Universal quantum semigroupoids
aa r X i v : . [ m a t h . QA ] A ug UNIVERSAL QUANTUM SEMIGROUPOIDS
HONGDI HUANG, CHELSEA WALTON, ELIZABETH WICKS, AND ROBERT WON
Abstract.
We introduce the concept of a universal quantum linear semigroupoid(UQSGd), which is a weak bialgebra that coacts on a (not necessarily connected) gradedalgebra A universally while preserving grading. We restrict our attention to algebraicstructures with a commutative base so that the UQSGds under investigation are facealgebras (due to Hayashi). The UQSGd construction generalizes the universal quantumlinear semigroups introduced by Manin in 1988, which are bialgebras that coact on aconnected graded algebra universally while preserving grading. Our main result is thatwhen A is the path algebra k Q of a finite quiver Q , each of the various UQSGds intro-duced here is isomorphic to the face algebra attached to Q . The UQSGds of preprojectivealgebras and of other algebras attached to quivers are also investigated. Introduction
The goal of this work is to examine the quantum symmetries of N -graded algebras, thatare not necessarily connected, within the framework of weak bialgebra coactions. All alge-braic structures here are k -linear, for k an arbitrary base field, and we reserve ⊗ to mean ⊗ k .In an algebraic quantum symmetry problem, one either: (1) fixes the type of algebrato study the symmetries of, and then proceeds with analyzing the Hopf-type algebra (ormonoidal category of (co)representations) that captures its symmetries, or (2) fixes theHopf-type algebra (or monoidal category of (co)representations), and then studies the typesof (co)module algebras that it captures the symmetries of. In this work, we pursue problem(1) for the k -algebras given below. Hypothesis 1.1 ( A, A ) . Let A be a locally finite, N -graded k -algebra A , that is, A has k -vector space decomposition L i ∈ N A i with A i · A j ⊆ A i + j , and dim k A i < ∞ . Furthersuppose that the degree 0 component A , which is a finite-dimensional k -subalgebra of A ,is a commutative and separable k -algebra. In particular, this implies that A is a Frobeniusalgebra over k .We say that A is connected if A = k , although we do not assume that condition here.The prototypical examples of algebras A whose symmetries we will examine are pathalgebras of finite quivers. Throughout, we fix the following notation. Notation 1.2 ( Q, Q , Q , s, t, k Q, e i , p, q, a, b ) . Let Q = ( Q , Q , s, t ) be a finite quiver (i.e.,a directed graph), where Q is a finite collection of vertices, Q is a finite collection of arrows,and s, t : Q → Q denote the source and target maps, respectively. We read paths of Q fromleft-to-right. Let k Q be the path algebra attached to Q , which is the k -algebra generated Mathematics Subject Classification.
Key words and phrases. path algebra, preprojective algebra, universal coaction, weak bialgebra. by { e i } i ∈ Q and { p } p ∈ Q , with multiplication given by m ( e i ⊗ e j ) = δ i,j e i for i, j ∈ Q and m ( p ⊗ q ) = δ t ( p ) ,s ( q ) pq for p, q ∈ Q , and with unit given by u (1 k ) = P i ∈ Q e i . The pathalgebra k Q is N -graded by path length, where for each ℓ ∈ N , ( k Q ) ℓ = k ( Q ℓ ), where Q ℓ consists of paths of length ℓ in Q . We usually use the letters a, b to denote paths in Q .Using path algebras as the prototypical examples of not necessarily connected k -algebras A in Hypothesis 1.1 is apt because if A is generated by A over A , then A is isomorphic toa quotient of some path algebra k Q . Namely, A is isomorphic to the path algebra on anarrowless quiver Q with | Q | = dim k A . Further, path algebras are free structures in thesense that they are tensor algebras: k Q ∼ = T k Q ( k Q ), where k Q is a k Q -bimodule; wewill return to this fact later in the introduction. Moreover, interesting examples of gradedquotients of path algebras include preprojective algebras [Rin98] and superpotential algebras(see, e.g., [BSW10]).Before we study quantum symmetries of the algebras A in Hypothesis 1.1, let us recallvarious notions of a universal bialgebra coacting on A in the case when A is connected,which are all due to Manin [Man88] (for the case when A is quadratic). First, we make thestanding assumption that will be used throughout this work often without mention. Hypothesis 1.3.
Let C be a monoidal category of corepresentations of an algebraic struc-ture H . If A is an algebra in C (i.e., if A is an H -comodule algebra), then we assume thateach graded component A i of A is an object in C (i.e., is an H -comodule). Namely, weassume that all coactions of H on A preserve grading, or are linear , in this work.Consider the following universal bialgebras that coact on a connected algebra A as inHypothesis 1.1, from either the left or right. Definition 1.4 (left UQSG, O left ( A ); right UQSG, O right ( A )) . [Man88, Chapter 4 and Sec-tions 5.1–5.8] Let A be a k -algebra as in Hypothesis 1.1 that is connected.(a) Let O := O left ( A ) be a bialgebra for which A is a left O -comodule algebra via left O -comodule map λ O : A → O ⊗ A . We call O left ( A ) the left universal quantum linearsemigroup (left UQSG) of A if, for any bialgebra H for which A is a left H -comodulealgebra via left H -comodule map λ H : A → H ⊗ A , there exists a unique bialgebramap π : O → H so that λ H = ( π ⊗ Id A ) λ O .(b) Let O := O right ( A ) be a bialgebra for which A is a right O -comodule algebra viaright O -comodule map ρ O : A → A ⊗ O . We call O right ( A ) the right universalquantum linear semigroup (right UQSG) of A if, for any bialgebra H for which A is a right H -comodule algebra via right H -comodule map ρ H : A → A ⊗ H , thereexists a unique bialgebra map π : O → H so that ρ H = (Id A ⊗ π ) ρ O .Other appearances of bialgebras that coact linearly and universally on algebraic struc-tures from one side include the universal bi/Hopf algebras that coact on (skew-)polynomialalgebras in [RRT02, LT07, CFR09], and the universal bi/Hopf algebras that coact on a su-perpotential algebra (or, equivalently that preserve a certain multilinear form) in [DVL90,BDV13, CWW19].Ideally, a universal bialgebra should behave ring-theoretically and homologically like thealgebra that it coacts on. But this is not the case even when the algebra is a polynomial ring NIVERSAL QUANTUM SEMIGROUPOIDS 3 in two variables; see Example 5.7(b). Namely, O left ( k [ x, y ]) is a non-Noetherian algebra ofinfinite Gelfand–Kirillov (GK) dimension, whereas k [ x, y ] is Noetherian of GK-dimension 2.Towards the goal above, one can consider a ‘smaller’ universal bialgebra introduced byManin, which coacts an algebra A universally from the left and right via ‘transposed’ coac-tions. Indeed, Manin inquired if such a universal bialgebra reflects the behavior of A (in theconnected and quadratic case) in [AST91, Introduction]. Definition 1.5 (transposed UQSG, O trans ( A )) . (cf., [Man88, Section 5.10, Chapters 6and 7]) Let A be a k -algebra as in Hypothesis 1.1 that is connected.(a) Let H be a bialgebra for which A is a left H -comodule algebra via left H -comodulemap λ HA , and for which A is a right H -comodule algebra via right H -comodule map ρ HA . We call A a transposed H -comodule algebra if, for the transpose of ρ HA ,( ρ HA ) T : (ev A ⊗ Id H ⊗ Id A )(Id A ∗ ⊗ ρ HA ⊗ Id A ∗ )(Id A ∗ ⊗ coev A ) : A ∗ → H ⊗ A ∗ , we obtain λ HA by identifying a basis of A with the dual basis of A ∗ .(b) Let O := O trans ( A ) be a bialgebra for which A is a transposed O -comodule algebravia left O -comodule map λ O and right O -comodule map ρ O . We call O trans ( A ) the transposed universal quantum linear semigroup (transposed UQSG) of A if, for anybialgebra H for which A is a transposed H -comodule algebra via left H -comodulemap λ H and a right H -comodule map ρ H , there exists a unique bialgebra map π : O → H so that λ H = ( π ⊗ Id A ) λ O and ρ H = (Id A ⊗ π ) ρ O .Other instances of bialgebras that coact linearly and universally on algebraic structures ina transposed manner include the universal bi/Hopf algebras that coact on skew-polynomialalgebras in [Tak90, AST91] (these are special cases of the construction in [Man88]), and theuniversal bi/Hopf algebras that coact on a superpotential algebras in [CWW19] (this is ageneralization of the construction in [Man88]).In order to study the quantum symmetries of an algebra A which satisfies Hypothesis 1.1,but is not necessarily connected, we use coactions of weak bialgebras, which are structuresthat have the underlying structure of an algebra and a coalgebra, with weak compatibilityconditions between these substructures [Definition 2.1]. For a weak bialgebra H , thereare two important coideal subalgebras, H s and H t , called the source and target counitalsubalgebras, that measure how far H is from being a bialgebra. Namely, H is a bialgebra ifand only if both H s and H t are the ground field k . These subalgebras are always separableand Frobenius (see Proposition 2.3(a)).Since we are considering quantum symmetries of algebras A whose degree 0 components A are commutative separable algebras, we will work within the framework of weak bial-gebras with commutative counital subalgebras, which are the same as V -face algebras by[Sch98, Theorem 4.3] and [Sch03, Theorems 5.1 and 5.5]. Here, V is a finite set. A keyexample of a V -face algebra is the weak bialgebra H ( Q ) attached to a finite quiver Q , whichwas introduced by Hayashi in [Hay93, Hay96]. In this case, V = Q and a presentation of H ( Q ) is provided in Example 2.6. Next, we propose a conjecture, which is a modification of[Hay99, Proposition 2.1] that remains unproved. HONGDI HUANG, CHELSEA WALTON, ELIZABETH WICKS, AND ROBERT WON
Conjecture 1.6.
Suppose that k is algebraically closed. If H is a finite-dimensional weakbialgebra with commutative counital subalgebras, then H is isomorphic to a weak bialgebraquotient of H ( Q ) for some finite quiver Q . This is akin to the result that every finite-dimensional algebra over an algebraically closedfield is isomorphic to a quotient of a path algebra of some finite quiver (see, e.g., [ASS06,Theorem II.3.7]).Returning to the study of the quantum symmetries of algebras A as in Hypothesis 1.1,we proceed by realizing such an algebra A as a comodule algebra over a weak bialgebra H (which will eventually have commutative counital subalgebras). For a weak bialgebra H , let H A (resp., A H ) denote the category of left (resp., right) H -comodule algebras. An exampleof an object in H A (resp., in A H ) is H t (resp., H s ) via comultiplication [Examples 2.15and 2.16]. Moreover, if an algebra A satisfying Hypothesis 1.1 belongs to H A (resp., A H ),then so does the subalgebra A [Remark 3.2]. We are now ready to introduce various notionsof a universal weak bialgebra coacting on A which are the focus of our work. Definition 1.7 (left UQSGd, O left ( A ); right UQSGd, O right ( A ); trans. UQSGd, O trans ( A )) . Let A be k -algebra as in Hypothesis 1.1.(a) Let O := O left ( A ) be a weak bialgebra so that A ∈ O A via left O -comodule map λ O with O t ∼ = A in O A . We call O left ( A ) the left universal quantum linear semigroupoid(left UQSGd) of A if, for any weak bialgebra H such that A ∈ H A via left H -comodulemap λ H with H t ∼ = A in H A , there exists a unique weak bialgebra map π : O → H so that λ H = ( π ⊗ Id A ) λ O .(b) Let O := O right ( A ) be a weak bialgebra so that A ∈ A O via right O -comodulemap ρ O with O s ∼ = A in A O . We call O right ( A ) the right universal quantum linearsemigroupoid (right UQSGd) of A if, for any weak bialgebra H such that A ∈ A H via right H -comodule map ρ H with H s ∼ = A in A H , there exists a unique weakbialgebra map π : O → H so that ρ H = (Id A ⊗ π ) ρ O .(c) Let O := O trans ( A ) be a weak bialgebra so that A ∈ O A and A ∈ A O so that A isa transposed O -comodule algebra, and with O t ∼ = A in O A and O s ∼ = A in A O .We call O trans ( A ) the transposed universal quantum linear semigroupoid (transposedUQSGd) of A if, for any weak bialgebra H such that A ∈ H A and A ∈ A H for which A is a transposed H -comodule algebra, and with H t ∼ = A in H A and H s ∼ = A in A H , there exists a unique weak bialgebra map π : O → H so that λ H = ( π ⊗ Id A ) λ O and ρ H = (Id A ⊗ π ) ρ O .Discussion about these definitions is provided in Remarks 3.3, 3.5–3.8, 3.11–3.14; the mostimportant observation is that, without the condition that the ‘base’ of the weak bialgebrais isomorphic to the ‘base’ of the comodule algebra, such universal weak bialgebras are notlikely to exist [Remark 3.3]. This brings us to our main result. Theorem 1.8.
For a finite quiver Q , the UQSGds O left ( k Q ) , O right ( k Q ) , and O trans ( k Q ) of the path algebra k Q exist, and each is isomorphic to Hayashi’s face algebra H ( Q ) as weakbialgebras. NIVERSAL QUANTUM SEMIGROUPOIDS 5
For example, if we take A to be the (connected, graded) free algebra k h t , . . . , t n i , i.e., thepath algebra on the n -loop quiver Q n -loop , then the UQSGds of A are the classical UQSGsof Definitions 1.4 and 1.5, and O left ( A ) ∼ = O right ( A ) ∼ = O trans ( A ) ∼ = H ( Q n -loop );see Example 4.20. But these isomorphisms need not hold if A is a proper quotient of k h t , . . . , t n i [Example 5.7]. In general, we have the following results for UQSGds of gradedquotient algebras of k Q . Proposition 1.9.
Let I ⊆ k Q be a graded ideal which is generated in degree or greater.If O ∗ ( k Q/I ) exists, then we have an isomorphism of weak bialgebras, O ∗ ( k Q/I ) ∼ = H ( Q ) / I , for some biideal I of H ( Q ) . Here, ∗ means ‘left’, ‘right’, or ‘trans’. Finally, in the case when I is generated in degree 2, i.e., when k Q/I is quadratic [Def-inition 5.8], we establish a non-connected generalization of [Man88, Theorem 5.10]. The quadratic dual ( k Q/I ) ! of the quadratic algebra k Q/I is reviewed in Definition 5.8.
Theorem 1.10.
If the quotient algebra k Q/I is quadratic, then we have that (a) O left ( k Q/I ) ∼ = O right (( k Q/I ) ! ) op , (b) O right ( k Q/I ) ∼ = O left (( k Q/I ) ! ) op , (c) O left ( k Q/I ) ∼ = O right ( k Q/I ) cop , (d) O trans ( k Q/I ) ∼ = O trans (( k Q/I ) ! ) op ,as weak bialgebras. The paper is organized as follows. We present background material and preliminaryresults on weak bialgebras, monoidal categories of corepresentations of weak bialgebras,and (examples of) comodule algebras over weak bialgebras in Section 2. We introducethe theory of universal quantum linear semigroupoids (of algebras as in Hypothesis 1.1) inSection 3, including Definition 1.7. Our main result, Theorem 1.8 on the UQSGds of pathalgebras, is established in Section 4. Examples and results about UQSGds of quotients ofpath algebras are presented in Section 5, including Proposition 1.9 and Theorem 1.10. Weend by providing directions for future investigation on universal quantum linear groupoids (i.e., universal weak Hopf algebras) in Section 6.
Acknowledgements.
The authors would like to thank Dmitri Nikshych for a helpfulexchange about material in Section 2, Pavel Etingof for posing Question 3.12 and otherinteresting comments, and James Zhang for inspiring Question 6.5. C. Walton is supportedby a research grant from the Alfred P. Sloan foundation and by NSF grant
Preliminaries
In this section, we provide background material and preliminary results on weak bialge-bras [Section 2.1], and on corepresentation categories of weak bialgebras and algebras within
HONGDI HUANG, CHELSEA WALTON, ELIZABETH WICKS, AND ROBERT WON them [Section 2.2]. We end by providing crucial examples of comodule algebras over weakbialgebras [Section 2.3].2.1.
Weak bialgebras.
To begin, recall that a k -algebra is a k -vector space A equippedwith a multiplication map m : A ⊗ A → A and unit map u : k → A satisfying associativityand unitality constraints. We reserve the notation 1 to mean 1 := 1 A := u (1 k ) . A k -coalgebra is a k -vector space C equipped with a comultiplication map ∆ : C → C ⊗ C and counit map ε : C → k satisfying coassociativity and counitality constraints. If ( C, ∆ , ε ) is a coalgebra,we use sumless Sweedler notation and write ∆( c ) := c ⊗ c for c ∈ C . Definition 2.1. A weak bialgebra over k is a quintuple ( H, m, u, ∆ , ε ) such that(i) ( H, m, u ) is a k -algebra,(ii) ( H, ∆ , ε ) is a k -coalgebra,(iii) ∆( ab ) = ∆( a )∆( b ) for all a, b ∈ H ,(iv) ε ( abc ) = ε ( ab ) ε ( b c ) = ε ( ab ) ε ( b c ) for all a, b, c ∈ H ,(v) ∆ (1) = (∆(1) ⊗ ⊗ ∆(1)) = (1 ⊗ ∆(1))(∆(1) ⊗ ⊗ ε (1) = 1. Instead,we have weak multiplicativity of the counit (condition (iv)) and weak comultiplicativity ofthe unit (condition (v)). Definition 2.2 ( ε s , ε t , H s , H t ) . Let (
H, m, u, ∆ , ε ) be a weak bialgebra. We define the source and target counital maps , respectively as follows: ε s : H → H, x ε ( x ) ε t : H → H, x ε (1 x )1 . We denote the images of these maps as H s := ε s ( H ) and H t := ε t ( H ). We call H s the source counital subalgebra and H t the target counital subalgebra of H (see Proposition 2.3).These subalgebras have special properties that we will need below. Proposition 2.3.
Let H and K be weak bialgebras. The following statements hold. (a) H s and H t are separable Frobenius (so, finite-dimensional) k -algebras. (b) ε s ( y ) = y for y ∈ H s , and ε t ( z ) = z for z ∈ H t . (c) If y ∈ H s and z ∈ H t , then yz = zy . (d) ∆( y ) = 1 ⊗ y = 1 ⊗ y for y ∈ H s , and ∆( z ) = 1 z ⊗ = z ⊗ for z ∈ H t . (e) H s (resp., H t ) is a left (resp., right) coideal subalgebra of H . We also have that H t = { ( ϕ ⊗ Id)∆(1) : ϕ ∈ H ∗ } , H s = { (Id ⊗ ϕ )∆(1) : ϕ ∈ H ∗ } . (f) ε t is an anti-isomorphism from H s to H t , i.e. H s ∼ = H op t as k -algebras. (g) H is a bialgebra if and only if dim k H s = 1 , if and only if dim k H t = 1 . (h) Any nonzero weak bialgebra morphism α : H → K preserves counital subalgebras,i.e. H s ∼ = K s and H t ∼ = K t as k -algebras. NIVERSAL QUANTUM SEMIGROUPOIDS 7
Proof. (a) This follows from [BCJ11, Corollary 4.4] and [BNS99, Proposition 2.11].(b), (c), (d), (e) These parts follow from [BNS99, Section 2.2] and [NV02, Proposi-tions 2.2.1 and 2.2.2].(f) This is an immediate consequence of [BCJ11, Propositions 1.15 and 1.18].(g) This is standard, and follows from (f) and [Nik02, Definition 3.1, Remark 3.2], forinstance.(h) The result for weak Hopf algebras is provided in [NV02, Proposition 2.3.3], and wegeneralize this to weak bialgebras as follows. Write ∆(1 H ) = P ni =1 w i ⊗ z i with { w i } ni =1 and { z i } ni =1 linearly independent. By part (e), H s = span k { w i } ni =1 . Using the linear inde-pendence of { w i } ni =1 , we have(2.4) n = dim k H s . Since z j (b) = ε t ( z j ) = P ni =1 ε ( w i z j ) z i and { z i } ni =1 are linearly independent, we also have(2.5) ε ( w i z j ) = δ i,j . Thereforedim k H s (2.4) = n (2.5) = P nj =1 ε H ( w j z j ) = ε H ((1 H ) (1 H ) ) ( ∗ ) = ε K ((1 K ) (1 K ) ) = dim k K s . Here, ( ∗ ) holds because the nonzero map α : H → K is an algebra and a coalgebra map;that is, 1 K = u K (1 k ) = αu H (1 k ) = α (1 H ) and ε K m K ∆ K (1 K ) = ε K m K ( α ⊗ α ) ∆ H (1 H ) = ε K α m H ∆ H (1 H ) = ε H m H ∆ H (1 H ) . Moreover, since α is a coalgebra map,∆(1 K ) = P ni =1 α ( w i ) ⊗ α ( z i ) . By part (e), K s = span { α ( w i ) } , i.e., α | H s : H s −→ K s is a surjective algebra morphism.Thus, α | H s is bijective. The proof for target subalgebras is similar. (cid:3) In this paper, the main weak bialgebras of interest are the following examples due toHayashi, see, e.g., [Hay96, Example 1.1]. Recall Notation 1.2.
Example 2.6 (Hayashi’s face algebra attached to a quiver) . For a finite quiver Q , we definethe weak bialgebra H ( Q ) as follows. As a k -algebra, H ( Q ) = k h x i,j , x p,q | i, j ∈ Q , p, q ∈ Q i ( R ) , for indeterminates x i,j and x p,q with relations R , given by:(2.7) x p,q x p ′ ,q ′ = δ t ( p ) ,s ( p ′ ) δ t ( q ) ,s ( q ′ ) x p,q x p ′ ,q ′ , (2.8) x s ( p ) ,s ( q ) x p,q = x p,q = x p,q x t ( p ) ,t ( q ) , for all p, p ′ , q, q ′ ∈ Q , and(2.9) x i,j x k,ℓ = δ i,k δ j,ℓ x i,j HONGDI HUANG, CHELSEA WALTON, ELIZABETH WICKS, AND ROBERT WON for all i, j, k, ℓ ∈ Q . (In fact, (2.7) follows from (2.8) and (2.9).) Then H ( Q ) is a unital k -algebra, with unit given by 1 H ( Q ) = P i,j ∈ Q x i,j . (2.10)Let k ≥ p p · · · p k , q q · · · q k ∈ Q k , where each p i , q i ∈ Q . Asshorthand, we define the symbols(2.11) x p ··· p k ,q ··· q k := x p ,q x p ,q · · · x p k ,q k . With this notation, as a vector space we can write H ( Q ) = L ℓ ≥ L a,b ∈ Q ℓ k x a,b . For a, b ∈ Q ℓ , the coalgebra structure is given by(2.12) ∆( x a,b ) = P c ∈ Q ℓ x a,c ⊗ x c,b and ε ( x a,b ) = δ a,b . It can be checked that this structure makes H ( Q ) a weak bialgebra.We record the following facts about H ( Q ). Proposition 2.13.
Let Q be a finite quiver. (a) For p , . . . , p k , q , . . . , q k ∈ Q , ε ( x p ,q · · · x p k ,q k ) = (cid:0) δ t ( p ) ,s ( p ) · · · δ t ( p k − ) ,s ( p k ) (cid:1) (cid:0) δ t ( q ) ,s ( q ) · · · δ t ( q k − ) ,s ( q k ) (cid:1) ( δ p ,q · · · δ p k ,q k ) . (b) For each j ∈ Q , define a j = P i ∈ Q x i,j and a ′ j = P i ∈ Q x j,i . Then { a j } j ∈ Q and { a ′ j } j ∈ Q are complete sets of primitive orthogonal idempotentsin H ( Q ) called the ‘face idempotents’ (see [Hay93] ). (c) As k -vector spaces, H ( Q ) s = L j ∈ Q k a j and H ( Q ) t = L j ∈ Q k a ′ j .Proof. (a) The equation clearly holds for k = 1. We will show this for k = 2; the rest followsby induction: ε ( x p ,q x p ,q ) (2.7) = ε ( δ t ( p ) ,s ( p ) δ t ( q ) ,s ( q ) x p ,q x p ,q ) (2.11) = δ t ( p ) ,s ( p ) δ t ( q ) ,s ( q ) ε ( x p p ,q q ) (2.12) = δ t ( p ) ,s ( p ) δ t ( q ) ,s ( q ) δ p p ,q q = δ t ( p ) ,s ( p ) δ t ( q ) ,s ( q ) δ p ,q δ p ,q . (b) This is straightforward to check.(c) We get ε s ( x a,b ) = δ a,b P i ∈ Q x i,t ( a ) and ε t ( x a,b ) = δ a,b P j ∈ Q x s ( b ) ,j for a, b ∈ Q ℓ . (cid:3) NIVERSAL QUANTUM SEMIGROUPOIDS 9
Corepresentation categories of weak bialgebras.
Here, we discuss the monoidalcategories of corepresentations of weak bialgebras, and algebras within these categories.
Definition 2.14. A monoidal category C = ( C , ⊗ , , α, l, r ) consists of: a category C ; abifunctor ⊗ : C ×C → C ; a natural isomorphism α X,Y,Z : ( X ⊗ Y ) ⊗ Z ∼ → X ⊗ ( Y ⊗ Z ) for each X, Y, Z ∈ C ; an object ∈ C ; and natural isomorphisms l X : ⊗ X ∼ → X, r X : X ⊗ ∼ → X for each X ∈ C , such that the pentagon and triangle axioms are satisfied (see [EGNO15,Equations 2.2, 2.10]).An example of a monoidal category is Vec k , the category of finite-dimensional k -vectorspaces, with ⊗ = ⊗ k , = k , and with the canonical associativity and unit isomorphisms. If H is a weak bialgebra, we can endow the category of right (or left) H -comodules with thestructure of a monoidal category as follows. Example 2.15 ([BCJ11, Nil98]) . For a weak bialgebra H = ( H, m, u, ∆ , ε ), the category M H of right H -comodules can be given the structure of a monoidal category: M H = ( Comod - H, ⊗ , = H s , α = α Vec k , l, r ) . Here, for
M, N ∈ M H , the monoidal product of M and N is defined to be M ⊗ N := (cid:8) m ⊗ n ∈ M ⊗ N | m ⊗ n = ε ( m [1] n [1] ) m [0] ⊗ n [0] (cid:9) . The counital subalgebra H s is naturally a right H -comodule since the image of ∆ | H s is asubspace of H s ⊗ H , and so ∆ | H s can be viewed as a map H s → H s ⊗ H . By [BCJ11,Theorem 3.1], H s is the unit object of the monoidal category M H . By [BCJ11, Section 3],the monoidal category M H has unit isomorphisms: l M : H s ⊗ M → M, x ⊗ m = ε ( x [1] m [1] ) x [0] ⊗ m [0] ε ( xm [1] ) m [0] ,r M : M ⊗ H s → M, m ⊗ x = ε ( x [1] m [1] ) m [0] ⊗ x [0] ε ( m [1] x ) m [0] , for all M ∈ M H . Example 2.16.
Likewise, for a weak bialgebra H = ( H, m, u, ∆ , ε ), the category H M ofleft H -comodules can be given the structure of a monoidal category: H M = ( H - Comod , ⊗ , = H t , α = α Vec k , l, r ) . To the best of our knowledge, the details of the monoidal structure of this category are notexplicitly stated in the literature, so we include them for the convenience of the reader. For
M, N ∈ H M , the monoidal product of M and N is defined to be M ⊗ N := (cid:8) m ⊗ n ∈ M ⊗ N | m ⊗ n = ε ( m [ − n [ − ) m [0] ⊗ n [0] (cid:9) . The restriction of the coproduct ∆ | H t , viewed as a map H t → H ⊗ H t makes H t a left H -comodule which is the unit object of the monoidal category H M . Explicitly, the unitisomorphisms of H M are given by: l M : H t ⊗ M → M, x ⊗ m = ε ( x [ − m [ − ) x [0] ⊗ m [0] ε ( xm [ − ) m [0] r M : M ⊗ H t → M, m ⊗ x = ε ( m [ − x [ − ) m [0] ⊗ x [0] ε ( m [ − x ) m [0] , for all M ∈ H M .Now we turn our attention to algebras in monoidal categories. Definition 2.17 ( Alg ( C )) . Let ( C , ⊗ , , α, l, r ) be a monoidal category. An algebra in C isa triple ( A, m, u ), where A is an object in C , and m : A ⊗ A → A , u : → A are morphismsin C , satisfying unitality and associativity constraints: m ( m ⊗ Id) = m (Id ⊗ m ) α A,A,A , m ( u ⊗ Id) = l A , m (Id ⊗ u ) = r A . A morphism of algebras ( A, m A , u A ) to ( B, m B , u B ) is a morphism f : A → B in C so that f m A = m B ⊗ B ( f ⊗ f ) and f u A = u B . Algebras in C and their morphisms form a category,which we denote by Alg ( C ).Algebras in Vec k are the same as k -algebras.Now we consider algebras that have the structure of a comodule over a weak bialgebra H . There are two related notions: we can consider the objects in Alg ( M H ) (or, Alg ( H M )),or we can consider k -algebras (i.e., objects of Alg ( Vec k )) which are also right (or, left) H -comodules such that the algebra and comodule structures are compatible as done below. Ineither case, we say that H coacts on algebra A if A is an comodule over H . Definition 2.18 ( H A , A H ) . Let H be a weak bialgebra.(a) Consider the category H A of left H -comodule algebras defined as follows. The objectsof H A are objects of Alg ( Vec k ),( A, m A : A ⊗ A → A, u A : k → A ) , with 1 A := u A (1 k ), so that the k -vector space A is a left H -comodule via λ A : A → H ⊗ A, a a [ − ⊗ a [0] , the multiplication map m A is compatible with λ A in the sense that(2.19) ( ab ) [ − ⊗ ( ab ) [0] = a [ − b [ − ⊗ a [0] b [0] ∀ a, b ∈ A ;the unit map u A is compatible with λ A in the sense that(2.20) λ A (1 A ) ∈ H s ⊗ A. The morphisms of H A are maps in Alg ( Vec k ) that are also H -comodule maps.(b) Consider the category A H of right H -comodule algebras defined as follows. Theobjects of A H are objects of Alg ( Vec k ),( A, m A : A ⊗ A → A, u A : k → A ) , with 1 A := u A (1 k ), so that the k -vector space A is a right H -comodule via ρ A : A → A ⊗ H, a a [0] ⊗ a [1] , the multiplication map m A is compatible with ρ A in the sense that( ab ) [0] ⊗ ( ab ) [1] = a [0] b [0] ⊗ a [1] b [1] ∀ a, b ∈ A ;the unit map u A is compatible with ρ A in the sense that ρ A (1 A ) ∈ A ⊗ H t . The morphisms of A H are maps in Alg ( Vec k ) that are also H -comodule maps.The categories H A and Alg ( H M ) (likewise, A H and Alg ( M H )) are essentially the same. NIVERSAL QUANTUM SEMIGROUPOIDS 11
Proposition 2.21. [WWW19, Theorem 4.5]
There is an isomorphism of categories between
Alg ( M H ) and A H , and between Alg ( H M ) and H A . (cid:3) In [WWW19, Theorem 4.5], the functors between
Alg ( M H ) and A H are given explicitly.For the isomorphism between Alg ( H M ) and H A , the proof should be adjusted using thestructures in Example 2.16 rather than Example 2.15.2.3. Examples.
Now we provide some examples of comodule algebras over weak bialgebras,which will be important in the rest of the paper.
Example 2.22.
Consider Hayashi’s face algebra H ( Q ) from Example 2.6. It is straightfor-ward to check that the path algebra k Q belongs to H ( Q ) A and to A H ( Q ) via the coactions: λ : k Q → H ( Q ) ⊗ k Q ρ : k Q → k Q ⊗ H ( Q ) e j P i ∈ Q x j,i ⊗ e i e j P i ∈ Q e i ⊗ x i,j q P p ∈ Q x q,p ⊗ p q P p ∈ Q p ⊗ x p,q , for j ∈ Q and q ∈ Q . See [WWW19, Example 4.10] for verification that k Q ∈ A H ( Q ) . Example 2.23.
Let Q •• be the quiver with two vertices and no arrows. Let D be thealgebra D = k h x, y i ( x = x, y = y, xy = yx = 0)so that 1 D = x + y (as an algebra, D ∼ = k Q •• ). Define a coproduct ∆ D on D by∆ D ( x ) = x ⊗ x + y ⊗ y, ∆ D ( y ) = x ⊗ y + y ⊗ x and a counit ε D by ε D ( x ) = 1 k , ε D ( y ) = 0 k . One can verify that this makes D a bialgebra.One can show that k Q •• is a transposed D -comodule algebra [Definition 1.5(a)] underthe left and right coactions: λ : k Q •• → D ⊗ k Q •• , e x ⊗ e + y ⊗ e , e y ⊗ e + x ⊗ e ; ρ : k Q •• → k Q •• ⊗ D, e e ⊗ x + e ⊗ y, e e ⊗ y + e ⊗ x. For our next example, we will need the following two lemmas. These lemmas are well-known, and their proofs are routine.
Lemma 2.24. If ( H, m H , u H , ∆ H , ε H ) and ( K, m K , u K , ∆ K , ε K ) are weak bialgebras, then H ⊕ K is a weak bialgebra with the following structure for all h, g ∈ H, k, l ∈ K :multiplication: ( h, k )( g, l ) := ( hg, kl ); unit: H ⊕ K := (1 H , K ); comultiplication: ∆ H ⊕ K (( h, k )) := ( h , ⊗ ( h ,
0) + (0 , k ) ⊗ (0 , k ); counit: ε H ⊕ K (( h, k )) := ε H ( h ) + ε K ( k ) . We also have that ( ε H ⊕ K ) t ( h, k ) = (( ε H ) t ( h ) , ( ε K ) t ( k )) , ( ε H ⊕ K ) s ( h, k ) = (( ε H ) s ( h ) , ( ε K ) s ( k )) . (cid:3) Lemma 2.25.
Suppose that V is a right H -comodule via ρ H : V → V ⊗ H, v v [0] ⊗ v [1] . Then V is a right ( H ⊕ K ) -comodule via ρ : V → V ⊗ ( H ⊕ K ) , v v [0] ⊗ ( v [1] , . Furthermore, if V is a right H -comodule algebra via ρ H , then V is a right ( H ⊕ K ) -comodulealgebra via ρ . A similar statement holds for left H -comodules and left H -comodule algebras. (cid:3) Example 2.26.
Let Q •• be the quiver with two vertices and no arrows, and recall thebialgebra D defined in Example 2.23. A presentation of D is given by D := k h y , , y , , y , , y , i ( y , = y , , y , = y , , y i,j y i,k = δ j,k y i,j , y j,i y k,i = δ j,k y j,i ) , with unit 1 D = y , + y , .Claim 1. k Q •• is a left and right ( D ⊕ D )-comodule algebra via linear coactions. Proof of Claim 1 . The coalgebra structure is given by∆ D ( y i,j ) = P k ∈ ( Q •• ) y i,k ⊗ y k,j , ε D ( y i,j ) = δ i,j , for all i, j ∈ ( Q •• ) . With this presentation, D left and right coacts linearly on k Q •• via k Q •• → D ⊗ k Q •• , e i P j ∈ ( Q •• ) y i,j ⊗ e j k Q •• → k Q •• ⊗ D, e i P j ∈ ( Q •• ) e j ⊗ y j,i . By Lemma 2.25 and Example 2.23, we have that the coactions λ : k Q •• → ( D ⊕ D ) ⊗ k Q •• ρ : k Q •• → k Q •• ⊗ ( D ⊕ D ) e i P j ∈ ( Q •• ) ( y i,j , ⊗ e j e i P j ∈ ( Q •• ) e j ⊗ ( y j,i , (cid:3) Claim 2. ( D ⊕ D ) t ∼ = k ( Q •• ) as algebras over k . Proof of Claim 2 . Consider the morphism ψ : k ( Q •• ) → ( D ⊕ D ) t , e (1 D , , e (0 , D ) . First, we will show that as a k -vector space, ( D ⊕ D ) t = Span k { (1 D , , (0 , D ) } . ByLemma 2.24, we have( ε D ⊕ D ) t (1 D ,
0) = (( ε D ) t (1 D ) ,
0) = (1 D , , ( ε D ⊕ D ) t (0 , D ) = (0 , ( ε D ) t (1 D )) = (0 , D ) . Therefore, Span k { (1 D , , (0 , D ) } ⊆ ( D ⊕ D ) t . To show the reverse inclusion, note that for a, b ∈ D we have( ε D ⊕ D ) t ( a, b ) = (( ε D ) t ( a ) , ( ε D ) t ( b )) = ( ε D ( a )1 D , ε D ( b )1 D ) , where the last equality holds because D is a bialgebra. Thus, k ( Q •• ) ∼ = dim( D ⊕ D ) t as k -vector spaces; here, dim( D ⊕ D ) t = dim k ( Q •• ) = 2. It is also clear that ψ preserves theunit and multiplication. Therefore, ψ is an isomorphism of k -algebras. NIVERSAL QUANTUM SEMIGROUPOIDS 13
Claim 3. k ( Q •• ) = ( D ⊕ D ) t as left ( D ⊕ D )-comodules, where k ( Q •• ) is a left ( D ⊕ D )-comodule via Claim 1, and ( D ⊕ D ) t is naturally a left ( D ⊕ D )-comodule via comultiplication[Example 2.16]. Proof of Claim 3 . By way of contradiction, suppose that we have an isomorphism ϕ : k ( Q •• ) → ( D ⊕ D ) t of left ( D ⊕ D )-comodules. Explicitly, the comodule structures aregiven by λ : k ( Q •• ) → ( D ⊕ D ) ⊗ k ( Q •• ) , e i P j ∈ ( Q •• ) ( y i,j , ⊗ e j ,λ t := ∆ D ⊕ D | ( D ⊕ D ) t : ( D ⊕ D ) t → ( D ⊕ D ) ⊗ ( D ⊕ D ) t , (1 D , (1 D , ⊗ (1 D , , (0 , D ) (0 , D ) ⊗ (0 , D ) . Since ( D ⊕ D ) t = Span k { (1 D , , (0 , D ) } , (see proof of Claim 2), we can write ϕ ( e i ) = α i (1 D ,
0) + β i (0 , D ) , for some α i , β i ∈ k . Since ϕ is a left ( D ⊕ D )-comodule map, (Id ( D ⊕ D ) ⊗ ϕ ) λ = λ t ϕ . Inparticular, P j ∈ ( Q •• ) ( y i,j , ⊗ ( α j (1 D ,
0) + β j (0 , D )) = P j ∈ ( Q •• ) ( y i,j , ⊗ ϕ ( e j )= (Id D ⊕ D ⊗ ϕ ) (cid:16)P j ∈ ( Q •• ) ( y i,j , ⊗ e j (cid:17) = (Id D ⊕ D ⊗ ϕ ) λ ( e i )= λ t ϕ ( e i )= λ t ( α i (1 D ,
0) + β i (0 , D ))= α i (1 D , ⊗ (1 D ,
0) + β i (0 , D ) ⊗ (0 , D ) . Notice that the left hand side is contained in ( D ⊕ ⊗ ( D ⊕ D ) t . Therefore, we musthave that β i = 0, since if not, the right hand side is not contained in ( D ⊕ ⊗ ( D ⊕ D ) t .Therefore, ϕ is not surjective and not an isomorphism of ( D ⊕ D )-comodules. (cid:3) Universal linear coactions on graded algebras
In this section, we introduce the notion of a weak bialgebra that coacts linearly anduniversally on a graded algebra A as in Hypothesis 1.1. The universal weak bialgebrascoacting on A are defined below in Definitions 3.4 and 3.10 below; we call them universalquantum linear semigroupoids . Recall here that A is N -graded k -algebra with dim k A i < ∞ for all i ∈ N , such that A is a commutative, separable (so, Frobenius) k -algebra (we discusshow the assumptions on A are used in Remarks 3.5 and 3.11 below). Moreover, we saythat A is connected if A = k , and that A is non-connected otherwise.To proceed, we reinterpret the standing assumption, Hypothesis 1.3 from the introduc-tion, as follows. Hypothesis 3.1. [ λ, λ i , ρ, ρ i ] Let H be a weak bialgebra, and recall the notion of a H -comodule algebra from Definition 2.18. From now on, we impose the assumptions below.(a) Each left H -comodule algebra structure on A will be linear in the sense that, forthe structure map λ := λ HA : A → H ⊗ A , the restriction λ | A i := λ i makes A i a left H -comodule for each i . (b) Each right H -comodule algebra structure on A will be linear in the sense that, forthe structure map ρ := ρ HA : A → A ⊗ H , the restriction ρ | A i := ρ i makes A i a right H -comodule for each i . Remark 3.2. If H left coacts linearly via λ on A , then A is a left H -comodule algebravia λ . By [WWW19, Theorem 4.5], we can view A as an object in the category H A [Definition 2.18]. The analogous statement holds for right coactions.Next, we discuss a na¨ıve notion of a weak bialgebra coacting universally on A , that is, bymerely replacing ‘bialgebra’ with ‘weak bialgebra’ in the definition of a universal quantumlinear semigroup [Definition 1.4]. This weak bialgebra fails to exist, even for an easy exampleof non-connected graded algebra A , as seen below. Remark 3.3.
Let A be an algebra satisfying Hypothesis 1.1. Suppose that there exists aweak bialgebra U := U ( A ) that left coacts on A so that, for every weak bialgebra H that leftcoacts on A , there exists a unique weak bialgebra map π : U → H so that ( π ⊗ Id A ) λ U = λ H .We will show that in general, such a weak bialgebra fails to exist.Let H be any nonzero weak bialgebra which left coacts on A . Then since there exists aweak bialgebra map π : U → H , we have that dim k U t = dim k H t by Proposition 2.3(h).Now take A = k Q •• as in Section 2.3, which is a comodule algebra over both the bialgebra D (Example 2.23) and also over the weak bialgebra D ⊕ D (Example 2.26). By the above, if wetake H = D , then we have dim k U t = dim k D t , and so by Proposition 2.3(g), dim k U t = 1.On the other hand, we can also substitute H by D ⊕ D and have dim k U t = dim k ( D ⊕ D ) t ;by Claim 2 of Example 2.26, dim k ( D ⊕ D ) t = dim k k ( Q •• ) = 2. Hence 1 = dim k U t = 2,which is a contradiction. Hence, U ( k Q •• ) does not exist.To remedy the non-existence issue in the remark above, we impose an extra hypothesisrelating A to the counital subalgebras of our universally coacting weak bialgebras. Thisis motivated by Claim 3 in Example 2.26. Our main result, Theorem 4.17 below, showsthat with this additional hypothesis, for any path algebra k Q , there exists a universal weakbialgebra coacting on k Q . Definition 3.4 (left UQSGd, O left ( A ); right UQSGd, O right ( A )) . Take a k -algebra A as inHypothesis 1.1.(a) Let O := O left ( A ) be a weak bialgebra that left coacts on A with A ∼ = O t in O A , sothat for any weak bialgebra H that left coacts on A with A ∼ = H t in H A , there isa unique weak bialgebra map π : O → H so that ( π ⊗ Id A ) λ O = λ H . We refer to O left ( A ) as the left universal quantum linear semigroupoid (left UQSGd) of A , andrefer to its coaction on A as universally base preserving .(b) Let O := O right ( A ) be a weak bialgebra that right coacts on A with A ∼ = O s in A O ,so that for any weak bialgebra H that right coacts on A with A ∼ = H s in H A , thereis a unique weak bialgebra map π : O → H so that (Id A ⊗ π ) ρ O = ρ H . We refer to O right ( A ) as the right universal quantum linear semigroupoid (right UQSGd) of A ,and refer to its coaction on A as universally base preserving .Here, the left (resp., right) H -coaction on A is given by λ (resp., ρ ) as in Remark 3.2, andthe left (resp., right) H -coaction on H t (resp., on H s ) is given by ∆ H as in Example 2.16. NIVERSAL QUANTUM SEMIGROUPOIDS 15
We make several remarks about the definition above.
Remark 3.5.
We use the assumption that A is Frobenius and separable (from Hypothe-sis 1.1) in the definition above and in Definition 3.10 below. Namely, for any weak bialgebra H , the counital subalgebras H s and H t are Frobenius and separable k -algebras [Proposi-tion 2.3(a)]. We do not need to require that A is commutative for Definition 3.4. Remark 3.6.
Note that the notion of universally base preserving coaction is weaker thanthe na¨ıve notion of a universal coaction discussed in Remark 3.3. Thus, the UQSGds inDefinition 3.4 are more likely to exist than the universal weak bialgebras in Remark 3.3.
Remark 3.7.
Observe that the universally base preserving condition takes a simple formwhen viewed through a categorical lens. By Proposition 2.21, we have categorical isomor-phisms
Alg ( H M ) ∼ = H A and Alg ( M H ) ∼ = A H , and by Examples 2.15 and 2.16, the unitobjects of the monoidal categories Alg ( H M ) , Alg ( M H ) are H t , H s , respectively. So, therequirement that H t ∼ = A in H A (resp., H s ∼ = A in A H ) is equivalent to requiring that A is isomorphic to the unit object of the monoidal category Alg ( H M ) (resp., Alg ( M H )). Remark 3.8.
Definition 3.4 generalizes Definition 1.4, the notion of a one-sided UQSG (or,universal bialgebra that coacts from one side). Indeed, take A a locally finite, connected N -graded algebra and suppose that O left ( A ) exists. Then, ( O left ( A )) t = A = k , as k -vectorspaces. So, O left ( A ) must also be a bialgebra by Proposition 2.3(g), and thus, we recoverthe left UQSG O left ( A ) of A when A is connected.To generalize the transposed UQSG from Definition 1.5(b) to the weak bialgebra setting,we need the following definitions. First, recall the transposed coaction from Definition 1.5(a)which we reinterpret below. Definition 3.9.
Suppose that H is a weak bialgebra coacting linearly on A on the left andright via coactions λ : A → H ⊗ A and ρ : A → A ⊗ H . Then for each i , H coacts fromthe left and right on A i via the restrictions λ i and ρ i . We call A a transposed H -comodulealgebra if for each i , there exists a basis { v ij } ≤ j ≤ dim A i for A i such that the coactions canbe written in the following form: λ i : A i → H ⊗ A i ρ i : A i → A i ⊗ Hv ij P ≤ k ≤ dim A i z ij,k ⊗ v ik v ij P ≤ k ≤ dim A i v ik ⊗ z ik,j , for some z ij,k ∈ H . Definition 3.10 (transposed UQSGd, O trans ( A )) . Let O := O trans ( A ) be a weak bialgebrasuch that A is a transposed O -comodule algebra with A ∼ = O t in O A and A ∼ = O s in A O ,so that for any weak bialgebra H for which A is a transposed H -comodule algebra with A ∼ = H t in H A and A ∼ = H s in A H , there exists a unique weak bialgebra map π : O → H such that ( π ⊗ Id A ) λ O = λ H and (Id A ⊗ π ) ρ O = ρ H . We call O trans ( A ) the transposeduniversal quantum linear semigroupoid (transposed UQSGd) of A . Remark 3.11.
We use the assumption that A is commutative in Definition 3.10. Namely,by Proposition 2.3(f): A ∼ = H t ∼ = H op s ∼ = A op0 as k -algebras. Question 3.12 (P. Etingof) . Can the assumption that A is commutative be removed byaltering Definition 3.10 (so that the results in the remainder of the paper are unaffected)? Remark 3.13.
For the same reasons as given in Remark 3.8, we can see that the abovedefinition is a generalization of the transposed UQSG from Definition 1.5(b).
Remark 3.14.
We only define the left/right/transposed UQSGd of A up to weak bialgebraisomorphism, and it is unique (up to weak bialgebra isomorphism) if it exists.Now we show that if the left and right UQSGd of A exist and are isomorphic to each other,then the transposed UQSGd of A exists and is isomorphic to the left (or right) UQSGd. Toproceed, consider the following terminology. Definition 3.15.
Let H be a weak bialgebra and let λ : A → H ⊗ A be a left coaction. Wecall this coaction inner-faithful if, whenever λ ( A ) ⊆ K ⊗ A for some weak subbialgebra K ,we must have that K = H . Right inner-faithful coactions are defined similarly. Lemma 3.16.
Take A as in Hypothesis 1.1, and suppose that O left ( A ) exists. (a) Suppose H is a weak bialgebra that left coacts on A with A ∼ = H t in H A . Then, H coacts on A inner-faithfully if and only if the weak bialgebra map π : O left ( A ) → H (that arises from Definition 3.4(a)) is surjective. (b) The weak bialgebra O left ( A ) left coacts on A inner-faithfully.Similar statements hold for right (resp., transposed) coactions and for the UQSGd O right ( A ) (resp., O trans ( A ) ).Proof. (a) If π is not surjective, then let K := im( π ) which is a proper weak subbialgebraof H . We get that K left coacts on A via λ K = ( π ⊗ Id A ) λ O left ( A ) : A → K ⊗ A . Therefore, H does not left coact on A inner-faithfully.Conversely, suppose that H does not coact on A inner-faithfully, and that there exists aproper weak subbialgebra K of H (via inclusion ι ) so that the coaction of K on A factorsthrough H on A . Then, ( π ⊗ Id A ) λ O left ( A ) = λ H = ( ι ⊗ Id A ) λ K . Now the im( π ) consists ofthe coefficients of λ K in K . So, im( π ) cannot be H , and π is not surjective.(b) This follows from part (a) by taking π = Id O left ( A ) . (cid:3) Proposition 3.17.
Suppose that O left ( A ) and O right ( A ) exist, and let O ( A ) := O left ( A ) . Suppose that O left ( A ) ∼ = O right ( A ) as weak bialgebras, and that their respective coactions on A are transpose. Then O trans ( A ) exists, and O trans ( A ) ∼ = O ( A ) as weak bialgebras.Proof. Assume that O ( A ) := O left ( A ) and O right ( A ) exist. For simplicity of proof, assumethat O ( A ) = O right ( A ) as weak bialgebras (instead of using an isomorphism). Now, supposethat we have a weak bialgebra H that left coacts and right coacts (via transposed coactions λ H : A → H ⊗ A, ρ H : A → A ⊗ H ) with the property that H s ∼ = A in A H and H t ∼ = A in H A . We will show that O ( A ) satisfies the universal property described in Definition 3.10;therefore, O trans ( A ) exists and O trans ( A ) ∼ = O ( A ) as weak bialgebras.Since O ( A ) := O left ( A ) and O right ( A ) exist, we have the following maps: λ L : A → O ( A ) ⊗ A, π L : O ( A ) → H, ρ R : A → A ⊗ O ( A ) , π R : O ( A ) → H. NIVERSAL QUANTUM SEMIGROUPOIDS 17 with the property that O ( A ) left coacts on A via λ L , O right ( A ) right coacts on A via ρ R , λ L and ρ R are transposed coactions, and π L and π R are the unique weak bialgebra mapssatisfying the following equations:( π L ⊗ Id A ) λ L = λ H , (Id A ⊗ π R ) ρ R = ρ H . We make the following definitions: λ := λ L : A → O ( A ) ⊗ A, ρ := ρ R : A → A ⊗ O ( A ) , π := π L : O ( A ) → H. We will prove that π is the unique weak bialgebra map such that (Id A ⊗ π ) ρ = ρ H . Thiswill imply that O ( A ) has the universal property of Definition 3.10, so we must have O ( A ) ∼ = O trans ( A ) as weak bialgebras. In fact, since π R is the unique weak bialgebra map such that(Id A ⊗ π R ) ρ = ρ H , it suffices to show that π = π R . Since λ H and ρ H are transposed coactions, for each i there exists a basis { v ij } ≤ j ≤ dim A i of A i such that the restricted coactions can be written in the following form: λ Hi : A i → H ⊗ A i ρ Hi : A i → A i ⊗ Hv ij P ≤ k ≤ dim A i z ij,k ⊗ v ik v ij P ≤ k ≤ dim A i v ik ⊗ z ik,j , for some z ij,k ∈ H . Since { v ij } is a basis for A i and the coactions λ, ρ are transpose, we canwrite λ i : A i → O ( A ) ⊗ A i ρ i : A i → A i ⊗ O ( A ) v ij P ≤ k ≤ dim A i y ij,k ⊗ v ik v ij P ≤ k ≤ dim A i v ik ⊗ y ik,j , for some y ij,k ∈ O ( A ).By the previous claims, we know that ( π ⊗ Id A ) λ = λ H and (Id A ⊗ π R ) ρ = ρ H . Therefore,for each v ij we have P ≤ k ≤ dim A i π ( y ij,k ) ⊗ v ik = ( π ⊗ Id A ) λ i ( v ij ) = λ Hi ( v ij ) = P ≤ k ≤ dim A i z ij,k ⊗ v ik . Since the { v ik } are a basis for A i , we know that π ( y ij,k ) = z ij,k for each i, j, k. Similarly, foreach v ij we have P ≤ k ≤ dim A i v ik ⊗ π R ( y ik,j ) = (Id A ⊗ π R ) ρ i ( v ij ) = ρ Hi ( v ij ) = P ≤ k ≤ dim A i v ik ⊗ z ik,j . Since the { v ik } are a basis for A i , we know that π R ( y ik,j ) = z ik,j for each i, j, k. Therefore,for each i, j, k , we have π R ( y ij,k ) = z ij,k = π ( y ij,k ) . Since the coactions are inner-faithful by Lemma 3.16, O ( A ) is generated as an algebra bythe y ij,k . (Else, there exists an algebra generator of O ( A ) not in the set { y ij,k } and the properweak subbialgebra generated by the y ij,k coacts on A , contradicting the inner-faithfulness ofthe coaction of O ( A ) on A .) Finally, π and π R are algebra maps, so we must have π = π R ,as desired. (cid:3) Universal quantum linear semigroupoids of a path algebra
In this section, we will prove our main theorem, Theorem 4.17, constructing the left,right, and transposed UQSGds of the path algebra k Q of a finite quiver Q . Furthermore, wewill show that all three are isomorphic to Hayashi’s face algebra H ( Q ) (Example 2.6). Notethat the coactions in the following hypothesis is a specific case of that in the Definition 3.9.In several of our results, we will assume one of the three hypotheses given below. Hypothesis 4.1.
Let Q be a finite quiver, and let ( H, m, u, ∆ , ε ) be a weak bialgebra.Consider the following formulas for each j ∈ Q and each q ∈ Q , λ : k Q → H ⊗ k Q ρ : k Q → k Q ⊗ He j P i ∈ Q y j,i ⊗ e i e j P i ∈ Q e i ⊗ y i,j q P p ∈ Q y q,p ⊗ p q P p ∈ Q p ⊗ y p,q for some elements y i,j ∈ H and y p,q ∈ H . We will consider three separate hypotheses inthe sequel.(a) Assume that k Q is a left H -comodule algebra via a coaction of the form λ .(b) Assume that k Q is a right H -comodule algebra via a coaction of the form ρ .(c) Assume that k Q is a transposed H -comodule algebra via λ and ρ .The following results will be of use in this section. Lemma 4.2.
Let H be a weak bialgebra which coacts on k Q as in Hypothesis 4.1. Considerthe following formulas for any i, j, k ∈ Q and p, q ∈ Q : ∆( y i,j ) = P k ∈ Q y i,k ⊗ y k,j , ε ( y i,j ) = δ i,j · k , (4.3) ∆( y p,q ) = P r ∈ Q y p,r ⊗ y r,q , ε ( y p,q ) = δ p,q · k , (4.4) y k,i y k,j = δ i,j y k,i (4.5) y i,k y j,k = δ i,j y i,k (4.6) y s ( p ) ,s ( q ) y p,q = y p,q (4.7) y p,q y t ( p ) ,t ( q ) = y p,q . (4.8)(a) If Hypothesis 4.1(a) holds, then H satisfies (4.3) , (4.4) , (4.6) , (4.7) , and (4.8) . (b) If Hypothesis 4.1(b) holds, then H satisfies (4.3) , (4.4) , (4.5) , (4.7) , and (4.8) . (c) If Hypothesis 4.1(c) holds, then H satisfies (4.3) to (4.8) .Proof. We will prove (a). The proof for (b) is similar and hence omitted, while (c) followsfrom (a) and (b).The formulas for ∆ and ε follow from the coassociativity and counitality of λ . Forexample, for i ∈ Q , P j ∈ Q ∆( y i,j ) ⊗ e j = (∆ ⊗ Id) λ ( e i ) = (Id ⊗ λ ) λ ( e i )= P k ∈ Q y i,k ⊗ λ ( e k ) = P j,k ∈ Q y i,k ⊗ y k,j ⊗ e j . Since the e j are linearly independent, we have that ∆( y i,j ) = P k ∈ Q y i,k ⊗ y k,j . Further,since e i = Id k Q ( e i ) = ( ε ⊗ Id k Q ) λ ( e i ) = P j ∈ Q ε ( y i,j ) e j , we conclude that ε ( y i,j ) = δ i,j · k .This proves (4.3); the proof for (4.4) is similar. NIVERSAL QUANTUM SEMIGROUPOIDS 19
Next, we will use the fact that k Q is a left H -comodule algebra. We have P k ∈ Q δ i,j y i,k ⊗ e k = λ ( δ i,j e i )= λ ( e i e j ) (2.19) = λ ( e i ) λ ( e j )= (cid:16)P k ∈ Q y i,k ⊗ e k (cid:17) (cid:16)P ℓ ∈ Q y j,ℓ ⊗ e ℓ (cid:17) = P k,ℓ ∈ Q y i,k y j,ℓ ⊗ e k e ℓ = P k ∈ Q y i,k y j,k ⊗ e k . Since the e k are linearly independent, we must have δ i,j y i,k = y i,k y j,k , that is, (4.6) holds.To show (4.7), notice that for p ∈ Q we have P q ∈ Q y p,q ⊗ q = λ ( p )= λ ( e s ( p ) p )= λ ( e s ( p ) ) λ ( p )= (cid:16)P i ∈ Q y s ( p ) ,i ⊗ e i (cid:17) (cid:16)P q ∈ Q y p,q ⊗ q (cid:17) = P i ∈ Q ,q ∈ Q y s ( p ) ,i y p,q ⊗ e i q = P q ∈ Q y s ( p ) ,s ( q ) y p,q ⊗ q. By linear independence of the set { q } q ∈ Q , we have y p,q = y s ( p ) ,s ( q ) y p,q for all p, q ∈ Q .We use the relation p = pe t ( p ) for p ∈ Q to prove (4.8) in the same manner as (4.7). (cid:3) The following proposition is a collection of identities that hold if we only assume theexistence of a left H -coaction making k Q an H -comodule algebra. Proposition 4.9.
Let H be a weak bialgebra which coacts on k Q as in Hypothesis 4.1(a).For each j ∈ Q , consider the elements η j := P i ∈ Q y i,j , θ j := P i ∈ Q y j,i . The following statements hold. (a)
For each j ∈ Q , η j and θ j are non-zero elements of H . (b) For each j ∈ Q , η j is an idempotent element of H s . (c) If k Q ∼ = ψ H t as left H -comodule algebras, then the following statements hold: (i) { η k } k ∈ Q is a k -basis of H s . (ii) 1 H = P i,j ∈ Q y i,j . (iii) For each k ∈ Q , ψ ( e k ) = θ k ; hence θ k ∈ H t . (iv) The set { θ i } i ∈ Q is a k -basis for H t of orthogonal idempotent elements. (v) The set { η j } j ∈ Q is a k -basis for H s of orthogonal idempotent elements. (vi) For all i, j, k, ℓ ∈ Q , y i,j y k,ℓ = δ i,k δ j,ℓ y i,j .Proof. (a) To show that η j is non-zero, we note that ε ( η j ) = ε (cid:16)P i ∈ Q y i,j (cid:17) (4.3) = 1 . A similar calculation shows that ε ( θ j ) = 1, so θ j is non-zero. (b) Let j ∈ Q . Then η j = P i,k ∈ Q y i,j y k,j (4.6) = P i ∈ Q y i,j = η j so η j is idempotent. Moreover, note that P i,j ∈ Q y i,j ⊗ e j = λ ( P i ∈ Q e i ) = λ (1 k Q ) (2.20) = ( ε s ⊗ Id k Q ) λ (1 k Q )= ( ε s ⊗ Id k Q ) λ ( P i ∈ Q e i ) = P j ∈ Q ε s ( P i ∈ Q y i,j ) ⊗ e j . Since the e j are linearly independent, for each j ∈ Q we have η j := P i ∈ Q y i,j ∈ H s .(c) Suppose that k Q ∼ = H t as left H -comodule algebras. Then there exists an algebraisomorphism ψ : k Q → H t which is also a map of left H -comodules. Hence,(4.10) (Id H ⊗ ψ ) λ = λ H t ψ, for λ H t = ∆ H | H t by Example 2.16.(i) Evaluating the left-hand side on 1 k Q , we have(Id ⊗ ψ ) λ (1 k Q ) = (Id ⊗ ψ ) λ ( P i ∈ Q e i ) = (Id ⊗ ψ )( P i,j ∈ Q y i,j ⊗ e j )= P j ∈ Q (cid:16)P i ∈ Q y i,j (cid:17) ⊗ ψ ( e j ) = P j ∈ Q η j ⊗ ψ ( e j ) . On the other hand, ψ is an algebra map, so ψ (1 k Q ) = 1 H . Thus, ∆ ψ (1 k Q ) = 1 ⊗ . Hence,(4.11) 1 ⊗ = P j ∈ Q η j ⊗ ψ ( e j ) . Since the distinct e j are linearly independent and ψ is an algebra isomorphism, the ψ ( e j ) arealso linearly independent. Now by Proposition 2.3(e), we conclude that the { η j } j ∈ Q span H s . By Proposition 2.3(f), we have dim k H t = dim k H s . Therefore dim k H s = dim k k Q = | Q | , so we have that { η j } j ∈ Q is a k -basis of H s .(ii) For any k ∈ Q we have P j ∈ Q η j ψ ( e k ) ⊗ ψ ( e j ) (4.11) = 1 ψ ( e k ) ⊗ = ∆ ψ ( e k ) (4.10) = (Id H ⊗ ψ ) λ ( e k ) = P j ∈ Q y k,j ⊗ ψ ( e j ) . Since the ψ ( e j ) are linearly independent, we have that(4.12) η j ψ ( e k ) = y k,j for each j, k ∈ Q .By (4.11),(4.13) 1 H = 1 ε (1 ) = P j ∈ Q η j ε ( ψ ( e j )) . Next, consider the following calculation: P k ∈ Q η k ⊗ ψ ( e k ) (4.11) = 1 ⊗ = ∆(1 H ) (4.13) = ∆( P j ∈ Q ε ( ψ ( e j )) η j )= ∆( P i,j ∈ Q ε ( ψ ( e j )) y i,j ) (4.3) = P i,j,k ∈ Q ε ( ψ ( e j )) y i,k ⊗ y k,j NIVERSAL QUANTUM SEMIGROUPOIDS 21 = P k ∈ Q ( P i ∈ Q y i,k ) ⊗ ( P j ∈ Q ε ( ψ ( e j )) y k,j )= P k ∈ Q η k ⊗ ( P j ∈ Q ε ( ψ ( e j )) y k,j ) . Since, by (i), the { η j } j ∈ Q are linearly independent, we must have(4.14) ψ ( e k ) = P j ∈ Q ε ( ψ ( e j )) y k,j for each k ∈ Q . Notice that y k,j (4.12) = η j ψ ( e k ) (4.14) = η j P ℓ ∈ Q ε ( ψ ( e ℓ )) y k,ℓ = P i,ℓ ∈ Q ε ( ψ ( e ℓ )) y i,j y k,ℓ . Multiplying both sides of the equation on the left by y k,j yields(4.15) y k,j (4.6) = ( y k,j ) = P i,ℓ ∈ Q ε ( ψ ( e ℓ )) y k,j y i,j y k,ℓ (4.6) = P i,ℓ ∈ Q ε ( ψ ( e ℓ )) δ i,k y k,j y k,ℓ = y k,j P ℓ ∈ Q ε ( ψ ( e ℓ )) y k,ℓ (4.14) = y k,j ψ ( e k ) . Now for each k ∈ Q , we get1 k (4.3) = ε ( y k,k ) (4.15) = ε ( y k,k ψ ( e k )) = ε ( y k,k ) ε (1 ψ ( e k )) (4.11) = P i ∈ Q ε ( y k,k η i ) ε ( ψ ( e i ) ψ ( e k )) ψ alg. map = P i ∈ Q ε ( y k,k η i ) ε ( δ i,k ψ ( e k ))= ε ( y k,k η k ) ε ( ψ ( e k ))= ε (cid:16)P i ∈ Q y k,k y i,k (cid:17) ε ( ψ ( e k )) (4.6) = ε (cid:16)P i ∈ Q δ i,k y k,k (cid:17) ε ( ψ ( e k )) (4.3) = ε ( ψ ( e k )) . Finally, 1 H (4.13) = P j ∈ Q ε ( ψ ( e j )) η j = P j ∈ Q η j = P i,j ∈ Q y i,j . (iii) For each k ∈ Q , we have θ k = P j ∈ Q y k,j (4.12) = P j ∈ Q η j ψ ( e k ) = ( P i,j ∈ Q y i,j ) ψ ( e k ) (ii) = ψ ( e k ) . (iv) Since { e i } i ∈ Q is a k -basis of k Q of orthogonal idempotent elements and ψ is analgebra isomorphism, { ψ ( e i ) } i ∈ Q is a k -basis of orthogonal idempotents of H t . By, part(iii), the claim follows. (v) Since k Q ∼ = ψ H t as algebras, by Proposition 2.3(f) we have H s ∼ = γ L i ∈ Q k . Let { E i } i ∈ Q be a set of primitive idempotents of L i ∈ Q k . By part (b), for each k ∈ Q , η k isan idempotent of H s . Hence, γ ( η k ) = P i ∈ I k E i for some subset I k of Q . Therefore, P k ∈ Q P i ∈ I k E i = P k ∈ Q γ ( η k ) = γ ( P i,k ∈ Q y i,k ) (ii) = γ (1 H ) = 1 L k = P i ∈ Q E i . As a result, we conclude that γ ( η k ) = E i for some i ∈ Q and for k = j , we have that γ ( η k ) = γ ( η j ). Thus, { η k } k ∈ Q is a set of orthogonal idempotents of H s .(vi) For each i, j, k, ℓ ∈ Q , we have: y i,j y k,ℓ (4.12) = η j ψ ( e i ) η ℓ ψ ( e k ) (iii) = η j θ i η ℓ θ k = θ i θ k η j η ℓ (iv) , (v) = δ i,k δ j,ℓ θ i η j (iii) , (4.12) = δ i,k δ j,ℓ y i,j , where the third equality holds by parts (b), (iii), and Proposition 2.3(c). (cid:3) The analogue of Proposition 4.9 for a weak bialgebra coaction on k Q satisfying Hypoth-esis 4.1(b) also holds, and follows by a similar proof. Proposition 4.16.
Let H be a weak bialgebra which coacts on k Q as in Hypothesis 4.1(b).For each j ∈ Q , consider the elements η j := P i ∈ Q y i,j , θ j := P i ∈ Q y j,i . The following statements hold. (a)
For each j ∈ Q , η j and θ j are non-zero elements of H . (b) For each j ∈ Q , θ j is an idempotent element of H t . (c) If k Q ∼ = φ H s as right H -comodule algebras, then the following statements hold: (i) { θ k } k ∈ Q is a k -basis of H t . (ii) 1 H = P i,j ∈ Q y i,j . (iii) For each k ∈ Q , φ ( e k ) = η k ; hence η k ∈ H s . (iv) The set { η j } j ∈ Q is a k -basis for H s of orthogonal idempotent elements. (v) The set { θ i } i ∈ Q is a k -basis for H t of orthogonal idempotent elements. (vi) For all i, j, k, ℓ ∈ Q , y i,j y k,ℓ = δ i,k δ j,ℓ y i,j . (cid:3) This brings us to the main result of the paper.
Theorem 4.17.
Let Q be a finite quiver with path algebra k Q . Then the universal quan-tum linear semigroupoids O left ( k Q ) , O right ( k Q ) , and O trans ( k Q ) exist, and they are eachisomorphic to H ( Q ) as weak bialgebras.Proof. Consider the left and right coaction of H ( Q ) on k Q presented in Example 2.22. Wewill show in full detail that O left ( k Q ) ∼ = H ( Q ) as weak bialgebras (under Hypothesis 4.1(a)),and briefly discuss the proof that O right ( k Q ) ∼ = H ( Q ) as weak bialgebras (under Hypothe-sis 4.1(b)). Then we have that O left ( k Q ) ∼ = H ( Q ) ∼ = O right ( k Q ) as weak bialgebras, and thatthe coactions of the left/right UQSGds are transposed via Example 2.22. Hence, Proposi-tion 3.17 yields O trans ( k Q ) ∼ = H ( Q ) as weak bialgebras (under Hypothesis 4.1(c)).To proceed, we will show that H ( Q ) satisfies the universal property of O left ( k Q ). Indeedwe have that H ( Q ) is a weak bialgebra that left coacts on k Q [Example 2.22]. Moreover, NIVERSAL QUANTUM SEMIGROUPOIDS 23 k Q ∼ = ( H ( Q )) t as left H ( Q )-comodule algebras: the algebra isomorphism, call it τ , holds byProposition 2.13 via e i a ′ i , which is a left comodule map due to the computation below: λ H ( Q ) t τ ( e i ) = λ H ( Q ) t ( a ′ i ) , = P k ∈ Q ∆ H ( Q ) ( x i,k ) (2.12) = P j,k ∈ Q x i,j ⊗ x j,k = P j ∈ Q (Id H ( Q ) ⊗ τ )( x i,j ⊗ e j ) = (Id H ( Q ) ⊗ τ ) λ k Q ( e i ) . Now, assume that H is a weak bialgebra which coacts from the left on k Q as in Hypoth-esis 4.1(a), recall Remark 3.2, and assume that there exists an isomorphism ψ : k Q ∼ −→ H t in H A . Recall that we have elements { y i,j } i,j ∈ Q and { y p,q } p,q ∈ Q in H , as well as idempotents { η i } i ∈ Q in H s and { θ i } i ∈ Q in H t , as in Lemma 4.2 and Proposition 4.9. Now consider themap π defined on the algebra generators of H ( Q ) and extended multiplicatively and linearly: π : H ( Q ) → H defined by x i,j y i,j for i, j ∈ Q , x p,q y p,q for p, q ∈ Q . We aim to show first that π is a weak bialgebra map (i.e., that π is an algebra map and acoalgebra map) satisfying ( π ⊗ Id k Q ) λ H ( Q ) = λ H , and that π is the only such weak bialgebramap H ( Q ) → H with this property. This would achieve the result that O left ( k Q ) ∼ = H ( Q )as weak bialgebras.To show that ( π ⊗ Id k Q ) λ H ( Q ) = λ H , note that for i ∈ Q ,( π ⊗ Id k Q ) λ H ( Q ) ( e i ) = ( π ⊗ Id k Q ) (cid:16)P j ∈ Q x i,j ⊗ e j (cid:17) = P j ∈ Q y i,j ⊗ e j = λ H ( e i ) . A similar calculation shows that ( π ⊗ Id k Q ) λ H ( Q ) ( p ) = λ H ( p ) for p ∈ Q . Since π, λ H ( Q ) and λ H are multiplicative, we must have ( π ⊗ Id k Q ) λ H ( Q ) = λ H .The unitality of π follows from the computation: π (1 H ( Q ) ) (2.10) = π ( P i,j ∈ Q x i,j ) = P i,j ∈ Q y i,j = 1 H . To prove that π is multiplicative, note that by Proposition 4.9(c)(vi), for all i, j, k, l ∈ Q , y i,j y k,l = δ i,k δ j,l y i,j . By (4.7) and (4.8) in Lemma 4.2, we have y s ( p ) ,s ( q ) y p,q = y p,q = y p,q y t ( q ) ,t ( p ) . So, we obtain that y p,q y p ′ ,q ′ (4.7) , (4.8) = y p,q y t ( p ) ,t ( q ) y s ( p ′ ) ,s ( q ′ ) y p ′ ,q ′ = δ t ( p ) ,s ( p ′ ) δ t ( q ) ,s ( q ′ ) y p,q y t ( p ) ,t ( q ) y p ′ ,q ′ (4.8) = δ t ( p ) ,s ( p ′ ) δ t ( q ) ,s ( q ′ ) y p,q y p ′ ,q ′ . (4.18)Now (2.7), (2.8) and (2.9) imply that π is multiplicative. Therefore, π is an algebra map. Next, we will show that π is also a coalgebra map, i.e., that ∆ H π = ( π ⊗ π )∆ H ( Q ) and ε H π = ε H ( Q ) . We will prove this for x p,q by induction on the length ℓ of the paths p, q ∈ Q .If ℓ = 0 ,
1, then the assertion holds by (4.3) and (4.4) in Lemma 4.2. Now take p = p · · · p ℓ − p ℓ and q = q · · · q ℓ − q ℓ paths of length ℓ with p i , q i ∈ Q . Then, for ℓ ≥ H π ( x p ,q · · · x p ℓ − ,q ℓ − x p ℓ ,q ℓ )= ∆ H ( y p ,q · · · y p ℓ − ,q ℓ − y p ℓ ,q ℓ )= ∆ H ( y p ,q · · · y p ℓ − ,q ℓ − ) ∆ H ( y p ℓ ,q ℓ )= (∆ H π )( x p ,q · · · x p ℓ − ,q ℓ − ) (∆ H π )( x p ℓ ,q ℓ ) induction = ( π ⊗ π )∆ H ( Q ) ( x p ,q · · · x p ℓ − ,q ℓ − ) ( π ⊗ π )∆ H ( Q ) ( x p ℓ ,q ℓ )= ( π ⊗ π )∆ H ( Q ) ( x p ,q · · · x p ℓ − ,q ℓ − x p ℓ ,q ℓ ) , where the first three equalities and the last equality hold because π , ∆ H , and ∆ H ( Q ) preservemultiplication. Further, we have( ε H π )( x p ,q · · · x p ℓ − ,q ℓ − x p ℓ ,q ℓ )= ε H ( y p ,q · · · y p ℓ − ,q ℓ − y p ℓ ,q ℓ )= ε H ( y p ,q · · · y p ℓ − ,q ℓ − H y p ℓ ,q ℓ ) = ε H ( y p ,q · · · y p ℓ − ,q ℓ − ) ε H (1 y p ℓ ,q ℓ ) , (4.3) = P i,j,k ∈ Q ε H ( y p ,q · · · y p ℓ − ,q ℓ − y i,k ) ε H ( y k,j y p ℓ ,q ℓ ) (4.18) = P i,j,k ∈ Q ε H ( y p ,q · · · y p ℓ − ,q ℓ − δ i,t ( p ℓ − ) δ k,t ( q ℓ − ) ) ε H ( δ k,s ( p ℓ ) δ j,s ( q ℓ ) y p ℓ ,q ℓ )= P k ∈ Q δ k,t ( q ℓ − ) δ k,s ( p ℓ ) ε H ( y p ,q · · · y p ℓ − ,q ℓ − ) ε H ( y p ℓ ,q ℓ )= δ t ( q ℓ − ) ,s ( p ℓ ) ε H ( y p ,q · · · y p ℓ − ,q ℓ − ) ε H ( y p ℓ ,q ℓ )= δ t ( q ℓ − ) ,s ( p ℓ ) ( ε H π )( x p ,q · · · x p ℓ − ,q ℓ − ) ( ε H π )( x p ℓ ,q ℓ ) induction = δ t ( q ℓ − ) ,s ( p ℓ ) ε H ( Q ) ( x p ,q · · · x p ℓ − ,q ℓ − ) ε H ( Q ) ( x p ℓ ,q ℓ )= δ t ( q ℓ − ) ,s ( p ℓ ) ε H ( Q ) ( x p ,q · · · x p ℓ − ,q ℓ − ) δ p ℓ ,q ℓ = δ t ( q ℓ − ) ,s ( q ℓ ) ε H ( Q ) ( x p ,q · · · x p ℓ − ,q ℓ − ) δ p ℓ ,q ℓ = δ t ( q ℓ − ) ,s ( q ℓ ) δ t ( p ) ,s ( p ) · · · δ t ( p ℓ − ) ,s ( p ℓ − ) δ t ( q ) ,s ( q ) · · · δ t ( q ℓ − ) ,s ( q ℓ − ) · δ p ,q · · · δ p ℓ − ,q ℓ − δ p ℓ ,q ℓ = δ t ( p ℓ − ) ,s ( p ℓ ) δ t ( q ℓ − ) ,s ( q ℓ ) δ t ( p ) ,s ( p ) · · · δ t ( p ℓ − ) ,s ( p ℓ − ) δ t ( q ) ,s ( q ) · · · δ t ( q ℓ − ) ,s ( q ℓ − ) · δ p ,q · · · δ p ℓ − ,q ℓ − δ p ℓ ,q ℓ = (cid:0) δ t ( p ) ,s ( p ) · · · δ t ( p ℓ − ) ,s ( p ℓ − ) δ t ( p ℓ − ) ,s ( p ℓ ) (cid:1) (cid:0) δ t ( q ) ,s ( q ) · · · δ t ( q ℓ − ) ,s ( q ℓ − ) δ t ( q ℓ − ) ,s ( q ℓ ) (cid:1) · (cid:0) δ p ,q · · · δ p ℓ − ,q ℓ − δ p ℓ ,q ℓ (cid:1) = ε H ( Q ) ( x p ,q · · · x p ℓ − ,q ℓ − x p ℓ ,q ℓ ) , as desired. Therefore, we have shown that π is a map of weak bialgebras. NIVERSAL QUANTUM SEMIGROUPOIDS 25
It remains to show that π is unique. Suppose that π ′ : H ( Q ) → H is a weak bialgebrahomomorphism such that ( π ′ ⊗ Id k Q ) λ H ( Q ) = λ H . Let i ∈ Q . Then( π ′ ⊗ Id k Q ) λ H ( Q ) ( e i ) = ( π ′ ⊗ Id k Q ) (cid:16)P j ∈ Q x i,j ⊗ e j (cid:17) = P j ∈ Q π ′ ( x i,j ) ⊗ e j while λ H ( e i ) = P j ∈ Q y i,j ⊗ e j . Since the e j are linearly independent in k Q , this implies that π ′ ( x i,j ) = y i,j for each i, j ∈ Q .By a similar argument, π ′ ( x p,q ) = y p,q for any p, q ∈ Q . Hence, π ′ and π agree on a set ofalgebra generators for H ( Q ), and since both π ′ and π are algebra homomorphisms, we havethat π ′ = π .To show that H ( Q ) satisfies the universal property of O right ( k Q ), one only needs tomake the following adjustments to the proof above: assume Hypothesis 4.1(b) in place ofHypothesis 4.1(a) (i.e., replace the left coaction λ with the right coaction ρ ); replace ψ with an isomorphism φ : k Q ∼ −→ H s in A H ; and employ Proposition 4.16 in place ofProposition 4.9 in the argument that π : H ( Q ) → H is an algebra map. Then, the result for O right ( k Q ) follows in a manner similar to that for O left ( k Q ) above. (cid:3) With Lemma 3.16, the following is a consequence of the theorem above.
Corollary 4.19.
The weak bialgebra H ( Q ) coacts on k Q inner-faithfully. (cid:3) We end this section with an example of our result above in the bialgebra case, thusobtaining a left/right/transposed UQSG as in Definitions 1.4 and 1.5.
Example 4.20.
Suppose that Q is a finite quiver with | Q | = 1 and | Q | = n for some n ∈ N , that is, Q is the n -loop quiver. Here, k Q is isomorphic to the free algebra k h t , . . . , t n i .Now Theorem 4.17 implies that, as bialgebras, O left ( k h t , . . . , t n i ) ∼ = O right ( k h t , . . . , t n i ) ∼ = O trans ( k h t , . . . , t n i ) ∼ = H ( Q n -loop ) , where H ( Q n -loop ) is defined in Example 2.6. Indeed, dim k ( H ( Q )) s = | Q | = 1 by Proposi-tion 2.13(c), so all of the structures above are bialgebras by Proposition 2.3(g). Moreover,one can check that H ( Q n -loop ) is isomorphic to the free algebra k h x t i ,t j | ≤ i, j ≤ n i .5. Universal quantum linear semigroupoids of quotients of path algebras
Let Q be a finite quiver and let I be a graded ideal of k Q . In this section, we studythe UQSGds of the quotient algebra k Q/I , showing that if they exist, they are each aquotient of H ( Q ) [Proposition 5.4]. Moreover, we generalize a result of Manin by showingthat a UQSGd of a quadratic quotient algebra is isomorphic to the opposite UQSGd of itsquadratic dual [Theorem 5.10]. We also provide several examples. To start, we need a fewwell-known facts. Definition 5.1.
Let (
H, m, u, ∆ , ε ) be a weak bialgebra. A biideal of H is a k -subspace I ⊆ H which is both an ideal and a coideal, that is: hI ⊆ I and Ih ⊆ I for any h ∈ H ;∆( I ) ⊆ I ⊗ H + H ⊗ I ; and ε ( I ) = 0. Lemma 5.2.
The kernel of a weak bialgebra map is a biideal.
Proof.
Let α : H → K be a weak bialgebra map. Since the kernel of an algebra map is anideal and the kernel of a coalgebra map is a coideal, ker α is a biideal. (cid:3) Lemma 5.3.
Suppose that H is a weak bialgebra and that I is a biideal. Then H/I can begiven the structure of a weak bialgebra as follows, for all h, k ∈ H : m H/I (( h + I ) ⊗ ( k + I )) = hk + I ; H/I = 1 H + I ; ∆ H/I ( h + I ) = ( h + I ) ⊗ ( h + I ) ; and ε H/I ( h + I ) := ε H ( h ) .Proof. The structures given above make
H/I both an algebra and a coalgebra. A straight-forward calculation verifies the compatibility conditions given in Definition 2.1. (cid:3)
Proposition 5.4.
Let Q be a finite quiver and let I ⊆ k Q be a graded ideal which is generatedin degree or greater. If O ∗ ( k Q/I ) exists (where ∗ means ‘left’, ‘right’, or ‘trans’), we have O ∗ ( k Q/I ) ∼ = H ( Q ) / I , for some biideal I of H ( Q ) . Remark 5.5. If I has generators in degree 0 or 1, then we can choose a smaller quiver Q ′ and an ideal I ′ of k Q ′ such that k Q ′ /I ′ ∼ = k Q/I as algebras and I ′ is generated in degree 2or greater. Proof of Proposition 5.4.
We will prove this statement for O left ( k Q/I ); the other statementsfollow similarly. By Lemma 5.2, it suffices to show that we have a weak bialgebra surjection π : H ( Q ) → O left ( k Q/I ), in which case, O left ( k Q/I ) ∼ = H ( Q ) / ker π .Let O := O left ( k Q/I ). For i ∈ Q and p ∈ Q , let e i , p denote the images of e i , p in k Q/I under the canonical quotient map k Q → k Q/I (regarding p as an element of k Q ).Since I is generated in degree 2 or greater, ( k Q/I ) ∼ = k Q as algebras, and dim k ( k Q/I ) =dim k k Q = | Q | . Hence, { e i } i ∈ Q is a basis of ( k Q/I ) and { p } p ∈ Q is a basis of ( k Q/I ) .We can write 1 k Q/I = P i ∈ Q e i . Then we have a linear coaction λ : k Q/I → O ⊗ k Q/Ie j P i ∈ Q y j,i ⊗ e i q P p ∈ Q y q,p ⊗ p for some elements y i,j , y p,q ∈ O . The result of Lemma 4.2(a) holds for this coaction. Namely,the proof is the same, except we replace k Q with k Q/I , elements of the form e i for i ∈ Q with e i , and arrows p ∈ Q (regarded as elements of k Q ) with p , making use ofthe fact that these elements of k Q/I still satisfy the fundamental relations e i e j = δ i,j e i , e s ( p ) p = p = p e t ( p ) , for i ∈ Q , p ∈ Q . Therefore, the results of Proposition 4.9(a),(b)also hold, since their proofs use the identities given in Lemma 4.2(a). Moreover, by thedefinition of a UQSGd, there exists a left O -comodule algebra structure on ( k Q/I ) suchthat O t ∼ = ( k Q/I ) in O A . Therefore, if we replace k Q with ( k Q/I ) in the statement andproof of Proposition 4.9(c), also replacing e i ∈ k Q with e i ∈ ( k Q/I ) , we obtain the sameresult.Now, imitating the proof of Theorem 4.17, we define a map π defined on the algebragenerators of H ( Q ) and extended multiplicatively and linearly: π : H ( Q ) → O defined by x i,j y i,j for i, j ∈ Q , x p,q y p,q for p, q ∈ Q . To show that π is an algebra map, we can simply follow the proof for Theorem 4.17, since thisproof only uses the results of Lemma 4.2(a) and Proposition 4.9. To show that π is a coalge-bra map, we again follow the proof for Theorem 4.17, replacing the paths p = p · · · p ℓ − p ℓ NIVERSAL QUANTUM SEMIGROUPOIDS 27 and q = q · · · q ℓ − q ℓ (for p i , q i ∈ Q ) with their images under the canonical quotient map k Q → k Q/I . This proof only uses the results of Lemma 4.2(a) and Proposition 4.9, theweak bialgebra structure of H ( Q ), and the fact that π is multiplicative, so the result stillholds. Therefore, π is a weak bialgebra map.Finally, we will show that π is surjective. By Lemma 3.16, the coaction of O on k Q/I isinner-faithful, and so O is generated as a weak bialgebra by the y i,j and y p,q for i, j ∈ Q and p, q ∈ Q . By the definition of π and the fact that π is a weak bialgebra map, we cansee that π is surjective. (cid:3) Every connected graded k -algebra which is finitely generated in degree one is isomorphicto k Q/I where Q is a finite quiver with | Q | = 1. For these algebras, we obtain the followingimmediate corollary. Corollary 5.6. If Q is a finite quiver with | Q | = 1 and | Q | = n , then O ∗ ( k Q/I ) is abialgebra quotient of the face algebra H ( Q n -loop ) from Example 4.20, where ∗ means ‘left’,‘right’, or ‘trans’. (cid:3) The next example is a special case of Proposition 5.4, which describes the UQSGdsexplicitly as a quotient of H ( Q ) when k Q/I is the polynomial ring k [ t , . . . , t n ]. Example 5.7.
Let A = k [ t , . . . , t n ]. We can describe A as a quotient of a path algebra k Q/I where, Q is a quiver with one vertex and n arrows t , . . . , t n , and I = ([ t i , t j ]) ≤ i Definition 5.8 ([GMV98, Section 2], [MV07, Section 1], [Gaw14]) . Let Q be a finite quiverand suppose I is a graded ideal of the path algebra k Q .(a) The opposite quiver Q op of Q is defined to be the quiver formed by ( Q op ) = Q and( Q op ) = Q ∗ , where Q ∗ is the arrow set consisting of reversed arrows of Q . For p ∈ Q , its reverse in Q ∗ is denoted by p ∗ . If a = p . . . p ℓ is a path of length ℓ in Q ,then we let a ∗ = p ∗ ℓ . . . p ∗ ∈ Q op . If f = P i α i a i is an element of k Q , the element f ∗ ∈ k Q op is defined to be P i α i a ∗ i .(b) We identify k Q op ℓ with ( k Q ℓ ) ∗ so that if { a , . . . , a d } is the basis of k Q ℓ consistingof paths of length ℓ , then { a ∗ , . . . , a ∗ d } is the dual basis.(c) We call the quotient algebra k Q/I quadratic if I is generated by elements of k Q .(d) The quadratic dual of the quadratic algebra k Q/I is defined to be( k Q/I ) ! = k Q op /I ⊥ op , where I ⊥ op is the ideal of k Q op generated by the orthogonal complement of the set I op := { f ∗ ∈ k Q op | f ∈ I ∩ k Q } in k Q op2 . Remark 5.9. As is our convention of Notation 1.2, we still read paths from left-to-right in Q op . Hence, in k Q op we have q ∗ p ∗ = ( pq ) ∗ for p, q ∈ Q (which is nonzero when s ( p ∗ ) = t ( p ) = s ( q ) = t ( q ∗ )). Note that identifying p ∈ k Q with p ∗ ∈ k Q op yields an anti-isomorphism of algebras and so k Q op ∼ = ( k Q ) op .For the face algebras H ( Q ) and H ( Q op ) attached to Q and Q op , respectively, the mapwhich sends x a,b ∈ H ( Q ) to x a ∗ ,b ∗ ∈ H ( Q op ) is an anti-isomorphism of algebras and anisomorphism of coalgebras. As weak bialgebras, H ( Q op ) ∼ = H ( Q ) op .The following theorem is a non-connected generalization of [Man88, Theorem 5.10]. Theorem 5.10. Let Q be a finite quiver and suppose I is an ideal such that k Q/I isquadratic. Then, we have that NIVERSAL QUANTUM SEMIGROUPOIDS 29 (a) O left ( k Q/I ) ∼ = O right (( k Q/I ) ! ) op , (b) O right ( k Q/I ) ∼ = O left (( k Q/I ) ! ) op , (c) O left ( k Q/I ) ∼ = O right ( k Q/I ) cop , (d) O trans ( k Q/I ) ∼ = O trans (( k Q/I ) ! ) op ,as weak bialgebras.Proof. We will only provide the proofs of parts (a) and (c), as other parts will hold bysimilar arguments. To start, suppose that Q = { p , . . . , p n } . Then, I = D r α := P ni,j =1 with t ( p i )= s ( p j ) c [ α ] i,j p i p j E α =1 ,...,m ⊆ k Q for some scalars c [ α ] i,j . Moreover, we have I ⊥ op = D r ∗ β := P nk,ℓ =1 with t ( p ∗ ℓ )= s ( p ∗ k ) d [ β ] k,ℓ p ∗ ℓ p ∗ k E β =1 ,..., | Q |− m ⊆ k Q op2 for some scalars d [ β ] k,ℓ . Here, P ni,j =1 with t ( p i )= s ( p j ) d [ β ] i,j c [ α ] i,j = 0 for each pair α, β .(a) By Proposition 5.4, we have that O left ( k Q/I ) = H ( Q ) / I for some biideal I of H ( Q ),with the coalgebra structure induced by H ( Q ) : ∆( x p i ,p k ) = P nw =1 x p i ,p w ⊗ x p w ,p k and ε ( x p i ,p k ) = δ i,k . We assert that I = * n X i,j,k,ℓ =1 t ( p i )= s ( p j ) , t ( p k )= s ( p ℓ ) c [ α ] i,j d [ β ] k,ℓ x p i ,p k x p j ,p ℓ + α =1 ,...,mβ =1 ,..., | Q |− m . Namely, there exists a basis of k Q consisting of elements { r α } α =1 ,...,m and { s γ } γ =1 ,..., | Q |− m so that the evaluation h r ∗ β , s γ i = δ β,γ for each β, γ = 1 , . . . , | Q | − m . Moreover, for each k, ℓ , we can write(5.11) p k p ℓ = P γ d [ γ ] k,ℓ s γ + P α e [ α ] k,ℓ r α for some scalars e [ α ] k,ℓ . (This can be checked by evaluation with r ∗ β .) Now, the left coactionof H ( Q ) on k Q , given by p i P k x p i ,p k ⊗ p k from Example 2.22, preserves the relation r α if and only if the following expression lies in O ⊗ I : P i,j c [ α ] i,j ( P k x p i ,p k ⊗ p k )( P ℓ x p j ,p ℓ ⊗ p ℓ )= P i,j,k,lt ( p i )= s ( p j ) t ( p k )= s ( p ℓ ) c [ α ] i,j x p i ,p k x p j ,p ℓ ⊗ p k p ℓ (5.11) = P γ,i,j,k,lt ( p i )= s ( p j ) t ( p k )= s ( p ℓ ) c [ α ] i,j d [ γ ] k,ℓ x p i ,p k x p j ,p ℓ ⊗ s γ + P α ′ ,i,j,k,lt ( p i )= s ( p j ) t ( p k )= s ( p ℓ ) c [ α ] i,j e [ α ′ ] k,ℓ x p i ,p k x p j ,p ℓ ⊗ r α ′ . Since { s γ } γ ∪ { r α ′ } α ′ is a basis of k Q , we must have that (cid:26) n X i,j,k,ℓ =1 t ( p i )= s ( p j ) t ( p k )= s ( p ℓ ) c [ α ] i,j d [ γ ] k,ℓ x p i ,p k x p j ,p ℓ = 0 (cid:27) α =1 ,...,mγ =1 ,..., | Q |− m are the generators of the relation space I for O left ( k Q/I ) as in Proposition 5.4. On the other hand, by Example 2.22 we have a right coaction of H ( Q op ) on k Q op given by p ∗ i P k p ∗ k ⊗ x p ∗ k ,p ∗ i . By a similar argument as above, this coaction preserves the relationsof I ⊥ op if and only if (cid:26) n X i,j,k,ℓ =1 t ( p ∗ j )= s ( p ∗ i ) t ( p ∗ ℓ )= s ( p ∗ k ) d [ β ] k,ℓ c [ η ] i,j x p ∗ j ,p ∗ ℓ x p ∗ i ,p ∗ k = 0 (cid:27) η =1 ,...,mβ =1 ,..., | Q |− m are the generators of the relation space of O right (( k Q/I ) ! ) as a quotient of H ( Q op ). Hence, O right (( k Q/I ) ! ) = H ( Q op ) / * n X i,j,k,ℓ =1 t ( p ∗ j )= s ( p ∗ i ) , t ( p ∗ ℓ )= s ( p ∗ k ) c [ α ] i,j d [ β ] k,ℓ x p ∗ j ,p ∗ ℓ x p ∗ i ,p ∗ k + α =1 ,...,mβ =1 ,..., | Q |− m , with ∆( x p ∗ i ,p ∗ k ) = P nw =1 x p ∗ i ,p ∗ w ⊗ x p ∗ w ,p ∗ k and ε ( x p ∗ i ,p ∗ k ) = δ i,k .Now, the desired isomorphism from O left ( k Q/I ) to O right (( k Q/I ) ! ) op is obtained by send-ing x p i ,p k to x p ∗ i ,p ∗ k .(c) By Proposition 5.4, we have that O right ( k Q/I ) = H ( Q ) / I ′ for some biideal I ′ of H ( Q ), with the coalgebra structure induced by H ( Q ) : ∆( x p k ,p i ) = P nw =1 x p k ,p w ⊗ x p w ,p i and ε ( x p k ,p i ) = δ i,k . Now, the right coaction of H ( Q ) on k Q , given by p i P k p k ⊗ x p k ,p i from Example 2.22, preserves each relation r α of I if and only if I ′ = * n X i,j,k,ℓ =1 t ( p i )= s ( p j ) , t ( p k )= s ( p ℓ ) c [ α ] i,j d [ β ] k,ℓ x p k ,p i x p ℓ ,p j + α =1 ,...,mβ =1 ,..., | Q |− m . Now, considering the presentation of O left ( k Q/I ) from part (a), the desired isomorphismfrom O left ( k Q/I ) to O right ( k Q/I ) cop is obtained by sending x p i ,p k to x p k ,p i . (cid:3) Example 5.12. Let A = k Q . Then A ! = k Q op / h k ( Q op ) i where h k ( Q op ) i is the ideal of k Q op generated by the space k ( Q op ) . By the above theorem, we have O left ( A ) ∼ = O right ( A ! ) op as weak bialgebras. Since, by Theorem 4.17, O left ( k Q ) ∼ = H ( Q ), we have that O right ( A ! ) op ∼ = H ( Q ). Further, H ( Q ) op ∼ = H ( Q op ), and so we conclude that O right ( A ! ) ∼ = H ( Q op ) . Similarly, we have that O left ( A ! ) ∼ = O trans ( A ! ) ∼ = H ( Q op ) as weak bialgebras.We end with a family of concrete examples of UQSGds for quadratic quotient pathalgebras– namely, those for preprojective algebras. NIVERSAL QUANTUM SEMIGROUPOIDS 31 Example 5.13. Let Q be the extended type A Dynkin quiver with | Q | ≥ 3, and considerits double Q formed by adding p ∗ for each p ∈ Q . For example, when | Q | = 3, Q = 31 2 p p p Q = 31 2 . p ∗ p p ∗ p p ∗ p The preprojective algebra on Q is defined to be the k -algebra,Π Q = k Q/ ( P i ∈ Q p i p ∗ i − P i ∈ Q p ∗ i − p i − ) . (Here, we index the vertices i by elements of Z / | Q | Z .) By [Wei19, Section 3], we have thatΠ Q ∼ = k Q/ ( p i p ∗ i − p ∗ i − p i − ) i ∈ Q as k -algebras. Therefore, any path in Q can be rewritten (in Π Q ) so that all of the nonstararrows occur, followed by all of the star arrows. We omit the details here, but we have that,as weak bialgebras, O left (Π Q ) ∼ = H ( k Q ) / I , O right (Π Q ) ∼ = H ( k Q ) / J , O trans (Π Q ) ∼ = H ( k Q ) / ( I + J ) , for I = * x p k ,p i x p ∗ k ,p i +1 − x p ∗ k − ,p i x p k − ,p i +1 , ( x p k ,p i x p ∗ k ,p ∗ i + x p k ,p ∗ i − x p ∗ k ,p i − ) − ( x p ∗ k − ,p i x p k − ,p ∗ i + x p ∗ k − ,p ∗ i − x p k − ,p i − ) ,x p k ,p ∗ i x p ∗ k ,p ∗ i − − x p ∗ k − ,p ∗ i x p k − ,p ∗ i − + i,k ∈ Q , J = * x p i ,p k x p i +1 ,p ∗ k − x p i ,p ∗ k − x p i +1 ,p k − , ( x p i ,p k x p ∗ i ,p ∗ k + x p ∗ i − ,p k x p i − ,p ∗ k ) − ( x p i ,p ∗ k − x p ∗ i ,p k − + x p ∗ i − ,p ∗ k − x p i − ,p k − ) ,x p ∗ i ,p k x p ∗ i − ,p ∗ k − x p ∗ i ,p ∗ k − x p ∗ i − ,p k − . + i,k ∈ Q . For further investigation: universal quantum linear groupoids In this section, we consider weak Hopf algebras that coact universally (and linearly) onan algebra A in Hypothesis 1.1, and propose directions for future research. First, let usrecall the notion of a universal coacting Hopf algebra, prompted by [Man88, Chapter 7]. Definition 6.1 (UQG) . Take A as in Hypothesis 1.1 and further assume that A is connected.Then a Hopf algebra is said to be a left (resp., right, transposed) universal quantum lineargroup (UQG) of A if it satisfies the conditions of Definition 1.4(a) (or, Definition 1.4(b),Definition 1.5(b)) by replacing ‘bialgebra’ with ‘Hopf algebra’.A general way of constructing a UQG from a UQSG is by taking the Hopf envelope asdiscussed briefly in [Man88, Section 7.5]. Other explicit constructions involve the quantumdeterminant (also known as the homological determinant ), which is a (typically) centralgroup-like element D of a UQSG that depends on the UQSG coaction on A . Here, one takesa UQSG, say O trans ( A ), and forms two Hopf algebras depending on whether the quantumdeterminant is trivial (i.e., equal to the unit) or is arbitrary: O transSL ( A ) := O trans ( A ) / ( D − , O transGL ( A ) := O trans ( A )[ D − ] . We refer to these universal Hopf algebras as UQGs of SL-type and of GL-type , respectively.Appearances of such Hopf algebras in the literature include those in [DVL90, Bic03, BDV13]for SL-type, [AST91, Tak90, Mro14] for GL-type, and [WW16, CWW19] for both types; seealso references therein.It is therefore natural to ask if this can be generalized to the framework of universalcoacting weak Hopf algebras. We recall the definition of a weak Hopf algebra below. Definition 6.2. A weak Hopf algebra is a sextuple ( H, m, u, ∆ , ε, S ), where ( H, m, u, ∆ , ε )is a weak bialgebra and S : H → H is a k -linear map called the antipode that satisfies thefollowing properties for all h ∈ H : S ( h ) h = ε s ( h ) , h S ( h ) = ε t ( h ) , S ( h ) h S ( h ) = S ( h ) . Note that if H is a weak Hopf algebra, the following are equivalent: H is a Hopf algebra;∆(1) = 1 ⊗ ε ( xy ) = ε ( x ) ε ( y ) for all x, y ∈ H ; S ( x ) x = ε ( x )1 for all x ∈ H ; and x S ( x ) = ε ( x )1 for all x ∈ H [BNS99, page 5].Now we define a universal weak Hopf algebra, similar to the manner that a UQG wasdefined above, for A not necessarily connected. Definition 6.3 (UQGd) . Take A as in Hypothesis 1.1. Then a weak Hopf algebra is said tobe a left (resp., right, transposed) universal quantum linear groupoid of A if it satisfies theconditions of part (a) (resp., (b), (c)) of Definition 1.7 by replacing ‘weak bialgebra’ with‘weak Hopf algebra’.This prompts the following series of questions. Question 6.4. Take A as in Hypothesis 1.1. In general:(1) When are the UQSGds O left ( A ), O right ( A ), O trans ( A ) weak Hopf algebras?(2) What is a ‘weak Hopf envelope’ (of a UQSGd)?Pertaining to the SL-type and GL-type constructions:(3) What is the quantum determinant D of the coaction of a UQSGd O left ( A ) (or, O right ( A ), O trans ( A )) on A ?(4) Is D invertible, and if so, are O left ( A )[ D − ], O right ( A )[ D − ], O trans ( A )[ D − ] weakHopf algebras that coact on A (universally) with arbitrary quantum determinant?(5) Are O left ( A ) / ( D − O right ( A ) / ( D − O trans ( A ) / ( D − 1) weak Hopf algebras thatcoact on A (universally) with trivial quantum determinant?Finally, as discussed in the introduction: Ideally, a universal (weak) bi/Hopf algebra should behave ring-theoreticallyand homologically like the algebra that it coacts on. This holds for transposed coactions on many connected graded algebras in Hypothesis 1.1;see, e.g., [AST91, WW16]. Likewise, the best candidates we have for the philosophy to holdfor coactions on algebras in Hypothesis 1.1 are the transposed UQSGds [Definition 3.10]and the transposed UQGds (defined above). Naturally, we inquire: NIVERSAL QUANTUM SEMIGROUPOIDS 33 Question 6.5. Take an algebra A in Hypothesis 1.1. In general, does the ring-theoreticand homological behavior of the transposed UQSGd and transposed UQGds of A reflectthat of A ? More specifically, if A has one of the following properties,(a) finite Gelfand-Kirillov dimension/nice Hilbert series,(b) Noetherian/coherent,(c) domain/prime/semiprime,(d) finite global dimension/finite injective dimension,(e) skew (or twisted) Calabi-Yau,do the transposed UQSGd and transposed UQGds of A satisfy the same property as well?Pertinent articles include the work of Gaddis, Reyes, Rogalski, and Zhang on (non-connected)graded skew Calabi-Yau algebras [RRZ14, RR18, RR19, GR18]. References [ASS06] Ibrahim Assem, Daniel Simson, and Andrzej Skowro´nski. 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