Hochschild and Cyclic Homology of Quantum Kummer Spaces
aa r X i v : . [ m a t h . K T ] J u l HOCHSCHILD AND CYCLIC HOMOLOGY OF THE QUANTUMKUMMER SPACES
SAFDAR QUDDUS
Abstract.
We study the quotient space obtained by the flip action on the quantum n-tori.The Hochschild, cyclic and periodic cyclic homology are calculated. Introduction
Spanier [S] studied the Kummer (non-smooth)manifolds obtained by the action of Z onthe 2 n dimensional torus. He concluded that the space is homeomorphic to RP n − . It has2 n double points and is simply connected with vanishing odd homology. Alternatively, alink of fixed-points after the quotient is homeomorphic to RP n − . The non-commutativegeometry currently does not have a well defined notion of “non-commutative knots/links”but we shall see that homologically the Z quotient of the n -quantum torus A Θ is similar tothe Kummer manifold/variety. The dimension of cyclic homology is the same as the Bettinumbers for the classical Kummer manifolds.The Hochschild homology for these orbifolds for the case n = 2 was done in [O] [B] and [Q].We here are inspired by the proof of [Q] and extending the methodology there into higherdimensions. It maybe noted that the periodic cyclic homology of A Θ ⋊ Z , the noncommu-tative smooth torus Z computed in a recent work [CTY] matches in dimension to whatwe have calculated in this article for the quantum/algebraic noncommutative torus with Z action. It is also expected that for “sufficiently” good Θ, the Z quotient of the non-commutative smooth n -torus A Θ shares similar Hochschild homological property but evenfor n = 2, such a computation was tricky[C]. What is known rather is that the odd periodichomology vanishes which does hint the vanishing of odd Hochschild homology. Other thanhaving a striking similarity with the smooth quotients, the Hochschild homology of the quan-tum tori themselves have been studied [W1] and [T]. Readers can refer to [BRT, Page 353]for the comparison table therein for various homological and algebraic properties betweenthe smooth non-commutative 2-torus and the quantum/algebraic non-commutative 2-torus.1. Statement
THEOREM 1.1.
Let Θ be a skew symmetric n × n matrix such that its entries are uni-modular but none are roots of unity.a) H ( A Θ ⋊ Z ) ∼ = C n +1 andb) H • ( A Θ ⋊ Z ) ∼ = C ( n • ) for • = 2 k for some k > and otherwise . Date : July 9, 2020.
Key words and phrases.
Homology, non-commutative N-torus, Hochschild. OROLLARY 1.2. dim C ( HC • ( A Θ ⋊ Z )) = X k ≤• (cid:18) n k (cid:19) + 2 n for • even, otherwise. COROLLARY 1.3.
The periodic homology are as follows:a) HP even ( A Θ ⋊ Z ) ∼ = C · n − andb) HP odd ( A Θ ⋊ Z ) = 0 . Strategy of the Proof
We shall study the Hochschild homology using the paracyclic spectral decomposition of thehomology of the crossed product algebra. [GJ] H • ( A Θ ⋊ Z ) ∼ = H • ( A Θ ) Z ⊕ H • ( − A Θ ) Z , where − A Θ is the algebra A Θ with (left) Z -twisted A e Θ module structure.Hence our proof will investigate each of the summands in the above decomposition byfirstly understanding the associated Hochschild homology and then locating the Z invariantcycles. We shall use Nest’s resolution(which is similar to Wambst’s resolution for quantumsymmetric algebras[W2]) and also Connes’ resolution as and when we find suitable.3. Nest’s Koszul Resolution Revisited
Nest [N] introduced a Koszul resolution for higher dimensional non-commutative tori. Webriefly describe his resolution below.The algebra A Θ for a skew symmetric complex matrix Θ is generated by unitaries ν , ν , . . . , ν n ,satisfying the commutation relations ν i ν j = λ ij ν j ν i , for 1 ≤ i, j ≤ n such that | λ ij | = 1.The enveloping algebra A e Θ is the algebraic tensor of A Θ and its opposite algebra. A e Θ = A Θ ⊗ A op Θ . An element of A e Θ is denoted by a ⊗ b o for a ∈ A Θ and b o ∈ A op Θ . We set V = C n withorthonormal basis e , e , . . . , e n . We have the standard bar resolution of A Θ is given asbelow: · · · b −→ Λ s ( A Θ ) b −→ Λ s − ( A Θ ) b −→ · · · b −→ A e Θ ǫ −→ A Θ , where Λ s ( A Θ ) = A e Θ ⊗ A ⊗ s Θ , ǫ is the augmentation map and b : Λ n ( A Θ ) → Λ n − ( A Θ ); a ⊗ a ⊗ · · · ⊗ a n n − X i =0 ( − i a ⊗ · · · a i a i +1 · · · a n + ( − n a n a ⊗ a ⊗ · · · a n − . We set E s = A e Θ ⊗ Λ s V and consider the following maps: h s : E s → Λ s ( A Θ );1 ⊗ e i ∧ e i ∧ · · · ∧ e i n X σ ∈ S s sgn ( σ )( ν σ ( i ) ν σ ( i ) · · · ν σ ( i s ) ) − ⊗ ν σ ( i ) ⊗ ν σ ( i ) ⊗ · · · ⊗ ν σ ( i s ) .α s : E s → E s − ; ⊗ e i ∧ e i ∧ · · · ∧ e i n s X k =1 ( − k (1 − ν − i k ⊗ ν oi k ) ⊗ e i ∧ e i ∧ · · · ∧ ˆ e i k ∧ · · · ∧ e i n .k s : Λ s ( A Θ ) → E s ; k (( ν π ν π · · · ν π s ) − ⊗ ν π ⊗ ν π ⊗ · · · ⊗ ν π s ) = X i >i >...i s ρ i (( ν π ) − ⊗ ( ν π )) ∧ ρ i (( ν π ) − ⊗ ( ν π )) ∧ · · · ∧ ρ i s (( ν π s ) − ⊗ ( ν π s )) . Where for E s ∼ = A Θ ⊗ Λ s V ⊗ A Θ has a graded product structure[N], for π = ( π , π , . . . π s ) ∈ Z s , ν π := ν π ν π . . . ν π s s and ρ i : Λ ( A Θ ) → E as defined as belowNote: The formula for ρ i (( ν π ) − ⊗ ν π ) in [N][Page 1050] has a misprint and the correctformula, which we shall use in our study is as follows: ρ i (( ν π ) − ⊗ ν π ) = ( ν π | >i ) − ( π i − X ′ s =0 ν − ki ⊗ e i ⊗ ν ki )( ν π | >i ) . where n X ′ i =0 = n X i =0 for n ≥ , n = − , − − X n +1 for n < − . and ν π | >p − := ν π p p . . . ν π n n .Though Nest gave this resolution for smooth non-commutative n -tori A Θ , but it is also aresolution of the quantum tori A Θ , the proof is easy and similar to the proof that Connes’resolution is a resolution for quantum 2-torus.[Q]4. Invariant cycles, H • ( A Θ ) Z Using the Nest resolution we can easily compute H ( A Θ ) explicitly, they zeroth cocyclesare the A Θ /im ( α ), where α ( a i ⊗ ⊗ e i ) = a i ⊗ (1 − ν − i ⊗ ν io ) = a i − ν − i a i ν i . Clearly, the zeroth Hochschild homology H ( A Θ ) is generated by the equivalence class ofelements supported at a ¯0 . These scalars are invariant under ν i ν − i , hence H ( A Θ ) Z = C .To compute H • ( A Θ ) Z for • > H • ( A Θ ) using the Nest’s Koszulresolution and then locate the invariant cycles. Wambst [W1] computed these k -Hochschildcycles of the quantum tori A Θ , they were generated by elements { ( x π ) − ⊗ x π } π ∈{ , } n with | π | = k . But in this article, we shall restrict ourselves with the notation of Nest [N].Let us consider E s in the Nests’ Koszul resolution, using the following map it is straight-forward to see that H s ( A Θ ) is generated by elements a ¯0 ⊗ e i ∧ · · · ∧ e i s where { i , . . . , i s } ⊂{ , , . . . , n } . We want to locate the Z Hochschild k -cycle using the Koszul resolution of A Θ . (1 ⊗ d s ) : A Θ ⊗ E s → A Θ ⊗ E s − . et a ¯0 ⊗ ⊗ e i ∧ · · · ∧ e i s ∈ ker (1 ⊗ d s ), to check if it is invariant under Z action we pushthe cycle into the bar complex using the map h s . · · · E E E A Θ · · · A ⊗ A ⊗ A ⊗ A Θ α k α k α k = id ∼ = b ′ h h b ′ b ′ h = id (1 ⊗ h s )( a ¯0 ⊗ ⊗ e i ∧ · · · ∧ e i s ) = a ¯0 ⊗ X σ ∈ S s sgn ( σ )( ν σ ( i ) . . . ν σ ( i s ) ) − ⊗ ν σ ( i ) ⊗ · · · ⊗ ν σ ( i s ) . Now, (1 ⊗ k s )( a ¯0 ⊗ X σ ∈ S s sgn ( σ )( ν − σ ( i ) . . . ν − σ ( i s ) ) − ⊗ ν − σ ( i ) ⊗ · · · ⊗ ν − σ ( i s ) )= sgn ( ψ ) a ¯0 ⊗ k s (( ν − ψ ( i ) . . . ν − ψ ( i s ) ) − ⊗ ν − ψ ( i ) ⊗ · · · ⊗ ν − ψ ( i s ) )Where ψ ∈ S k is the permutation such that ψ ( i ) > ψ ( i ) > · · · ψ ( i s ). We have used thefact that ρ i (( ν π ) − ⊗ ν π ) = 0 if ( π ) i = 0. ρ ψ j ( ν ψ ( i j ) ⊗ ν − ψ ( i j ) ) = − ( ν ψ j ⊗ e ψ j ⊗ ν − ψ j ) . Therefore, (1 ⊗ k s )( a ¯0 ⊗ X σ ∈ S s sgn ( σ )( ν − σ ( i ) . . . ν − σ ( i s ) ) − ⊗ ν − σ ( i ) ⊗ · · · ⊗ ν − σ ( i s ) )= ( − s sgn ( ψ ) sgn ( ψ ) − a ¯0 ⊗ ⊗ e i ∧ · · · ∧ e i s . Hence for s even all the s -cycles are Z invariant and for s odd none are. This was exactlythe case in [Q, Page 329, 331], for the 1-cycles and the 2-cycle of the quantum 2-torus with SL ( Z ) action. We have the following: LEMMA 4.1. H • ( A Θ ) Z = C ( n • ) if • = 2 k , 0 otherwise. Twisted invariant cycles, H • ( − A Θ ) Z To calculate H • ( − A Θ ) we need to consider the Z twisted Koszul chain complex. For s = 0we can explicitly see that the Z twisted zeroth cycle. H ( − A Θ ) = − A Θ / − α . where − α ( a i ⊗ ⊗ e i ) = a i ⊗ (1 − ν − i ⊗ ν io ) = a i − ν i a i ν i . Hence H ( − A Θ ) is generated by the equivalence class of elements of the form ( a β ) β ∈{ , } n .Therefore H ( − A Θ ) = C n . Under the action of Z , an element ( a β ) is mapped to homolo-gous element ( a − β ) hence H ( − A Θ ) Z = C n . Observe that for a ∈ H n ( − A Θ ) = ker (1 ⊗ d n ), a = ν i aν i for all i . Hence H n ( − A Θ ) = 0.We shall proceed by induction, we shall induct on the dimension of the torus.We state that for the n - dimensional quantum torus( n > H • ( − A Θ ) = 0 for all 0 < • < n .As we noted earlier, the above statement holds for the case n = 2. Let us assume that for ll torus of dimensions less than n it holds. We shall prove that H • ( − A Θ ) = 0 for all0 < • < n . The proof is now divided into two cases:5.1. Case I: • > .LEMMA 5.1. In this case we shall prove that if H • ( − A Θ ) = 0 for n − > • > then H • ( − A n Θ ) = 0 for all n > • > .Proof. We notice that (cid:0) ns (cid:1) = (cid:0) n − s (cid:1) + (cid:0) n − s − (cid:1) . Hence we have the following identification.( π , π ) : − A Θ ⊗ E ns = − A Θ ⊗ E n − s ⊕ − A Θ ⊗ E n − s − ∧ e n Therefore to show that A Θ ⊗ E n • is acyclic at s it is enough to show that − A Θ ⊗ E ns +1 M − A Θ ⊗ E ns ∧ e n ⊗ − α s +1 −−−−−−→ − A Θ ⊗ E n − s M − A Θ ⊗ E n − s − ∧ e n ⊗ − α s −−−−→ − A Θ ⊗ E n − s − M − A Θ ⊗ E n − s − ∧ e n is middle exact.We notice that the map 1 ⊗ − α ns does mix the direct summands. Explicitly,(1 ⊗ − α s )( − A Θ ⊗ E n − s ) ⊂ − A Θ ⊗ E n − s − ;(1 ⊗ − α s )( − A n Θ ⊗ E n − s − ∧ e n ) ⊂ − A n Θ ⊗ E n − s − M − A Θ ⊗ E n − s − ∧ e n . Hence for γ ∈ ker (1 ⊗ − α ns ), π ( γ ) ∈ ker (1 ⊗ − α n − s − ) ⊂ − A Θ ⊗ E n − s − ∧ e n . But, since byhypothesis H s − ( − A Θ ) = 0, there exists µ ∈ − A Θ ⊗ E n − s − such that(1 ⊗ − α n − s )( µ ) = π ( γ ) . Therefore, π (1 ⊗ − α ns +1 )( µ ∧ e n ) = π ( γ ).We are now left to prove that there exists a µ ∈ − A Θ ⊗ E n − s , such that π (1 ⊗ − α ns +1 )( µ ) = π ( γ ). A kernel relation over − A Θ ⊗ E n − s − having the indices e r ∧ e r ∧ · · · ∧ e r s − such that r i = n for any i looks like ψ p V p + ψ p V p + · + ψ p n − s V p n − s + ψ p n V e n = 0 . Where V e i := (1 − ν i ⊗ ν − i ), p i = e j for any i, j and ψ k ∈ − A Θ . It can be observed that ψ p n = − c p V p − c p V p − · · · − c p s − V p n − s , where c p k are the coefficients of e r ∧ e r ∧ · · · ∧ e r s − ∧ e p k ∧ e n ∈ E ns ∧ e n . Since for a ∈ − A Θ , aV i V n = aV n V i for all i , the above kernelrelation is hence reduced to one of the following form ψ ′ p V p + ψ ′ p V p + · + ψ ′ p n − s V p n − s = 0 . Which has a solution by induction hypothesis, H s ( − A Θ ) = 0 for the n − (cid:3) .2. Case II: • = 1 . We prove for this case by induction over the dimension of torus andusing the techniques of [Q]. The ker (1 ⊗ d s ) ⊂ − A Θ ⊗ E s are represented as diagrams inthe (cid:0) ns (cid:1) -dimensional lattice space. For γ ∈ ker (1 ⊗ d s ), consider a its diagram, Diag ( γ ).Without loss of generality we may assume that Diag ( γ ) is a connected sub-lattice of Z ( ns )assembled by (cid:0) ns (cid:1) dimensional polytopes. The case 2 s = n = 2 corresponds to the quantum2-torus. Here s = 1, hence consider an arbitrary 1-cocycle γ and its Diag ( γ ) ⊂ Z n . Weshall prove that γ is homologous to 0 in a similar way as we did in [Q]. We firstly changethe basis of the Koszul resolution, while the basis of Connes’ Koszul resolution is 1 ⊗ u and 1 ⊗ u for the 2 dimensional case and (1 ⊗ u i ⊗ u i ⊗ . . . ⊗ u i s ) in general, the basis forthe Nest’s Koszul resolution is anti symmetrised (cid:0) ( u i u i . . . u i s ) − ⊗ u i ⊗ u i ⊗ . . . u i s (cid:1) . While Nest’s resolution is computationally convenient, Connes’ basis is more convenient fordiagrammatic approach [Q]. In this subsection, we shall consider the Generalized Connes’Koszul resolution. An element of − A Θ is finitely supported in Z n , hence there exists l > Diag ( γ ) ⊂ B l (¯0). The hyperplanes x = l and x = − l contain Diag ( γ ) betweenthem.A connected component of Diag ( γ ) is an assembly of n -hypercubes with no edges, i.e.diagram of the following form does not exist.A typical kernel diagram in Z looks like as below. It is cubes connected by the kernelrelation, the bullet dots represents a non-zero element of A ⊕ n Θ . LEMMA 5.2. H ( − A Θ ) = 0 .Proof. We prove by induction, let us assume that for all tori of dimension less than n the firstHochschild homology vanishes. Consider Diag ( γ ) ∩ { x = l } , we choose the a l i ⊗ e ∧ e i suchthat the projection of (1 ⊗ α n )( a l i ⊗ e ∧ e i ) on the hyperplane { x = l } kills Diag ( γ ) ∩{ x = l } .This can be done by ordering the non-zero lattice points in the i th dimension and using e ∧ e i with appropriate coefficients to kill them. Thus, what remains is a cycle which isnot supported on { x = l } and is homologous to γ . Repeating this for each hyper plane { x i = d } , d < l we end up with a diagram which represents a cycle γ d homologous to γ and lies entirely in { x = d } for some d ≥ − l . But we observe that γ d is also a 1-cyclefor n − (cid:3) roof of Theorem 1.1. It follows from Lemma 4.1, Lemma 5.1 and Lemma 5.2. (cid:3) Cyclic Homology
Connes introduced an
S, B, I long exact sequence relating the Hochschild and cyclic homol-ogy of an algebra A , ... B −→ HH n ( A ) I −→ HC n ( A ) S −→ HC n − ( A ) B −→ HH n − ( A ) I −→ ... .The cyclic homology Proof of Corollary 1.2 and 1.3.
The Z action on A Θ commutes with the map − α , weobtain the following exact sequence ... B −→ ( HH n ( − A Θ )) Z I −→ ( HC n ( − A Θ )) Z S −→ ( HC n − ( − A Θ )) Z B −→ ( HH n − ( − A Θ )) Z I −→ ... .Using the above long exact sequence we deduce the cyclic homology of A Θ ⋊ Z [Corollary1.2]. The Z invariant periodic cyclic homology for − A Θ is HP even ( − A Θ ) Z = C n and HP odd ( − A Θ ) Z = 0. Similarly for the untwisted case HP even ( A Θ ) Z has dimension X • =2 k (cid:18) n • (cid:19) ,hence; HP even ( A Θ ) Z = C n − and HP odd ( A Θ ) Z = 0 . By using the paracyclic spectral decomposition we have: HP even ( A Θ ⋊ Z ) ∼ = HP even ( A Θ ) Z ⊕ HP even ( − A Θ ) Z = C n ⊕ C n − = C · n − and HP odd ( A Θ ⋊ Z ) = 0 . (cid:3) References [B] Baudry J.; Invariants du tore quantique, Bull. Sci. Math., 134 (2010), 531-547.[BRT] Berest, Y., Ramadoss, A, Tang, X.; The Picard group of a noncommutative algebraic torus. J.Noncommut. Geom. 7 (2013), no. 2, 335356.[C] A. Connes: Noncommutative differential geometry, IHES Publ. Math., 62 (1985), 257-360.[CTY] Chakraborty S., Tang X., Yao Y.; Smooth Connes–Thom isomorphism, cyclic homology, and equi-variant quantisation, preprint, arXiv:1907.09051.[GJ] E. Getzler and J.D.S. Jones: The cyclic homology of crossed product algebras, J. Reine Angew. Math.,445 (1993), 161-174.[N] Nest, R.; Cyclic cohomology of noncommutative tori; Canad. J. Math. 40 (1988), no. 5, 1046-1057.[O] A. Oblomkov: Double affine Hecke algebras of rank 1 and affine cubic surfaces, Int. Math. Res. Not.,no. 18 (2004), 877-912.[Q] Quddus S.; Hochschild and cyclic homology of the crossed product of algebraic irrational rotationalalgebra by finite subgraoups of SL (2 , Z ), J. Algebra 447 (2016), 322-366.[S] Spanier, E.; The homology of Kummer manifolds ; Proc. Amer. Math. Soc. 7 (1956), 155-160.[T] Takhtadjian L.; Non-commutative homology of quantum tori; Funkt. Anal. Pril., 24 (2) (1989), pp.75-76.[W1] Wambst, M.; Hochschild and cyclic homology of the quantum multiparametric torus; J. Pure Appl.Algebra 114 (1997), no. 3, 321-329. W2] Wambst, M.; Complexes de Koszul quantiques.[Quantum Koszul complexes] Ann. Inst. Fourier (Greno-ble) 43 (1993), no. 4, 1089-1156.