes if and only if either so that by linear independence of the Bi , Eλ (Tαβ α = γ or, α = β and each Ai is traceless, and hence, if and only α = γ or, α = β α and Mαβ ∈ sl(nα ) ⊗ M(m2 )γβ ×(m1 )αβ (C), as required.
Since the left action and right action commute, we must therefore have that Hαβ ∼ = Hmαβ Eαβ for some mαβ ∈ Z≥0 . Taking the direct sum of the Hαβ , we therefore see that H is unitarily equivalent to Hm for m = (mαβ ) ∈ MS (Z≥0 ), that is, [H] = bimod(m). We denote the inverse map bimod−1 : Bimod(A) → MS (Z≥0 ) by mult. 2. Let H be an A-bimodule. Then the multiplicity matrix of A is the matrix mult[H] ∈ MS (Z≥0 ). From now on, without any loss of generality, we shall assume that an Abimodule H with multiplicity matrix m is Hm itself.
Thus we have an isomorphism comp : L(H, H ) → b α,β,γ,δ∈A Mnγ ×nα (C) ⊗ Mmγδ ×mαβ ⊗ Mnδ ×nβ (C) γδ )α,β,γ,δ∈Ab. Note that when H = H , T is self-adjoint if given by comp(T ) := (Tαβ αβ γδ ∗ = (Tαβ ) for all α, β, γ, δ ∈ A. 4]). Let H and H be A-bimodules with multiplicity matrices m and m , respectively. 4) comp(LLR A (H, H )) = b α,β∈A 1nα ⊗ Mmαδ ×mαβ (C) ⊗ Mnδ ×nβ (C), Mnγ ×nα (C) ⊗ Mmγβ ×mαβ (C) ⊗ 1nβ , 1nα ⊗ Mmαβ ×mαβ (C) ⊗ 1nβ . Proof. Observe that T ∈ L(H, H ) is left, right, or left and right A-linear if γδ is left, right, or left and right A.