On the Smoothness of the Noncommutative Pillow and Quantum Teardrops
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2014), 015, 8 pages On the Smoothness of the Noncommutative Pillowand Quantum Teardrops (cid:63)
Tomasz BRZEZI ´NSKIDepartment of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK
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Received December 03, 2013, in final form February 09, 2014; Published online February 14, 2014http://dx.doi.org/10.3842/SIGMA.2014.015
Abstract.
Recent results by Kr¨ahmer [
Israel J. Math. (2012), 237–266] on smooth-ness of Hopf–Galois extensions and by Liu [arXiv:1304.7117] on smoothness of generalizedWeyl algebras are used to prove that the coordinate algebras of the noncommutative pilloworbifold [
Internat. J. Math. (1991), 139–166], quantum teardrops O ( WP q (1 , l )) [ Comm.Math. Phys. (2012), 151–170], quantum lens spaces O ( L q ( l ; 1 , l )) [ Pacific J. Math. (2003), 249–263], the quantum Seifert manifold O (Σ q ) [ J. Geom. Phys. (2012),1097–1107], quantum real weighted projective planes O ( RP q ( l ; ± )) [ PoS Proc. Sci. (2012),PoS(CORFU2011), 055, 10 pages] and quantum Seifert lens spaces O (Σ q ( l ; − )) [ Axioms (2012), 201–225] are homologically smooth in the sense that as their own bimodules theyadmit finitely generated projective resolutions of finite length. Key words: smooth algebra; generalized Weyl algebra; strongly graded algebra; noncom-mutative pillow; quantum teardrop; quantum lens space; quantum real weighted projectiveplane
Dedicated to Marc Rieffel on the occasion of his 75th birthday.
This note is intended to illustrate the claim that (often) a q -deformation of a non-smooth clas-sical variety or an orbifold produces an algebra which has properties of the coordinate algebraof a non-commutative smooth variety or manifold. More precisely, we say that an algebra B (over an algebraically closed field K ) is homologically smooth or simply smooth provided that asa B -bimodule it has a finitely generated projective resolution of finite length; see [20, Erratum].We prove that several classes of examples of coordinate algebras of q -deformed orbifolds arehomologically smooth. To achieve this aim we use techniques developed in [13], which are appli-cable to principal comodule algebras [6], and those of [14], which are applicable to generalizedWeyl algebras [1]. We summarize these presently.In [13, Corollary 6] Kr¨ahmer gives a criterion for smoothness of quantum homogeneous spaces,which through an immediate extension to a more general class of Hopf–Galois extensions andthen specification to strongly group-graded algebras, provides us with a tool for showing thesmoothness of the noncommutative pillow algebra studied in [2], the quantum lens space algebras O ( L q ( l ; 1 , l )) introduced in [12], the quantum teardrop algebras [5], the coordinate algebrasof quantum real weighted projective planes O ( RP q ( l ; − )) defined in [3], the quantum Seifertmanifold O (Σ q ) [7] and the quantum Seifert lens spaces O (Σ q ( l ; − )) [4]. More specifically, let G be a group (with the neutral element e ). A G -graded algebra A = ⊕ g ∈ G A g is said to be strongly (cid:63) a r X i v : . [ m a t h . QA ] F e b T. Brzezi´nski graded if, for all g, h ∈ G , A g A h = A gh . For such algebras, Kr¨ahmer’s criterion for smoothnesstakes the following form. Criterion 1.
Let A be a strongly G -graded algebra and set B = A e . If the enveloping algebra E A := A ⊗ A op of A is left Noetherian of finite global dimension, then the enveloping algebraof B is also left Noetherian of finite global dimension. Consequently, B is a homologically smoothalgebra. Although [13, Corollary 6] is formulated for quantum homogeneous spaces obtained via sur-jective homomorphism of Hopf algebras with a cosemisimple codomain, the proof extends imme-diately to all faithfully flat Hopf–Galois extensions B ⊆ A or principal comodule algebras suchthat B is a direct summand of A as a B -bimodule. In the case of G -graded algebras, B = A e isa direct summand of A as a B -bimodule, and [16, Proposition AI.3.6] ensures that a strongly G -graded algebra is a principal K G -comodule algebra. In particular A is projective and faithfullyflat as a left and right B -module. The way Criterion 1 is stated indicates its iterative naturewhich is implicit in the proof of [13, Corollary 6]. Note in passing that the assumption aboutthe dimension of E A is not necessary to conclude that E B is left Noetherian.An effective way of checking whether a graded G -algebra A is strongly graded is describedin [16, Section AI.3.2]: Lemma 1. A = ⊕ g ∈ G A g is strongly graded if and only if there exists a function ω : G → A ⊗ A such that ( a ) for all g ∈ G , ω ( g ) ∈ A g − ⊗ A g , ( b ) for all g ∈ G , µ ◦ ω ( g ) = 1 , where µ is the multiplication map of A . Furthermore, if G is a cyclic group, then conditions ( a ) and ( b ) need only be checked fora generator g of G . If ω ( g ) = (cid:80) i ω (cid:48) i ⊗ ω (cid:48)(cid:48) i , satisfies ( a ) and ( b ), then ω is defined by setting ω ( e ) = 1 ⊗ ω (cid:0) g n +1 (cid:1) = (cid:88) i ω (cid:48) i ω ( g n ) ω (cid:48)(cid:48) i , for all n > . A function ω satisfying conditions ( a ) and ( b ) in Lemma 1 is a predecessor of a strong connectionform on a principal comodule algebra; see [6, 8, 11].Let R be an algebra, let p be an element of the centre of R and let π be an automorphism of R .The ( degree-one ) generalized Weyl algebra R ( π, p ) is the extension of R by generators x + , x − subject to the relations, for all r ∈ R , x − x + = p, x + x − = π ( p ) , x ± r = π ± ( r ) x ± ;see [1]. In [14, Theorem 4.5] Liu gives the following criterion of smoothness of a generalizedWeyl algebra over the polynomial algebra. Criterion 2.
Let R = K [ a ] be a polynomial algebra and an automorphism π : K [ a ] → K [ a ] bedetermined by π ( a ) = κa + χ . Then the generalized Weyl algebra R ( π, p ) is homologically smoothwith homological dimension if and only if the polynomial p ∈ K [ a ] has no multiple roots. Furthermore, Liu proves that if the smoothness Criterion 2 is satisfied, then A = R ( π, p )is a twisted Calabi–Yau algebra of dimension 2 with the Nakayama (twisting) automorphism ν : A → A given by ν ( x ± ) = κ ± x ± and ν ( a ) = a . This means that the Hochschild cohomologyof A with values in its enveloping algebra is trivial in all degrees except degree 2, where it isequal to A with the A -bimodule structure a · b · a (cid:48) = abν ( a (cid:48) ).The reader should observe that, except for some special cases, the algebras described herein-after are not smooth whenever the deformation parameters λ or q are equal to 1. By noting thisthey will fully grasp the main message of this note, namely that deformation may (and quiteoften does) result in smoothing classically singular objects.n the Smoothness of the Noncommutative Pillow and Quantum Teardrops 3 Throughout we work with associative complex ∗ -algebras with identity. We write E A for theenveloping algebra A ⊗ A op of A . We often use the q -Pochhammer symbol which, for an inde-terminate x and a complex number q , is defined as( x ; q ) n := n − (cid:89) m =0 (cid:0) − q m x (cid:1) . Let λ = e πiθ , where θ is an irrational number. Recall that the coordinate ∗ -algebra O ( T θ ) ofthe noncommutative torus is generated by unitaries U, V , such that
U V = λV U ; see [18]. Theinvolutive algebra automorphism given by σ : O (cid:0) T θ (cid:1) → O (cid:0) T θ (cid:1) , U (cid:55)→ U ∗ , V (cid:55)→ V ∗ , makes O ( T θ ) into a Z -graded algebra. The fixed point (or degree-zero) subalgebra O ( P θ ) isgenerated by U + U ∗ and V + V ∗ . It has been introduced and studied from a topologicalpoint of view in [2] (see also [9, Section 3.7]) as a deformation of the coordinate algebra of the pillow orbifold [19, Chapter 13] (an orbifold rather than manifold since, classically, the Z -actiondetermined by the automorphism σ is not free). Theorem 1. O ( T θ ) is a strongly Z -graded algebra and the noncommutative pillow algebra O ( P θ ) is homologically smooth. Proof .
Setˆ x = U − U ∗ , ˆ y = V − V ∗ , ˆ z = U V ∗ − U ∗ V. Note that σ (ˆ x ) = − ˆ x , σ (ˆ y ) = − ˆ y and σ (ˆ z ) = − ˆ z , i.e. all these are homogeneous elementsof O ( T θ ) with the Z -degree 1. A straightforward calculation affirms that these elements satisfythe following relationˆ x + ˆ y − ¯ λ ˆ z − ˆ xz ˆ y = 2 (cid:0) ¯ λ − (cid:1) , where z = U V ∗ + U ∗ V ∈ O ( P θ ). Therefore, the mapping ω : Z → O ( T θ ) ⊗ O ( T θ ), defined as ω (0) = 1 ⊗ ω (1) = 12 (cid:0) ¯ λ − (cid:1) (cid:0) ˆ x ⊗ ˆ x + ˆ y ⊗ ˆ y − ¯ λ ˆ z ⊗ ˆ z − ˆ xz ⊗ ˆ y (cid:1) , satisfies conditions ( a ) and ( b ) in Lemma 1, and O ( T θ ) is a strongly Z -graded algebra.Both O ( T θ ) and EO ( T θ ) can be understood as iterated skew Laurent polynomial rings andhence they are left Noetherian by [15, Theorem 1.4.5]. Furthermore, the global dimension of thelatter is less than or equal to 4 by [15, Theorem 7.5.3]. Therefore, the noncommutative pillowalgebra O ( P θ ) is homologically smooth by Criterion 1. (cid:4) In short, Theorem 1 means that for the irrational θ (or, more generally, for any real θ ∈ (0 , \ { } ) the action of Z on the noncommutative torus is free despite the fact that thecorresponding action on the classical level is not free. The set of fixed points corresponds toa manifold rather than an orbifold. T. Brzezi´nski Here we deal with three (classes of) complex ∗ -algebras given in terms of generators and relations.The coordinate algebra of the quantum three-sphere, O ( S q ), is generated by α and β suchthat αβ = qβα, αβ ∗ = qβ ∗ α, ββ ∗ = β ∗ β,αα ∗ = α ∗ α + (cid:0) q − − (cid:1) ββ ∗ , αα ∗ + ββ ∗ = 1 , (1)where q ∈ (0 , l , the coordinate algebra of the quantum lensspace O ( L q ( l ; 1 , l )) is a ∗ -algebra generated by c and d subject to the following relations: cd = q l dc, cd ∗ = q l d ∗ c, dd ∗ = d ∗ d, cc ∗ = (cid:0) dd ∗ ; q (cid:1) l , c ∗ c = (cid:0) q − dd ∗ ; q − (cid:1) l , see [12]. Finally, for a positive integer l , the coordinate algebra of the quantum teardrop O ( WP q (1 , l )) is the ∗ -algebra generated by a and b subject to the following relations a ∗ = a, ab = q − l ba, bb ∗ = q l a (cid:0) a ; q (cid:1) l , b ∗ b = a (cid:0) q − a ; q − (cid:1) l ;see [5]. These algebras form a tower O ( WP q (1 , l )) (cid:44) → O ( L q ( l ; 1 , l )) (cid:44) → O ( S q ) with embeddings a (cid:55)→ dd ∗ , b (cid:55)→ cd and c (cid:55)→ α l , d (cid:55)→ β , respectively. We thus can and will think of O ( WP q (1 , l ))and O ( L q ( l ; 1 , l )) as subalgebras of O ( S q ). O ( S q ) is a Z l -graded algebra with grading givenby deg( α ) = 1, deg( α ∗ ) = l −
1, deg( β ) = deg( β ∗ ) = 0, and the above embedding identifiesthe degree-zero part of O ( S q ) with O ( L q ( l ; 1 , l )). By [5, Theorem 3.3], the latter is a strongly Z -graded algebra with grading provided by deg( c ) = deg( d ∗ ) = 1, deg( c ∗ ) = deg( d ) = − O ( WP q (1 , l )).That O ( WP q (1 , l )) is homologically smooth can be argued as follows. O ( S q ) is a coordinatealgebra of the quantum group SU(2) and thus EO ( S q ) is left Noetherian and has a finite globaldimension; see [10]. Hence, if it were a strongly Z l -graded algebra, then EO ( L q ( l ; 1 , l )) would beleft Noetherian and would have a finite global dimension (so, in particular O ( L q ( l ; 1 , l )) would behomologically smooth) by Criterion 1. Since, in turn O ( L q ( l ; 1 , l )) is a strongly graded algebra,Criterion 1 would imply smoothness of the teardrop algebra O ( WP q (1 , l )). This arguing leads to: Theorem 2. O ( S q ) is a strongly Z l -graded algebra with the degree-zero subalgebra isomorphicto O ( L q ( l ; 1 , l )) . Consequently, both O ( L q ( l ; 1 , l )) and O ( WP q (1 , l )) are homologically smoothalgebras. Proof .
The case l = 1 is dealt with in [13], the remaining cases follow from Lemma 2.
For all integers l > , there exist elements ω (1) ∈ O ( S q ) l − ⊗ O ( S q ) such that µ ( ω (1)) = 1 . Proof .
Set: ω (1) = x α l − ⊗ α ∗ l − + l − (cid:88) p =1 y p a p − α ∗ ⊗ α, where x , y , . . . , y l − ∈ C are to be determined and a = ββ ∗ = dd ∗ . Then ω (1) ∈ O ( S q ) l − ⊗O ( S q ) . Using (1) one finds that α m α ∗ m = (cid:0) a ; q (cid:1) m =: m (cid:88) p =0 c mp a p , (2)n the Smoothness of the Noncommutative Pillow and Quantum Teardrops 5where c mp are the appropriate q -binomial coefficients (defined by the second equality in (2)). Inview of (1), the condition µ ( ω (1)) = 1 leads to x l − (cid:88) p =0 c l − p a p + l − (cid:88) p =0 y p +1 a p − q − l − (cid:88) p =1 y p a p = 1 . By comparing the powers of a , this is converted into an inhomogeneous system of l equationswith unknown x , y , . . . , y l − , whose determinant is∆ l :1 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c l − · · · c l − − q − · · · c l − − q − · · · c l − l − · · · − q − c l − · · · − q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ∆ l :1 can be evaluated by expanding by the first column to give∆ l :1 = ( − l − (cid:0) q − l − c l − + q − l − c l − + · · · + c l − l − (cid:1) = (cid:0) − q (cid:1) l − l − (cid:89) p =1 (cid:0) − q p (cid:1) (cid:54) = 0 . The final equality follows from the definition of the q -binomial coefficients (2). This proves theexistence of ω (1) as stated. (cid:4) Since 1 is a generator of Z l , Lemma 2 ensures the existence of mappings ω : Z l → O ( S q ) ⊗O ( S q ) that satisfy conditions ( a ) and ( b ) in Lemma 1. Hence O ( S q ) is a strongly Z l -gradedalgebra, and the second assertion of the theorem follows by Criterion 1. (cid:4) Therefore, for any q ∈ (0 ,
1) the action of Z l on the quantum three-sphere described aboveis free despite the fact that the corresponding action on the classical level is not free (unless,obviously, l = 1). The fixed points correspond to a manifold rather than an orbifold. RP q ( l ; − )and quantum Seifert lens spaces For a positive integer l , the coordinate ∗ -algebra O ( RP q ( l ; − )) of the odd quantum weightedreal projective plane is generated by a , b , c − which satisfy the relations: a = a ∗ , ab = q − l ba, ac − = q − l c − a, b = q l ac − , bc − = q − l c − b,bb ∗ = q l a (cid:0) a ; q (cid:1) l , b ∗ b = a (cid:0) q − a ; q − (cid:1) l , b ∗ c − = q − l (cid:0) q − a ; q − (cid:1) l b,c − b ∗ = q l b (cid:0) a ; q (cid:1) l , c − c ∗− = (cid:0) a ; q (cid:1) l , c ∗− c − = (cid:0) q − a ; q − (cid:1) l , see [3]. To prove homological smoothness of O ( RP q ( l ; − )) we make use of Criterion 1 and builda tower of strongly graded algebras with O ( RP q ( l ; − )) as the foundation.The coordinate ∗ -algebra of the quantum Seifert manifold O (Σ q ) is generated by a centralunitary ξ and elements ζ , ζ such that ζ ζ = qζ ζ , ζ ζ ∗ = ζ ∗ ζ + (cid:0) q − − (cid:1) ζ ξ, ζ ζ ∗ + ζ ξ = 1 , ζ ∗ = ζ ξ. (3) T. Brzezi´nskiIt has been shown in [7, proof of Proposition 5.2] that O (Σ q ) can be understood as the degree-zero part of a Z -grading of O ( S q )[ u, u − ], where u − = u ∗ and O ( S q ) is the coordinate ∗ -algebraof the equatorial Podle´s sphere [17], generated by z and self-adjoint z such that z z = qz z , z z ∗ = z ∗ z + (cid:0) q − − (cid:1) z , z z ∗ + z = 1 . (4)The Z -grading of O ( S q )[ u, u − ] is determined by setting, for all monomials w of degree k inthe basis { z r z s , z ∗ r z s | r, s ∈ N } of O ( S q ), deg( wu m ) = ( k + m ) mod 2. O (Σ q ) can be identifiedwith the degree-zero part of O ( S q )[ u, u − ] by ∗ -embedding ζ i (cid:55)→ z i u , ξ (cid:55)→ u − . Thanks to thelast of equations (4), the function ω : Z → O (cid:0) S q (cid:1)(cid:2) u, u − (cid:3) ⊗ O (cid:0) S q (cid:1)(cid:2) u, u − (cid:3) , (cid:55)→ ⊗ , (cid:55)→ z ⊗ z ∗ + z ⊗ z , satisfies conditions (a) and (b) in Lemma 1, hence O ( S q )[ u, u − ] is a strongly Z -graded algebra.Since EO ( S q ) is Noetherian, and there is a surjective ∗ -algebra homomorphism O ( S q ) → O ( S q ), α (cid:55)→ z , β (cid:55)→ z ∗ , both EO ( S q ) and EO ( S q )[ u, u − ] and hence also EO (Σ q ) are Noetherian.As explained in [4], O (Σ q ) is a Z l -graded algebra with grading given bydeg( ζ ) = 1 , deg( ζ ∗ ) = l − , deg( ζ ) = deg( ξ ) = 0 . The degree-zero part of O (Σ q ) is isomorphic to the ∗ -algebra O (Σ q ( l ; − )) generated by x , y andcentral unitary z subject to the following relations y ∗ = yz, xy = q l yx, xx ∗ = (cid:0) y z ; q (cid:1) l , x ∗ x = (cid:0) q − y z ; q − (cid:1) l . The embedding of O (Σ q ( l ; − )) into O (Σ q ) is given by x (cid:55)→ ζ l , y (cid:55)→ ζ and z (cid:55)→ ξ . Thesimilarity of relations (3) and (1) leads immediately to equations (2) with α replaced by ζ and a = ζ ξ . This allows one to use the same arguments as in Lemma 2 to prove that there exist x , y , . . . , y l − ∈ C such that ω (1) = x ζ l − ⊗ ζ ∗ l − + l − (cid:88) i =1 y i a i − ζ ∗ ⊗ ζ ∈ O (cid:0) Σ q (cid:1) l − ⊗ O (cid:0) Σ q (cid:1) l , has the required property µ ( ω (1)) = 1. Therefore, O (Σ q ) is a strongly graded Z l -algebra.Finally, it is proven in [4] that O (Σ q ( l ; − )) is a strongly Z -graded algebra with gradinggiven by deg( x ) = deg( y ) = 1, deg( x ∗ ) = − z ) = −
2. The degree-zero subalgebraof O (Σ q ( l ; − )) can be identified with the coordinate algebra of weighted real projective plane O ( RP q ( l ; − )) via the map a (cid:55)→ y z , b (cid:55)→ xyz and c − (cid:55)→ x z .Summarizing, we have presented in this section a tower of ∗ -algebras O (cid:0) RP q ( l ; − ) (cid:1) (cid:44) → O (cid:0) Σ q ( l ; − ) (cid:1) (cid:44) → O (cid:0) Σ q (cid:1) (cid:44) → O (cid:0) S q (cid:1)(cid:2) u, u − (cid:3) . (5)The second, third and fourth terms are strongly group graded algebras. Each antecedent termis the degree-zero part of the subsequent one. Since the enveloping algebra of O ( S q )[ u, u − ] isNoetherian, so are the enveloping algebras of all its predecessors. By [14, Corollary 4.6] theglobal dimension of EO ( S q ) is finite, hence so is the global dimension of EO ( S q )[ u, u − ], and,by Criterion 1, the global dimensions of enveloping algebras of all its predecessors in (5). Thisproves the following Theorem 3.
The algebras O (Σ q ) , O ( RP q ( l ; − )) and O (Σ q ( l ; − )) are homologically smooth. n the Smoothness of the Noncommutative Pillow and Quantum Teardrops 7 RP q ( l ; +)and teardrops (revisited) Let k be a natural number and l be a positive integer. Write A ( k, l ) for the ∗ -algebra generatedby a and b subject to the following relations a ∗ = a, ab = q − kl ba, bb ∗ = q kl a k (cid:0) a ; q (cid:1) l , b ∗ b = a k (cid:0) q − a ; q − (cid:1) l . If k and l are coprime then A ( k, l ) is the coordinate algebra of the quantum weighted projectiveline or the quantum spindle O ( WP q ( k, l )) introduced in [5]. The special case k = 1 is simply the quantum teardrop ; see Section 2.2. For l odd, A (0 , l ) is the coordinate algebra of the quantumweighted even real projective plane O ( RP q ( l ; +)) introduced in [3]. The following theorem isa consequence of Criterion 2. Theorem 4.
The algebras A ( k, l ) are homologically smooth ( of dimension if and only if k = 0 , . Proof .
We only need to observe that each A ( k, l ) is a generalized Weyl algebra over the poly-nomial algebra C [ a ] given by the automorphism π ( a ) = q l a , element p = a k l (cid:81) m =1 (1 − q − m a ) andgenerators x − = b , x + = b ∗ . Since p has no multiple roots if and only if k = 0 ,
1, the assertionfollows by Criterion 2. (cid:4)
Furthermore, for k = 0 , A ( k, l ) are twisted Calabi–Yau algebras with the twisting auto-morphism ν ( b ) = q − l b , ν ( b ∗ ) = q l b ∗ and ν ( a ) = a . Hence they enjoy the Poincar´e duality inthe sense of Van den Bergh [20]. Acknowledgements
I would like to thank Ulrich Kr¨ahmer for discussions, Li-Yu Liu for bringing reference [14] tomy attention, and Piotr M. Hajac and the referees for helpful comments. I am grateful toFields Institute for Research in Mathematical Sciences in Toronto, where these results were firstpresented, for creating excellent research environment and for support.
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