Pointed Hopf actions on quantum generalized Weyl algebras
aa r X i v : . [ m a t h . R A ] J a n POINTED HOPF ACTIONS ON QUANTUM GENERALIZED WEYL ALGEBRAS
JASON GADDIS AND ROBERT WON
Abstract.
We study actions of pointed Hopf algebras in the Z -graded setting. Our main result classifiesinner-faithful actions of generalized Taft algebras on quantum generalized Weyl algebras which respect the Z -grading. We also show that generically the invariant rings of Taft actions on quantum generalized Weylalgebras are commutative Kleinian singularities. Introduction
The goal of this work is to study Hopf actions in the setting of Z -graded algebras. The Weyl algebra isan example of such an algebra, but the Weyl algebra has no finite quantum symmetry [9]. Instead, we studygeneralized Weyl algebras (GWAs), specifically quantum GWAs. By work of Su´arez-Alvarez and Vivas, theautomorphism groups of quantum GWAs are well-understood [17]. Here we study actions of (generalized)Taft algebras on quantum GWAs that respect their Z -grading.Actions of Taft algebras on various families of algebras have been considered in a variety of works. Actionson finite-dimensional algebras with a single generator were studied by Montgomery and Schneider [15]. Theseresults were recently generalized by Cline [7]. Furthermore, Centrone and Yasumura studied actions ongeneral finite-dimensional algebras [3], while Bahturin and Montgomery classified actions on matrix algebras[2]. Kinser and Walton considered Taft algebra actions on path algebras of quivers [12]. In [11], the presentauthors along with Yee classified actions on quantum planes and quantum Weyl algebras. The first authoralong with Cline considered more general actions on quantum affine spaces, quantum matrices, and quantizedWeyl algebras [8]. More generally, Chen, Wang, and Zhang studied actions of nonsemisimple Hopf algebrason Artin–Schelter regular algebras in characteristic p [6].This work provides a unique perspective on Taft algebra actions. In particular, we obtain actions onquantum planes k q [ u, v ] that respect the grading deg( u ) = 1 and deg( v ) = −
1, and these actions are distinctfrom those obtained in [11]. Our primary goal is to classify actions by generalized Taft algebras on quantumGWAs. In some sense, this is a natural extension of work on Hopf actions in the setting of Artin–Schelterregular algebras [5, 13]. By a result of Liu [14], these GWAs are twisted Calabi–Yau, and so this work maybe seen as an extension of the study of quantum symmetries to the Z -graded setting.In Section 2, we provide background information on (pointed) Hopf algebras and (quantum) generalizedWeyl algebras. Since the GWAs A we consider have base ring A = k [ t ] (the commutative polynomial ring Mathematics Subject Classification.
Primary 16T05, Secondary 16W50.
Key words and phrases.
Generalized Taft algebra, generalized Weyl algebra, Hopf algebra actions. n one variable), in Section 3 we consider generalized Taft actions on k [ t ]. We provide a full classificationof inner-faithful actions in this setting without any assumptions on linearity of the action (Proposition 3.4).Our main results, Theorems 4.18 and 4.22, fully classify inner-faithful actions of generalized Taft algebras onquantum GWAs at a root of unity q = ± Z -grading. Section 4 is devoted to proving thisclassification. In Section 5 we compute the invariants for Taft actions on quantum GWAs under Taft algebraactions and show that generically the invariant rings are (commutative) Kleinian singularities (Theorem 5.1).2. Background
Throughout, all algebras are associative k -algebras over a base field k of characteristic zero.2.1. Hopf algebras.
A Hopf algebra is a bialgebra with antipode. Given a Hopf algebra H , we denote thecomultiplication, counit, and antipode by ∆, ǫ , and S , respectively. An element g ∈ H is grouplike if ∆( g ) = g ⊗ g . Given grouplikes g, h ∈ H , an element x ∈ H is ( g, h ) -skew-primitive provided ∆( x ) = g ⊗ x + x ⊗ h .For any ( g, h )-skew-primitive element x , ǫ ( x ) = 0 and S ( x ) = − g − xh − .An algebra A is a left H -module algebra if A is a left H -module with action h ⊗ a h ( a ) such that h (1 A ) = ǫ ( h )1 A and h. ( ab ) = P h ( a ) h ( b ) for all a, b ∈ A and h ∈ H . Equivalently, A is an algebra objectin the monoidal category of left H -modules.We say that a Hopf algebra H is pointed if every simple H -comodule is one-dimensional. A partial butextensive classification of finite-dimensional pointed Hopf algebras has been obtained by Andruskiewitschand Schneider [1]. Amongst the most fundamental examples of pointed Hopf algebras are the Taft algebras.Let θ ∈ k , let m, n ∈ N such that m > m | n , and let λ ∈ k be a primitive m th root of unity. Thealgebra T n ( λ, m, θ ) := k h x, g | g n − , x m − θ ( g m − , gx − λxg i is a generalized Taft algebra [16]. It is a Hopf algebra in which g is grouplike and x is ( g, m = n and γ = 0, we say that T n ( λ ) = T n ( λ, n,
0) is a
Taft algebra . We will focus on the coradicallygraded generalized Taft algebra T n ( λ, m, H be a Hopf algebra and M an H -module. We say M is an inner-faithful H -module (or that theaction is inner-faithful) if IM = 0 for every Hopf ideal I of H . If A is an H -module algebra, then we saythat A is inner-faithful if it is inner-faithful as an H -module. By [7, Corollary 3.7], a T = T n ( λ, m, M is inner-faithful if and only if h g i acts faithfully and x ( M ) = 0. If m = n , so T is a Taft algebra, then x ( M ) = 0 whenever h g i does not act faithfully (see [12, Lemma 2.5],[2, Corollary 3.2]). Thus, in this case,the faithfulness condition on h g i is superfluous.2.2. Generalized Weyl algebras.Definition 2.1.
Let R be a k -algebra, let σ = ( σ , . . . , σ p ) be a p -tuple of commuting automorphisms of R ,and let h = ( h , . . . , h p ) be a p -tuple of central regular elements in R such that σ i ( h j ) = h j for all distinct , j ∈ { , . . . , p } . With this data, we define the generalized Weyl algebra (GWA) of rank p as the k -algebragenerated over R as an algebra by u = ( u , . . . , u p ) and v = ( v , . . . , v p ) with relations u i r = σ i ( r ) x i v i r = σ − i ( r ) y i for all i ∈ { , . . . , p } u i v i = σ i ( h i ) v i u i = h i for all i ∈ { , . . . , p } [ u i , u j ] = [ v i , v j ] = [ u i , v j ] = 0 for all distinct i, j ∈ { , . . . , p } . We denote this algebra by R ( u , v , σ, h ) . A rank p GWA R ( u , v , σ, h ) is naturally Z p -graded by setting deg( r ) = for all r ∈ R , deg( u i ) = e i , anddeg( v i ) = − e i . Definition 2.2.
We say the GWA R ( u , v , σ, h ) is a quantum GWA if R = k [ t ] and there exist scalars q = ( q , . . . , q p ) ∈ ( k \{ , } ) p such that σ i ( t ) = q i t for all i ∈ { , . . . , p } . Let ord( q i ) denote the order of q i in the multiplicative group k × . We use the notation R ( u, v, σ, h ) for a GWA of rank one. Our primary focus will be on quantum GWAsof rank one; later we extend these results to higher rank quantum GWAs. The quantum planes are quantumGWAs over k [ t ] with h = t . The quantum Weyl algebras are quantum GWAs over k [ t ] with h = t −
1. In [11],Yee and the authors classified linear actions of Taft algebras on quantum planes and quantum Weyl algebras.This was extended to linear actions of certain generalized Taft algebras by Cline and the first author [8].In this work, we consider actions of a different type. In particular, we consider actions that respect the Z -grading on quantum GWAs. More specifically, if A is a quantum GWA, then we consider actions bygeneralized Taft algebras T such that A and A − i ⊕ A i are T -modules for every i ∈ N .2.3. Automorphisms of quantum GWAs.
We recall the description of the automorphism group of aquantum GWA given by Su´arez-Alvarez and Vivas [17].Let A = k [ t ]( u, v, σ, h ) be a quantum GWA and assume that h = P h i t i is not a unit. Let supp( h ) = { i | h i = 0 } ⊂ Z and ℓ = gcd { i − j | h i h j = 0 } . If h is a monomial, set C ℓ = k × and otherwise let C ℓ bethe subgroup of k × consisting of ℓ th roots of unity. If ( γ, µ ) ∈ C ℓ × k × and i ∈ supp( h ), then there is anautomorphism η γ,µ of A such that η γ,µ ( t ) = γt, η γ,µ ( v ) = µv, η ( u ) = µ − γ i u. (2.3)The definition of η γ,µ is independent of the choice of i (see [10, Remark 2.4]), hence we typically take i = deg t ( h ).Let G be the subgroup of Aut( A ) generated by { η γ,µ | ( γ, µ ) ∈ C ℓ × k × } . When q = −
1, then Aut( A ) = G [17, Theorem B]. If q = −
1, then there is an order 2 automorphism Ω defined byΩ( t ) = − t, Ω( v ) = u, Ω( u ) = v. hen there is a short exact sequence,0 −→ G −→ Aut( A ) −→ Z / Z −→ . In this case, every automorphism of A is either some η γ,µ or else Ω ◦ η γ,µ [10, Proposition 2.7]. Work in [4]classifies the automorphism groups for certain higher rank quantum GWAs.3. Generalized Taft actions on k [ t ]Before we can study generalized Taft actions on quantum GWAs, we first consider actions on the polyno-mial ring k [ t ]. Throughout this section, let m and n be positive integers such that m > m | n . Let λ be a primitive m th root of unity. Set T = T n ( λ, m,
0) = k h x, g | g n − , x m , gx − λxg i . For γ ∈ k \{ , } and k ∈ N , define the k th γ -number to be[ k ] γ = 1 − γ k − γ = k − X j =0 γ j . Observe that if γ is an s th root of unity, and k ≡ m (mod s ), then [ k ] γ = [ m ] γ . For f ∈ k [ t ], let δ γ ( f )denote the γ -derivative of f , defined by δ γ ( t k ) = [ k ] γ t k − and extended linearly. The γ -derivative satisfies a γ -analog of the Leibniz rule, namely δ γ ( f ( t ) g ( t )) = δ γ ( f ( t )) g ( t ) + f ( γt ) δ γ ( g ( t )) = δ γ ( f ( t )) g ( γt ) + f ( t ) δ γ ( g ( t )) . We remark that δ γ is nilpotent of index s . If k < s , then δ ord( γ ) γ ( t k ) = 0 and if k ≥ s then because [ s ] γ = 0we have δ ord( γ ) γ ( t k ) = ord( γ ) − Y i =0 [ i ] γ t k − ord( γ ) = 0 . In order to classify T actions on k [ t ], we first need two technical lemmas. Lemma 3.1.
Suppose k [ t ] is a left T -module algebra and suppose that g ( t ) = γt for some primitive s th rootof unity γ ∈ k × . Write x ( t ) = φ ∈ k [ t ] where φ = P di =0 φ i t i with φ d = 0 . If γ = 1 , then φ = 0 . In general, x ( t k ) = t k − φ k − X j =0 γ j = [ k ] γ φt k − = φδ γ ( t k ) . (3.2) Proof.
We first consider the case when γ = 1. Since k [ t ] is a T -module algebra, then0 = ( gx − λxg )( t ) = (1 − λ ) φ, and since λ = 1, we have φ = 0.In general, when γ = 1, we have x ( t ) = g ( t ) x ( t ) + x ( t ) t = γtφ + φt = ( γ + 1) φt. Then (3.2) follows by induction. (cid:3) emma 3.3. Let γ = 1 be a primitive s th root of unity and let φ ∈ k [ t ] be a polynomial of degree d . Supposethat k [ t ] is a left T -module algebra where g ( t ) = γt and x ( t ) = φ . Let f ∈ k [ t ] be a polynomial of degree k ≥ , all of whose nonzero terms occur in degrees which are equivalent to k modulo s . Then x ( f ) = [ k ] γ φ e f where e f is a polynomial of degree k − all of whose nonzero terms occur in degrees which are equivalent to k − modulo s .Proof. This follows from Lemma 3.1. (cid:3)
Suppose k [ t ] is a T -module algebra. Since g ∈ T is grouplike then it acts as an automorphism on k [ t ].That is, g ( t ) = γt + κ for some γ ∈ k × and κ ∈ k . If γ = 1, then after replacing t by t + κγ − , we have g ( t ) = γt . Since g has finite order, then necessarily either g is the identity or else γ = 1. Hence, in thenext result we make the assumption that g acts by scalar multiplication on t . However, we do not make anyassumption about the action of x . The next result fully classifies inner-faithful generalized Taft actions on k [ t ]. Proposition 3.4.
Let T = T n ( λ, m, be a generalized Taft algebra. Let γ ∈ k \ { , } and φ ∈ k [ t ] be anonzero polynomial of degree d . If k [ t ] is a T -module algebra with g ( t ) = γt and x ( t ) = φ , then(1) γ is a root of unity of order m,(2) λ = γ d − and so gcd( d − , m ) = 1 , and(3) supp( φ ) ⊆ { d, d − m, d − m, . . . } .Furthermore, the action is inner-faithful if and only if m = n .Conversely, setting g ( t ) = γt and x ( t ) = φ subject to (1)–(3) and extending naturally via the coproductdefines an action which makes k [ t ] a T -module algebra.Proof. Suppose first that k [ t ] is a T -module algebra with g ( t ) = γt and x ( t ) = φ . Since g n = 1, we have g n ( t ) = γ n t = t and hence γ is a root of unity of order s | n . Furthermore we have0 = ( gx − λxg )( t ) = g ( φ ( t )) − λx ( γt ) = φ ( γt ) − λγφ ( t ) . Writing φ = P di =0 φ i t i , we must have P di =0 ( γ i − λγ ) φ i t i = 0. Therefore, λ = γ i − for all i ∈ supp( φ ) . (3.5)In particular, λ = γ d − . Hence, ord( λ ) = m must divide s and so gcd( d − , s ) = s/m .If d ≡ s ), then by (3.5), λ = γ − and so γ is a root of unity of order m . Otherwise, d >
0. ByLemma 3.3 and induction, we compute that x m ( t ) = m − Y i =0 [1 + i ( d − γ ! f (3.6) or some f ∈ k [ t ], f = 0. Since x m ( t ) = 0, then there exists some 0 ≤ i ≤ m − i ( d − γ = 0.This happens if and only if 1 = γ i ( d − = γλ i . Therefore, s | m and since m | s , we conclude that m = s . The action is inner-faithful if and only if g acts faithfully and x ( A ) = 0. This happens if and only if s = m = n .Conversely, suppose that conditions (1)–(3) hold. We will show that k [ t ] admits the structure of a T -module algebra where g ( t ) = γt and x ( t ) = φ , with the action extended to all of k [ t ] naturally via thecoproduct of T . It suffices to show that g n ( t ) = t , ( gx − λxg )( t ) = 0, and x m ( t ) = 0.Since γ is a root of order m and m | n , it is clear that g n ( t ) = γ n t = t . Since we are extending the actionto all of k [ t ] via the coproduct of T , then as above,( gx − λxg )( t ) = φ ( γt ) − λγφ ( t )and this is 0 by conditions (2) and (3).Finally, if d ≡ m ), then x ( t ) = φ has degree d and so by the same computation as in Lemma 3.3, x ( t ) = 0. Since m >
1, we have x m ( t ) = 0. Otherwise, d > f ∈ k [ t ]. Since gcd( d − , m ) = 1, one of the m factors in the product isequal to 0, and hence x m ( t ) = 0. (cid:3) Generalized Taft actions on quantum generalized Weyl algebras
Fix a quantum GWA A = k [ t ]( u, v, σ, h ) where σ ( t ) = qt for some root of unity q = 1 and a generalizedTaft algebra T = T n ( λ, m, T on A that are Z -graded in the sensethat T ( A ) ⊂ A and T ( A − i ⊕ A i ) ⊂ A − i ⊕ A i for all i ∈ N . Hence, these actions extend the actions of T on k [ t ] studied above. This is also a natural restriction because it generalizes group actions that are Z -gradedup to a group automorphism of Z . By the classification of Su´arez-Alvarez and Vivas [17, Proposition 2.7],every automorphism of A is in fact Z -graded in this sense.Since g ∈ T is grouplike, then it acts on A as an automorphism of the form η γ,µ as in equation (2.3) or,in the case q = −
1, as either η γ,µ or Ω ◦ η γ,µ . To be Z -graded, considering the action of x on A , we musthave x ( t ) ∈ A and x ( u ) , x ( v ) ∈ A − ⊕ A . We summarize our standing hypotheses as follows. Hypothesis 4.1.
Let h = P Di =0 h i t i ∈ k [ t ] be a polynomial of degree D > , let q ∈ k × be a root of unity oforder ord( q ) > , and let σ ∈ Aut( k [ t ]) be defined by σ ( t ) = qt . We consider the rank one quantum GWA A = k [ t ]( u, v, σ, h ) = k [ t ] h u, v i ut = qtu vu = h = P Di =0 h i t i vt = q − tv uv = σ ( h ) = P Di =0 h i q i t i . Let m and n be positive integers such that m > and m | n , and let λ be a primitive m th root of unity. Let T = T n ( λ, m, be a generalized Taft algebra and assume that A is a Z -graded T -module algebra in the sensethat T ( A ) ⊂ A and T ( A − i ⊕ A i ) ⊂ A − i ⊕ A i for all i ∈ N . That is: for some µ ∈ k × and γ an s th root of unity, g acts as an automorphism of the form η γ,µ as inequation (2.3) or, in the case q = − , as either η γ,µ or Ω ◦ η γ,µ , and • x acts on A via x ( t ) = φ ( t ) = d X i =0 φ i t i , x ( u ) = α u + α v, x ( v ) = α u + α v, (4.2) where for each ≤ i ≤ d , φ i ∈ k with φ d = 0 and α ij ∈ k [ t ] for i, j ∈ { , } . For most of this section we make no restriction on q but assume that g acts as an automorphism η γ,µ asin equation (2.3). If q = −
1, then this is automatic. For the case q = −
1, we consider the case where g actsas Ω ◦ η γ,µ in Theorem 4.22. Lemma 4.3.
Assume Hypothesis 4.1 with g acting as an automorphism of the form η γ,µ . Then α = α =0 . Moreover, if γ = 1 , then x ( t ) = 0 and hence the action of T of A is not inner-faithful.Proof. Since T acts on A , then the action of x necessarily preserves the relations of A . In particular,0 = x ( vu − h ) = ( µσ − ( α ) + α ) h + µσ − ( α ) v + α u − x ( h )0 = x ( uv − σ ( h )) = ( µ − γ D σ ( α ) + α ) σ ( h ) + α v + µ − γ D σ ( α ) u − x ( σ ( h )) . That α = α = 0 now follows from the Z -grading on A .Suppose γ = 1. Then x ( t ) = 0 by Lemma 3.1. Now x ( u ) = α u and so x m ( u ) = α m u . Thus, x m ( u ) = 0if and only if α = 0. Similarly, x m ( v ) = 0 if and only if α = 0. In this situation, the action of T is notinner-faithful. (cid:3) Suppose that T and A satisfy Hypothesis 4.1. If g acts as an automorphism η γ,µ , then in light ofLemma 4.3, the following equations are satisfied:0 = x ( vu − h ) = ( µσ − ( α ) + α ) h − x ( h ) (4.4)0 = x ( uv − σ ( h )) = ( µ − γ D σ ( α ) + α ) σ ( h ) − x ( σ ( h )) (4.5)0 = ( gx − λxg )( u ) = g ( α u ) − λx ( µ − γ D u ) = µ − γ D ( g ( α ) − λα ) u (4.6)0 = ( gx − λxg )( v ) = g ( α v ) − λx ( µv ) = µ ( g ( α ) − λα ) v (4.7)0 = ( gx − λxg )( t ) = g ( φ ( t )) − λx ( γt ) = φ ( γt ) − λγφ ( t ) (4.8)0 = x ( ut − qtu ) = ( µ − γ D φ ( qt ) − qφ ( t )) u + α q (1 − γ ) tu (4.9)0 = x ( vt − q − tv ) = ( µφ ( q − t ) − q − φ ( t )) v + α q − (1 − γ ) tv (4.10)0 = x n ( t ) = x n ( u ) = x n ( v ) . (4.11)Since we are primarily interested in inner-faithful actions, henceforth, we will assume γ = 1. emma 4.12. Assume Hypothesis 4.1 with g acting as η γ,µ and with γ = 1 . Then the following hold:(1) µ is a root of unity and ord( µ ) | n ,(2) m = s ,(3) supp( φ ) is a single equivalence class modulo m ,(4) λ = γ d − and so gcd( d − , m ) = 1 ,(5) supp( h ) is a single equivalence class modulo m .Proof. The automorphism η γ,µ has order dividing n if and only if ord( γ ) | n and ord( µ ) | n , which givesstatement (1). Under Hypothesis 4.1, A = k [ t ] is a T -module algebra with g ( t ) = γt and x ( t ) = φ . Hence,Proposition 3.4 yields statements (2)–(4).If h is a monomial, then (5) is trivial. Otherwise, recall that ℓ = gcd { i − j | h i h j = 0 } , and for η γ,µ to bean automorphism of A , we must have γ ℓ = 1. Therefore ℓ must be a multiple of m and so the support of h is also a single equivalence class modulo m . (cid:3) In the remainder of this section, we abuse derivative notation and write φ ( k ) = d − k X i =0 φ i + k t i and h ( k ) = D − k X i =0 h i + k t i . Lemma 4.13.
Assume Hypothesis 4.1 with g acting as η γ,µ . For all f ∈ A = k [ t ] , x ( f ) = φδ γ ( f ) . Inparticular, x ( h ) = [ D ] γ φh (1) and x ( φ ) = [ d ] γ φφ (1) .Proof. The first statement follows directly from the linearity of δ γ and Lemma 3.1. For the second statement,we need only recall from Lemma 4.12 that the support of h and φ consist of a single equivalence class moduloord( γ ) and so δ γ ( h ) = [ D ] γ h (1) and δ γ ( φ ) = [ d ] γ φ (1) . (cid:3) Next, we determine the parameters α and α from equation (4.2). Lemma 4.14.
Assume Hypothesis 4.1 with g acting as η γ,µ and with γ = 1 . Then the following hold.(1) We have α = 1 − µq − d − γ φ (1) (4.15) α = 1 − µ − γ D q d − − γ φ (1) . (4.16) Hence, for i = 1 , , α ii is supported on a single equivalence class modulo m and if α ii = 0 , then thatequivalence class is d − .(2) If α = 0 or α = 0 , then t | φ .(3) If the support of α is modulo m , then x m ( u ) = 0 if and only if α = 0 . If the support of α is not modulo m , then x m ( u ) = 0 if and only if µq − d is an m th root of unity (not necessarilyprimitive).
4) If the support of α is modulo m , then x n ( v ) = 0 if and only if α = 0 . If the support of α is not modulo m , then x n ( v ) = 0 if and only if µq − d is an m th root of unity (not necessarilyprimitive).Proof. Throughout we prove the statements for v and α . The statements for u and α follow similarly.As mentioned after Lemma 4.3, under these hypotheses, the T -action on A satisfies equations (4.4)–(4.11).By (4.10) and Lemma 4.12, α q − (1 − γ ) t = q − φ ( t ) − µφ ( q − t )= q − d X i =0 φ i t i − µ d X i =0 φ i ( q − t ) i = q − d X i =0 (1 − µq − i ) φ i t i . So, either α = 0, in which case µ = q i − for all i ∈ supp( φ ), or else t | φ and α = d X i =1 (1 − µq − i )1 − γ φ i t i − . This proves (2). Since the support of φ is a single equivalence class modulo ord( γ ), then it follows fromthe above computation that the support of α is also. Note ord( γ ) = m by Lemma 4.12 (2).First, suppose that the support of α is 0 modulo m . Then δ γ ( α ) = 0. By an induction argument, foreach k ≥ x k ( v ) = α k v and so the first part of (4) follows.Now suppose that the support of α is not 0 modulo m . Set β = 1 and for k >
0, let β k = d X i =1 (1 − γ ( k − d − µq − i )1 − γ φ i t i − , so that β = α . We claim that x ( β k · · · β β v ) = β k +1 · · · β β v. (4.17)Clearly this is true for k = 0. Then x ( β k +1 · · · β v ) = g ( β k +1 ) x ( β k · · · β v ) + x ( β k +1 ) β k · · · β v = ( γ d − β k +1 )( β k +1 · · · β v ) + ([ d − γ φβ (1) k +1 ) β k · · · β v = ( γ d − β k +1 + [ d − γ φ (1) ) β k +1 · · · β v = d X i =1 γ d − (1 − γ k ( d − µq − i ) + (1 − γ d − )1 − γ φ i t i − ! β k +1 · · · β v = β k +2 · · · β v. Now (4.17) implies that x m ( v ) = β m · · · β v . ence, x m ( v ) = 0 if and only if β k = 0 for some k = 1 , . . . , m . Moreover, β k = 0 if and only if γ ( k − d − = µ − q i − for all i ∈ supp( φ ). This implies that µ − q i − = µ − q d − for all i ∈ supp( φ ), which reduces to q d = q i .Hence we obtain (4.15), completing our proof of (1).Finally, recall from Lemma 4.12 (4) that γ d − is a primitive m th root of unity and so the γ ( k − d − range over the m th roots of unity (other than 1) exactly once. It follows that x m ( v ) = 0 if and only if µq d − is an m th root of unity. Since ord( γ ) = m , then this reduces to x m ( v ) = 0 if and only if µq − d is an m th root of unity which completes our proof of (4). (cid:3) We are now ready to state our main theorem.
Theorem 4.18.
Fix a GWA A = k [ t ]( u, v, σ, h ) with defining polynomial h of degree D > , and definingautomorphism σ ( t ) = qt where q ∈ k × is a root of unity, q = 1 . Let T = T n ( λ, m, be a generalized Taftalgebra.Suppose that A is an inner-faithful Z -graded T -module algebra where g acts as an automorphism η γ,µ asin equation (2.3) with γ = 1 , and x ( t ) = φ ( t ) ∈ k [ t ] . Then(1) the automorphism η γ,µ has order n ,(2) supp( h ) is a single equivalence class modulo m ,(3) φ ( t ) is a nonzero polynomial of degree d whose support is a single equivalence class modulo m ,(4) ord( γ ) = m and λ = γ d − ,(5) µq − d is an m th root of unity,(6) x ( u ) = 1 − µ − γ D q d − − γ φ (1) u =: α u , and(7) x ( v ) = 1 − µq − d − γ φ (1) v =: α v . Conversely, suppose that η γ,µ ∈ Aut( A ) with γ = 1, φ ( t ) ∈ k [ t ], and the parameters h and q satisfyconditions (1)–(5). Let g act on A via η γ,µ . Define an action of x on A by setting x ( t ) = φ ( t ), x ( u ) to be asin (6), x ( v ) to be as in (7), and extending to all of A via the coproduct in T . Then this gives A the structureof an inner-faithful Z -graded T -module algebra. Proof.
Suppose A is an inner-faithful Z -graded T -module where g acts as an automorphism η γ,µ with γ = 1.Since the T -action is Z -graded, we must have x ( t ) = φ ( t ) for some φ ( t ) ∈ k [ t ]. Then conditions (1)–(7)follow from Lemmas 4.12 and 4.14.Conversely, suppose that η γ,µ ∈ Aut( A ) with γ = 1, φ ( t ) ∈ k [ t ], and the parameters h and q satisfyconditions (1)–(5). We will show that A is a Z -graded T -module algebra where g acts via η γ,µ and x actsvia x ( t ) = φ ( t ), x ( u ) is as in (6) and x ( v ) is as in (7) (and the action of x is extended to all of A via thecoproduct). ince we are assuming that g acts via an automorphism of order n , it suffices to show that equations(4.4)–(4.11) hold for the T -action on A .By Lemma 4.13 and conditions (6) and (7), we have( µσ − ( α ) + α ) h = (cid:18) µσ − (cid:18) − µ − γ D q d − − γ φ (1) (cid:19) + (cid:18) − µq − d − γ (cid:19) φ (1) (cid:19) h = (1 − γ ) − (cid:0) µq − d (1 − µ − γ D q d − ) + (1 − µq − d ) (cid:1) φ (1) h = (1 − γ ) − (cid:0) − γ D (cid:1) φ (1) h = [ D ] γ φ (1) h = x ( h ) . Thus, (4.4) is satisfied. One checks similarly that (4.5) is satisfied.By conditions (4) and (6), α ∈ k [ t ] is a polynomial of degree d − m . Hence, by condition (3), we a have g ( α ) = λα and so equation (4.6) is satisfied. Asimilar argument for α shows that equation (4.7) is satisfied.By conditions (4) and (3), we have φ ( γt ) = γ d φ ( t ) = λγφ ( t ) and so equation (4.8) is satisfied. Theproof of Lemma 4.14 shows that if conditions (6) and (7) hold, then equations (4.9) and (4.10) are satisfied.Finally, by condition (3) and Lemma 3.3, we have x m ( t ) = 0. By conditions (6) and (7) and Lemma 4.14, x m ( u ) = x m ( v ) = 0. Hence, equation (4.11) is satisfied.By [7, Corollary 3.7], the T -action on A is be inner-faithful if and only if g acts an automorphism oforder n and x does not act identically as 0 on A . Hence, by conditions (1) and (3), A is an inner-faithful T -module. (cid:3) Remark 4.19. If m = n in Theorem 4.18, then T n ( λ, m, is the Taft algebra T n ( λ ) . Then ord( µ ) | n andso Theorem 4.18 (5) reduces to the condition that q − d is an n th root of unity. Let A be a rank p quantum GWA. We denote by A i the rank one quantum GWA subalgebra of A generated by x i , y i , t . In the following corollary we show that if a generalized Taft algebra T on each A i , wecan construct an action of T on A . Corollary 4.20.
Let A be a quantum GWA of rank p and let T = T n ( λ, m, be a generalized Taft algebra.Suppose each A i is an inner-faithful Z -graded T -module with g acting on each A i via an automorphism of theform η γ i ,µ i ∈ Aut( A i ) with γ i = 1 . Then T acts on A . That is, A is an inner-faithful Z p -graded T -modulealgebra.Proof. By the hypotheses, g acts as some η γ i ,µ i on A i , with γ i , µ i ∈ k × satisfying the conditions of Theorem4.18. Thus, ( gx − λxg )( t ) = ( gx − λxg )( u i ) = ( gx − λxg )( v i ) = 0and x m ( t ) = x m ( u i ) = x m ( v i ) = 0 for each i . Moreover, g ( r ) = x ( r ) = 0 for each defining relation r of A i .It remains only to check that g and x satisfy the commutation relations between the A i . hoose i, j with 1 ≤ i < j ≤ p . To simplify the exposition, we take i = 1 and j = 2. Set g ( u ) = µ − γ D u , g ( v ) = µ v , x ( u ) = α u , x ( v ) = α v ,g ( u ) = µ − γ D u , g ( v ) = µ v , x ( u ) = β u , x ( v ) = β v , where α ii , β ii satisfy the appropriate conditions of Theorem 4.18. Then x ( v v − v v ) = (( µ v )( β v ) + ( α v ) v ) − (( µ v )( α v ) + ( β v ) v )= (cid:0) ( µ q − d − β + (1 − µ q − d ) α (cid:1) v v = 0 ,x ( u u − u u ) = (cid:0) ( µ − γ D u )( β u ) + ( α u ) u (cid:1) − (cid:0) ( µ − γ D u )( α u ) + ( β u ) u (cid:1) = (cid:0) ( µ − γ D q d − − β + (1 − µ − γ D q d − ) α (cid:1) u u = 0 ,x ( u v − v u ) = (cid:0) ( µ − γ D u )( β v ) + ( α u ) v (cid:1) − (( µ v )( α u ) + ( β v ) u )= (cid:0) ( µ − q d − γ D − β + (1 − q − d µ ) α (cid:1) u v = 0 . It follows that T acts on A preserving the Z p grading. (cid:3) We end this section by specializing to the case q = − g acts as someΩ ◦ η γ,µ . That is g ( t ) = − γt, g ( v ) = µu, g ( u ) = µ − γ D v. (4.21) Theorem 4.22.
Fix a GWA A = k [ t ]( u, v, σ, h ) with defining polynomial h of degree D > , and definingautomorphism σ ( t ) = − t . Let T = T n ( λ, m, be a generalized Taft algebra. Suppose that A is an inner-faithful Z -graded T -module algebra where g acts as an automorphism Ω ◦ η γ,µ . Suppose that the action of x is given by (4.2) . Then(1) λ = γ = − ,(2) φ = 0 ,(3) σ ( α ) = ± α , α = − α , α = − µσ ( α ) , α = µ − σ − ( α ) , and(4) σ ( h ) = γ D h .Conversely, defining an action of g as in (4.21) and an action of x by (4.2) subject to conditions (1)–(4)and extending naturally via the coproduct on T defines an action which makes A an inner-faithful Z -graded T -module algebra. roof. Suppose that A is an inner-faithful Z -graded T -module algebra where g acts as an automorphismΩ ◦ η γ,µ . Since A is a T -module algebra, the action of x must preserve the relations of A so0 = x ( vu − h ) = ( µσ ( α ) + α ) u + µσ ( α ) σ ( h ) + α h − x ( h ) , (4.23)0 = x ( uv − σ ( h )) = ( µ − γ D σ − ( α ) + α ) v + µ − γ D σ − ( α ) h + α σ ( h ) − x ( σ ( h )) . (4.24)By the Z -grading on A , we see that α = − µσ ( α ) and α = − µ − γ D σ − ( α ). Further, since therelations of T must act as 0 on A , we have0 = ( gx − λxg )( u ) = − γ D ( σ − ( α ) − λσ ( α )) u + µ − γ D ( α − λα ) v (4.25)0 = ( gx − λxg )( v ) = − γ D ( σ ( α ) − λσ − ( α )) v + µ ( α − λα ) u (4.26)0 = ( gx − λxg )( t ) = g ( φ ) − λx ( − γt ) = φ ( − γt ) + λγφ ( t ) . (4.27)By (4.25) and (4.26), α = λα = λ α , so either λ = ± α ij = 0 for all i, j . By Hypothesis 4.1, λ = 1.Further we have 0 = x ( ut + tu ) = ( µ − γ D ) vφ + φu − (1 + γ ) t ( α u + α v ) (4.28)0 = x ( vt + tv ) = µuφ + φv − (1 + γ ) t ( α u + α v ) . (4.29)By (4.25), (4.28), and (4.29), φ = (1 + γ ) tα = − (1 + γ ) tα = − φ. Thus, φ = 0 and γ = −
1. Note this implies that g ( t ) = t .By (4.24), 0 = µ − γ D σ − ( α ) h + α σ ( h ) = α ( σ ( h ) − γ D h ) , so σ ( h ) = γ D h .Now we need 0 = x ( u ) = ( α + α α ) u + α ( α + α ) v = ( α + α α ) u x ( v ) = α ( α + α ) u + ( α + α α ) v = ( α + α α ) v. This gives 0 = α + α α = α + ( − µ − σ − ( α )( − µσ ( α )) = σ − ( σ ( α ) − α ) , so σ ( α ) = ± α .Conversely, let A be a quantum GWA with q = −
1, let T = T ( − , ,
0) be the Sweedler Hopf algebra.Suppose g acts on A as an automorphism Ω ◦ η − ,µ for some µ ∈ k × . Suppose x ( t ) = 0, x ( u ) = α u + α v ,and x ( v ) = α u + α v with the α ij ∈ k [ t ] satisfying (3). Finally, suppose σ ( h ) = ( − D h where D = deg t ( h ). e extend the action naturally via the coproduct on T . It is then easy to check that (4.23)–(4.29) are satisfied.Thus, this defines an an action of T on A . (cid:3) Fixed rings by Taft actions
For a Hopf algebra H and an H -module algebra A , the fixed ring of A by H is A H = { a ∈ H | h.a = ǫ ( h ) a for all h ∈ H } . This generalizes the usual definition of the fixed ring of a group action. In this section we consider fixedrings of quantum GWAs by Taft algebra actions.Let A be a quantum GWA and set η = η γ,µ . Suppose gcd(ord( γ ) , ord( µ )) = 1 and ord( γ ) | D . Then A h η i is again a quantum GWA generated by t ord( γ ) , x ord( µ ) , and y ord( µ ) [10, Corollary 3.5]. However, if T = T n ( λ )is a Taft algebra and A an inner-faithful Z -graded T -module algebra such that g acts as some η γ,µ with γ = 1, then gcd(ord( γ ) , ord( µ )) = 1 in general (in fact this holds if and only if µ = 1). Hence, the abovemethod will not apply when computing A T = ( A h g i ) h x i . Theorem 5.1.
Fix a GWA A = k [ t ]( u, v, σ, h ) with defining polynomial h of degree D > and definingautomorphism σ ( t ) = qt where q ∈ k × is a root of unity, q = 1 . Let T = T n ( λ ) be a Taft algebra. Supposethat A is an inner-faithful Z -graded T -module algebra where g acts as η γ,µ ∈ Aut( A ) with γ = 1 . Then(1) For all k , ( A T ) k = ( A k ) T . In particular, ( A T ) = ( A ) T = k [ t n ] .(2) Suppose k > and x ( v ) = 0 . If q k = 1 , then ( A T ) − k = 0 . If q k = 1 , then f ( t ) v k ∈ ( A T ) − k if andonly supp( f ( t )) ⊂ { a ∈ N | γ a µ k = 1 } .(3) Suppose k > and x ( u ) = 0 . If q k = 1 , then ( A T ) k = 0 . If q k = 1 , then g ( t ) u k ∈ ( A T ) k if and onlyif supp( g ( t )) ⊂ { b ∈ N | γ b + kD µ − k = 1 } .(4) A T is a (commutative) Kleinian singularity.Proof. (1) By the definition of η γ,µ and Lemma 4.3, the actions of both g and x respect the grading on A andhence ( A T ) k = ( A k ) T for all k ∈ Z . Clearly, ( A ) h g i = k [ t n ] and x ( t n ) = 0 by (3.2). Thus, ( A ) T = k [ t n ].(2) By Theorem 4.18, x ( v ) = α v where α = − µq − d − γ φ (1) . Thus, x ( v ) = (cid:18) − µq − d − γ (cid:19) (cid:16) µφ (1) ( q − t ) + φ (1) ( t ) (cid:17) v . But q is an n th root of unity and supp( φ ) (and hence supp( φ (1) )) is a single equivalence class mod n . Itfollows that φ (1) ( q − t ) = q − d φ (1) ( t ). Set ω = µq − d . Then x ( v ) = α [2] ω v . An induction argument thenshows that x ( v k ) = α [ k ] ω v k . uppose t a v k ∈ A T . Then t a v k = g ( t a v k ) = γ a µ k t a v k . Thus, γ a µ k = 1. Now, x ( t a v k ) = g ( t a ) x ( v k ) + x ( t a ) v k = γ a t a ( α [ k ] ω v k ) + φ [ a ] γ t a − v k = (cid:18) γ a − γ a µ k q k (1 − d ) − γ + [ a ] γ (cid:19) φ (1) t a v k = (cid:18) − q k (1 − d ) − γ (cid:19) φ (1) t a v k . Hence, x ( t a v k ) = 0 if and only if q k (1 − d ) = 1. Since gcd( d − , n ) = 1, then this is equivalent to q k = 1.By linearity, it follows that f ( t ) v k ∈ A T then supp( f ( t )) is a single equivalence class modulo n .(3) This is similar to (2).(4) First consider the case α , α = 0. Let k be the minimal positive integer such that q k = 1. Choose a, b ∈ N minimal such that γ a µ k = 1 and γ b + kD µ − k = 1. By (1)-(3), A T is generated as an algebra by T = t n , V = t a v k , and U = t b u k . Clearly U and V commute with T since q n = 1. Moreover, inductionshows that U V = t a + b k Y i =1 σ i ( h ) =: H and V U = t a + b k − Y i =0 σ − i ( h ) . Since σ − i ( h ( t )) = h ( q − i t ) = q − Di h ( t ) = q D ( k − i ) h ( t ) = h ( q k − i t ) = σ k − i ( h ( t )) , (5.2)then it follows that U V = V U . Hence, A T ∼ = k [ U, V, T ] / ( U V − H ).Now suppose α = α = 0. Let t a v k ∈ A T , then0 = x ( t a v k ) = g ( t a ) x ( v k ) + x ( t a ) v k = [ a ] γ t a v k , so n | a . Assuming this, we have g ( t a v k ) = γ a µ k t a v k = µ k t a v k , so ord( µ ) | k . It follows that A T is generatedby T = t n , V = v k , and U = u k , where k = ord( µ ). As above, U and V commute with T since q n = 1. Theargument in (5.2) shows that U V = V U . Hence, A T ∼ = k [ U, V, T ] / ( U V − H ) as before. (cid:3) The main result of Theorem 5.1 is consistent with several other related results in the literature. The fixedring of a quantum plane or quantum Weyl algebra by a Taft algebra acting linearly and inner-faithfully wasshown to be a commutative polynomial ring [11, Lemma 2.1]. Similar results exist in the case of generalizedTaft actions [8, Proposition 5.6].However, in the setting of Theorem 4.22 (i.e., when g acts via some Ω ◦ η γ,µ ), the fixed ring is not alwayscommutative. In the notation of that theorem, take D to be even and σ ( α ) = α . By [10, Lemma 3.7], A h g i is generated by T = t and W = µu + v . In fact, A h g i ∼ = k − [ W, T ], the ( − x ( T ) = 0 and one checks that x ( W ) = 0 so in fact A T ∼ = k − [ W, T ]. In light of this, weask the following question. uestion 5.3. Under what conditions is the fixed ring of a twisted Calabi–Yau algebra by a Taft actioncommutative?
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Email address : [email protected] (Won) Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195, USA Email address : [email protected]@math.washington.edu