A cogroupoid associated to preregular forms

We construct a family of cogroupoids associated to preregular forms and recover the Morita-Takeuchi equivalence for Artin-Schelter regular algebras of dimension two, observed by Raedschelders and Van den Bergh. Moreover, we study the 2-cocycle twists of pivotal analogues of these cogroupoids, by developing a categorical description of preregularity in any tensor category that has a pivotal structure.


Introduction
This paper examines superpotentials associated to Artin-Schelter (AS) regular algebras and their universal quantum groups via the construction of certain bi-Galois objects using the language of cogroupoids.Superpotentials, or their duals, preregular forms, can be associated to any N -Koszul AS-regular algebra [15] and play an important role in noncommutative algebra, noncommutative algebraic geometry, and quantum groups, for example, via the classification of algebras [9,27,28].Quantum groups associated to these objects were introduced independently by Dubois-Violette and Launer [16] and by Wang [38].Later, Bichon and Dubois-Violette [7] gave an explicit presentation of this quantum group by generators and relations.By [11], when the superpotential algebra is N -Koszul and AS-regular, this quantum group coincides with Manin's universal quantum group (c.f.[26]), the Hopf algebra that universally coacts on the underlying algebra.These quantum groups and their generalizations, which consists of a wide class of Hopf algebras, including the coordinate rings O(GL n ) and their quantum analogues O q (GL n ) (as in [10]), have recently been studied in [11,37].
Schauenburg showed in [34] that the categories of comodules over two Hopf algebras H and L are monoidally equivalent, called Morita-Takeuchi equivalent, if and only if there exists an H-L-bi-Galois object between them.Later, Bichon [6] introduced the notion of a cogroupoid to provide a categorical context for Hopf-(bi)Galois objects.An understanding of the structure of cogroupoids is useful since it enhances the classical theory by allowing categorical arguments on Hopf-(bi)Galois objects.Other applications of cogroupoids include: explicit construction of new resolutions from old ones in homological algebra, invariant theory, monoidal equivalences between categories of Yetter-Drinfeld modules with applications to bialgebra cohomology and Brauer groups [5,6].Here, we construct a cogroupoid whose objects are determined by preregular forms.
Proposition A (Lemma 3.1.3,Lemma 3.1.5,Definition 3.1.6).For any integer m ≥ 2, there is a cogroupoid GL m , whose objects are given by all m-preregular forms.In particular, for any m-preregular form f , the Hopf algebra GL m (f, f ) is the universal quantum group associated to f , as given in [11].
Since Dubois-Violette showed that any N -Koszul AS-regular algebra is a superpotential algebra associated to some preregular form [15], our construction provides an explicit way to establish the Morita-Takeuchi equivalence between Manin's universal quantum groups associated to N -Koszul AS-regular algebras.In particular, we recover a special case of a result of Raedschelders and Van den Bergh from [31] stating the Morita-Takeuchi equivalence between Manin's universal quantum groups associated to AS-regular algebras of the same dimension.Our method does not depend on the categorical approach of the Tannaka-Krein formalism, but relies instead on the non-vanishing of certain bi-Galois objects between any N -Koszul ASregular algebras.

Theorem B (Theorem 3.2.2). Manin's universal quantum groups associated to any two AS-regular algebras of dimension two are Morita-Takeuchi equivalent.
Moreover, we study SL m -type universal quantum groups under 2-cocycle twisting.As a necessary condition, we introduce the notion of a preregular morphism, a generalization of a preregular form, in any rigid tensor category that has a pivotal structure.Pivotal (also called sovereign) categories have been important in topological quantum field theory.The pivotal structure allows the definition of quantum dimension, which can be used to produce numerical invariants of 3-manifolds and knots [20,23].When the (co)representation category of a Hopf algebra is pivotal, the Hopf algebra is called (co)pivotal (or (co)sovereign, as in [3]).In our paper, we employ the pivotal structure to define a Hom-space operator D m V which resembles the cyclic permutation of tensor products of vector spaces V ⊗m .This enables us to define the notion of preregulariy on morphisms in a categorical context.
Definition C (Definition 4.1.1).Let C be a pivotal tensor category.For any integer m ≥ 2 and V ∈ ob(C), a morphism f : V ⊗m → 1 is called preregular if (1) f is non-degenerate, namely there is a surjection π : ).We observe that the inverse duals of these Hom-space operators D m V are exactly those E m V * , where V * denotes the dual of V , used to define generalized Frobenius-Schur indicators in an arbitrary pivotal category [29,30].As a consequence, any preregular morphism is an eigenvector of the above operator D m V satisfying some nondegeneracy conditions.Moreover, the dual of a preregular morphism, which we refer to as a twisted superpotential, is invariant under the operator E m V * .We use this generalization to construct another cogroupoid associated to preregular forms.
Proposition D (Definition 4.2.2,Lemma 4.2.4).For any integer m ≥ 2, there is a cogroupoid SL m , whose objects are given by all m-preregular forms.In particular, for any m-preregular form f , the Hopf algebra is the universal copivotal Hopf algebra associated to f , as given in [3,7].Here, D is the quantum determinant of GL m (f, f ).
Using the cogroupoid SL m , we obtain the following result on the 2-cocycle twists of the universal quantum groups of preregular forms considered in [7], which are copivotal Hopf algebras.
Theorem E (Theorem 4.2.9).Let m ≥ 2 be an integer and V be a finite-dimensional k-vector space.Let f be an m-preregular form on V and σ be a left 2-cocycle on SL m (f, f ).Then the twisted map f σ (see Definition 4.2.7) is also an m-preregular form on V and the universal quantum groups

Preliminaries
Throughout the paper, let k be a base field with ⊗ taken over k unless stated otherwise.All categories are k-linear and all algebras are associative over k.We use the Sweedler's notation for the coproduct in a coalgebra B: ∆(h) = h 1 ⊗ h 2 for any h ∈ B. When a Hopf algbera H (right) coacts on an algebra A, we denote the coaction ρ : The category of all (resp.finite-dimensional) right B-comodules is denoted by comod(B) (resp.comod fd (B)).
In this section, we present some background on superpotential algebras associated to preregular forms, cogroupoids, and 2-cocycle twists.In [15,Theorem 4.3], Dubois-Violette proved that every N -Koszul ASregular algebra of finite global dimension d, generated by n elements in degree one, is a twisted superpotential algebra.In this paper, AS-regular algebras will refer to Gorenstein algebras with finite global dimension; one may view them as "nice" noncommutative analogues of polynomial rings.Note that we do not require AS-regular algebras to have finite Gelfand-Kirillov dimension.
2.1.Superpotential algebras and Manin's universal quantum groups.We use the definitions from [15] of an algebra associated to a preregular form.Definition 2.1.1.Let 2 ≤ N ≤ m be integers and V be an n-dimensional k-vector space.
(1) An m-linear form f on V is called preregular if it satisfies the following conditions: Given a preregular form f on a vector space V with fixed basis {v 1 , . . ., v n }, we will typically denote by ) Let f be an m-preregular form on V , and {v 1 , . . ., v n } be a fixed basis of V .The superpotential algebra associated to f , denoted by A(f, N ), is the k-algebra generated by n generators x 1 , . . ., x n subject to the relations 1≤j1,...,jN ≤n For any m-preregular form f on V , it is straightforward to check that there is an associated for every possible 1 ≤ i 1 , . . ., i m ≤ n.Henceforth, an algebra A will be called a superpotential algebra if there are some choice of integers m and N with 2 ≤ N ≤ m, and an m-preregular form f so that A ∼ = A(f, N ).
Here, any superpotential algebra can be considered as a graded algebra by assigning degree 1 to its generators.
Remark 2.1.2.Using the notation from Definition 2.1.1,let c : V ⊗m → V ⊗m be the linear map defined by An element s ∈ V ⊗m is a twisted superpotential if there is P ∈ GL(V ) so that Given a twisted superpotential s ∈ V ⊗m , the superpotential algebra associated to s is defined as where T V is the tensor algebra on V , By identifying V ⊗m ∼ = ((V * ) ⊗m ) * , where (−) * denotes the k-dual, there is a one-to-one correspondence between m-preregular forms on V * and twisted superpotentials in V ⊗m : Furthermore, the associated algebras A(f, N ) and A(s, N ) are isomorphic for f associated to s under the above correspondence [11,Lemma 2.4].
Next, we review Manin's construction of the universal quantum group aut(A) associated to any superpotential algebra A = A(f, N ) as described in [26].Note that the original definition was only given under the assumption that A is a quadratic algebra, but it can be generalized to any graded algebra.
Definition 2.1.3.Let A be a Z-graded algebra. ( commutes.Similarly, we can define aut r (A) by using the universal right coaction on A preserving the grading of A.
By [1, Example 4.8(1)-( 2)], we know aut l (A) always exists if A is locally finite, namely, when dim k A i < ∞ for all i ∈ Z.In particular, when A = A(f, N ) is a superpotential algebra, aut l (A) and aut r (A) always exist.
The bialgebra structure of aut l (A) is given by 2.2.Cogroupoids.We now discuss Morita-Takeuchi equivalence in the context of the universal quantum groups associated to preregular forms using the language of cogroupoids introduced by Bichon [6].These provide a categorical framework for bi-Galois objects discussed by Schauenburg [34].
For a X,Y ∈ C(X, Y ), we use Sweedler's notation to write From its definition, a cocategory with one object is just a bialgebra.In particular, C(X, X) is a bialgebra for any The following proposition describes properties of the "antipodes" in cogroupoids.For other properties of cogroupoids, we refer the reader to [6].In a cogroupoid C, the bialgebra C(X, X) is a Hopf algebra for any X ∈ ob(C), with the antipode map S X,X described here.
(2) For any Z ∈ ob(C) and a Y,X ∈ C(Y, X), The following is Bichon's reformulation of Schauenburg's result [34] about bi-Galois objects for Morita-Takeuchi equivalences in terms of cogroupoids.Theorem 2.2.4.[6, Theorem 2.10] Let H and L be Hopf algebras.The following assertions are equivalent.
(1) There exists a k-linear equivalence of monoidal categories comod(H) 2.3.Pivotal tensor categories.We now recall some concepts and notation from the theory of tensor categories.We use (C, ⊗, Φ, 1, r, ℓ) to denote a k-linear tensor category, with a bifunctor ⊗ : called the right and left unit constraints for all X, Y, Z ∈ ob(C).These structure maps are subject to the Pentagon and Triangle axioms [22, XI.2.1].If two objects X, Y ∈ ob(C) are obtained by tensoring together the same sequence of objects with two different arrangements of parentheses, one can then construct a natural isomorphism between them by composing several instances of the tensor products of Φ, Φ −1 and the identity.Any above isomorphism is unique by Mac Lane's coherence theorem [25], and will be denoted by Φ ?: X → Y .By [35], we may always assume that the unit object 1 of C is strict, namely X ⊗ 1 = X = 1 ⊗ X and the left and right unit constraints ℓ X and r X are just the identity maps for every X ∈ ob(C).
A left dual of an object V ∈ ob(C) is an object V * together with two morphisms ev : We say C is left rigid if every object of C admits a left dual.A right dual of an object and right rigidity can be defined similarly for C. Suppose C is left rigid.Then (−) * is a contravariant monoidal functor together with a monoidal structure ζ : commutes, where the natural isomorphism ξ V : F (V * ) → F (V ) * , called the duality transformation, is uniquely determined by (F , ξ) (see [29, §1]).A strict pivotal category is a strict monoidal category with a pivotal structure in which both the monoidal functor structure ξ of (−) * and the pivotal structure j are identities.By [29, Theorem 2.2], every pivotal category is equivalent, as a pivotal category, to a strict one.
2.4.2-cocycle twists.Schauenburg proved in [34] that Morita-Takeuchi equivalences for Hopf algebras are in bijection with bi-Galois objects.A subset of these equivalences correspond to 2-cocycle twists, which were introduced by Doi and Takeuchi [13,14] (in fact, when the Hopf algebra is finite-dimensional, every Morita-Takeuchi equivalence arises from a 2-cocycle).In this section, we give a brief overview of 2-cocycle twists, and in Section 4 we discuss 2-cocycle twists of preregular forms and their associated universal pivotal cogroupoids.
for all x, y, z ∈ H.The convolution inverse of σ is usually denoted by σ −1 .
Given a 2-cocycle σ : H ⊗ H → k, let H σ denote the coalgebra H endowed with the original unit and deformed product for any x, y ∈ H.In fact, H σ is a Hopf algebra with the deformed antipode S σ given in [13,Theorem 1.6].
We call H σ the 2-cocycle twist of H by σ.It is well-known that two Hopf algebras are 2-cocycle twists of each other if and only if there exists a bicleft object between them (e.g., see [34]).Now, suppose σ : H ⊗H → k is a left 2-cocycle on H.It is well-known that there is a monoidal equivalence between the category of comodules of H and that of H σ given by (F, ξ) : comod(H) where F is identity as functor on objects together with the monoidal functor structure Moreover, the set of all left 2-cocycles on a Hopf algebra H gives rise to an associated 2-cocycle cogroupoid H as follows.
Example 2.4.2.[6, Definition 3.14] Let H be any Hopf algebra.The 2-cocycle cogroupoid of H, denoted by H, is defined as follows: (1) ob(H) = Z 2 (H), which is the set of all left 2-cocycles on H.
(2) For any σ, τ ∈ Z 2 (H), the algebra H(σ, τ ) is defined such that H(σ, τ ) = H as vector spaces with the new multiplication Example 2.4.3.As in Example 2.4.2, let H be any Hopf algebra.Take H Z to be the full subcogroupoid of H, where the objects correspond to those 2-cocycles arising from twisting pairs (using the language of [21]).
In this case, for a 2-cocyle σ on H, the Hopf algebra H Z (σ, σ) is equal to a twist (in the sense of [2,40]) of H by a graded automorphism (see [8,Remark 2.9] or [21, Theorem E]).

The cogroupoids associated to preregular forms
In this section, we introduce a family of cogroupoids associated to preregular forms and discuss their properties.In Theorem 3.2.2, using the language of cogroupoids, we show the Morita-Takeuchi equivalence for AS-regular algebras of dimension two.
3.1.Construction of the cogroupoid GL m .Let V be a k-dimensional vector space and W be an ldimensional vector space over k with fixed bases {v 1 , ..., v k } of V and {w 1 , ..., w l } of W .For any integer m ≥ 2, let e be an m-linear preregular form on V and f be an m-linear preregular form on W . Recall that we write e (v i1 , . . . We denote the generators of GL m (e, f ) by a e,f ij , b e,f ij , and D e,f ±1 when multiple preregular forms are involved, and omit the superscripts when the context is clear.In particular, if W = V and f = e, then we simply write GL m (e) = GL m (e, e).Remark 3.1.2.We note that GL m (e) is the algebra H(e) associated to a preregular form e, as defined in [11,Definition 5.1].It is a Hopf algebra with the Hopf structure given in [11,Proposition 5.8].When the superpotential algebra A(e, N ) is N -Koszul and Gorenstein, GL m (e) is Manin's universal quantum group coacting on A(e, N ) (see [11,Theorem 5.33]).
Lemma 3.1.3.For any integer m ≥ 2, GL m forms a k-cocategory, where the objects are m-linear preregular forms on k-vector spaces.In particular, for any vector spaces U, V, W with dim U = p, dim V = q, and dim W = r, for any m-linear preregular forms e on U , f on V and g on W , there exist algebra maps and ε e : GL m (e) → k such that ε e (a e,e ij ) = ε e (b e,e ji ) = δ ij , for 1 ≤ i, j ≤ p, and ε e ((D e,e ) ±1 ) = 1.
Proof.Since ∆ = ∆ f e,g is already defined on the generators of GL m (e, g), it suffices to verify that it preserves all relations: Similarly, we may also show that On the other hand, one can check that Hence ∆ f e,g : GL m (e, g) → GL m (e, f ) ⊗ GL m (f, g) is a well-defined algebra map.Note that ε : GL m (e) → k is a well-defined algebra map since GL m (e) is a Hopf algebra [11,Proposition 5.8].It remains to show that the diagrams in Definition 2.2.1 commute, which is straightforward on the generators.
The following result is similar to [11,Lemma 5.6] in the context of cogroupoids, and we leave its proof to the reader.Lemma 3.1.4.Let V and W be k-vector spaces of dimension k and l, respectively, Consider two mpreregular forms e and f on V and W , respectively, together with invertible matrices P ∈ GL k (k) = GL(V ) and Q ∈ GL l (k) = GL(W ) as in (2.1).Then the following equalities hold in GL m (e, f ): Lemma 3.1.5.For any integer m ≥ 2, the cocategory GL m forms a cogroupoid, with ∆ and ε defined as in Lemma 3.1.3,and the algebra map is defined by the formulas Proof.Set dim V = k and dim W = l.We first show that S e,f : GL m (e, f ) → GL m (f, e) op preserves the relations in GL m (e, f ) and hence it is a well-defined algebra map.One can see that = S e,f (f j1...jm D) , and similarly, Moreover, it is clear that S e,f preserves the relation DD The last equality holds in GL m (f, e) by Lemma 3.1.4.It remains to show the commutativity of the two diagrams in Definition 2.2.2, which is straightforward to check on generators.
We summarize the discussion above in the following definition/theorem, which follows similarly to [6, Definitions 2.1 and 2.4].Definition 3.1.6.For any integer m ≥ 2, the cogroupoid GL m is defined as follows: (1) ob(GL m ) = {e : V ⊗m → k | e is a preregular form on some finite-dimensional vector space V }.
(2) For e, f ∈ ob(GL m ), GL m (e, f ) is the algebra defined in Definition 3.1.1.
(3) The structural maps ∆ We present an alternate criterion for the Morita-Takeuchi equivalence of universal quantum groups, using the language of cogroupoids.
Proposition 3.2.1.Let 2 ≤ N, N ′ ≤ m be three integers, and V, W be two finite-dimensional k-vector spaces.Let e and f be two m-preregular forms on V and W , respectively, such that the associated superpotential algebras A = A(e, N ) and B = A(f, N ′ ) are two N and N ′ -Koszul AS-regular algebras.If the algebra GL m (e, f ) = 0, then the universal quantum groups aut l (A) and aut l (B) are Morita-Takeuchi equivalent.
In the following, we give an application of Proposition 3.2.1 for the case when m = 2 and e : V ⊗2 → k and f : W ⊗2 → k are two preregular forms.As a consequence, it provides another proof of a special case of [31, Theorem 7.2.3] for AS-regular algebras of dimension two.Recall that in the classification of AS-regular algebras of dimension two [41, Theorem 0.1], being AS-regular is equivalent to being a superpotential algebra.Theorem 3.2.2.Let A and B be any two AS-regular algebras of dimension two.Then aut l (A) and aut l (B) are Morita-Takeuchi equivalent.
Proof.By the classification of AS-regular algebras of dimension 2 in [41, Theorem 0.1], A and B can be presented as superpotential algebras A = A(e, 2) and B = A(f, 2), respectively, for some 2-preregular forms e and f .Hence, by Remark 3.1.2and Proposition 3.2.1, it suffices to show that the bi-Galois object GL 2 (e, f ) between aut l (A) = GL 2 (e) and aut l (B) = GL 2 (f ) is nonzero.
Suppose e and f are preregular forms on vector spaces V and W , respectively, of dimensions k and l.We fix a basis {v 1 , . . ., v k } for V and write e as a matrix It is easy to check that e is a preregular form if and only if E ∈ GL(V ) = GL k (k); in this case, the twisting matrix for e is given by P = E −T E. Similarly, we denote the matrix F ∈ GL(W ) = GL l (k) associated to the preregular form f , and the twisting matrix for f is given by Q = F −T F.
By (3.1), the k-algebra GL 2 (e, f ) is presented by (2kl + 2) generators subject to the relations By Lemma 3.1.4,we also have B A = I l×l .Hence This implies that B T = D −1 E T A F −T , and so B = D −1 F −1 A T E. Hence GL 2 (e, f ) is the quotient of the free algebra k A, D ±1 by the relations Recall the algebra B(E, F) defined in [4, Definition 3.1].One checks directly that For arbitrary F ∈ GL(W ) with corresponding preregular form f , let k = 2 and such that q 2 +tr(F T F −1 )+1 = 0. Denote e q as the preregular form corresponding to E q .Then GL 2 (e q , f ) = 0 by [4,Proposition 3.4].Now in view of [6,Proposition 2.15], to show that GL 2 (e, f ) = 0 it suffices to show that GL 2 (e q , e q ′ ) = 0 for q, q ′ ∈ k × , which follows from the lemma below.Lemma 3.2.3.Let e and f be two 2-preregular forms on the same vector space V .Then GL 2 (e, f ) = 0.
Proof.Use the notation from the previous proof.Note that from the presentation given above, by setting the degree of each a ij to be 1, the degree of each b ij to be −1, and the degree of D ±1 to be ±2, GL 2 (e, f ) is a Z-graded algebra.Let M ∈ GL(V ) = GL k (k).We construct a nonzero graded representation U for GL 2 (e, f ) based on the matrix M in the following way: (1) As a vector space, set U := d∈Z U d , where each U d is defined to be the 1-dimensional vector space k.
(2) Set M 0 := M and inductively define  4) The action of D ±1 on U d will be defined as the multiplication by 1 from U d → U d±2 .By the equalities , for all d ∈ Z, these actions respect the relations (3.2).Hence U is a nonzero representation for GL 2 (e, f ), and so this algebra is itself nonzero.
The following result is a straightforward consequence of the proof of Theorem 3.2.2.Corollary 3.2.4.The cogroupoid GL 2 is connected.
Example 3.2.5.Let E and F be identity matrices on k and l-dimensional vector spaces V and W , respectively, with l and k ≥ 2. In this case, it can be checked that the following collection of relations forms a noncommutative Gröbner basis (see e.g., [33, and for all triples 1 ≤ h, i, j ≤ l, the relations It follows that there exists a basis for the algebra GL 2 (e, f ) consisting of all monomials in the variables A, B, D, and D −1 which do not contain any leading term in the above list of relations.In particular, we can observe concretely that this algebra is nonzero.This calculation follows similarly to the argument given by Cohn in [12,Section 5] for the related algebra with generators A and B and relations AB = I k×k , BA = I l×l , which was originally constructed by Leavitt [24].Note, however, that the additional relations for GL 2 (e, f ) lead to a more complicated normal form.

Preregular forms in pivotal tensor category and their 2-cocycle twists
In this section, we introduce categorical descriptions of superpotentials and preregular forms in pivotal tensor categories.We also study their realization in comodule categories over copivotal Hopf algebras.We refer the reader to Section 2.3 for some background on such categories.4.1.Twisted superpotentials and preregular forms in pivotal tensor categories.Consider a klinear Hom-finite tensor category (C, ⊗, Φ, 1, r, ℓ) with pivotal structure j : id C → (−) * * .For any V ∈ ob(C), we denote by V ⊗n the n-fold tensor product of V with rightmost parentheses.This means V ⊗0 = 1, and V ⊗(n+1) = V ⊗ V ⊗n .Following the notation in [29], there is a unique isomorphism: that is inductively defined by Φ (1) : V → V is the identity, and for n ≥ 1 We now generalize the notion of a prereregular form to its categorical analogue.Definition 4.1.1.Let C be a tensor category with pivotal structure j.For any integer m ≥ 2 and object V in C, we define the following.
(1) A morphism f : Here, the operator D m V : Hom C (V ⊗m , 1) → Hom C (V ⊗m , 1) is defined as for any f ∈ Hom C (V ⊗m , 1).See Section 2.3 for the definition of Φ ? .We simply say f : Example 4.1.2.Let C be the tensor category Vec k of all finite-dimensional k-vector spaces.It is clear that Vec k is pivotal with the pivotal structure given by the natural identification V ∼ = V * * for any finite-dimensional vector space V over k.Remark the operator Moreover, f is an m-preregular form of character q on V if and only if it is an m-preregular form in the sense of Definition 2.1.1 with the matrix P ∈ GL(V ) given by P = diag(q, . . ., q).
To connect the generalization of a preregular form from Definition 4.1.1 to the notion of superpotential, we recall some Hom-space operators used in the definition of higher Frobenius-Schur indicators, which play the same role as the cyclic condition in Definition 2.1.1(1)(b) of a preregular form.Remark that superpotentials in the categorical context are slightly different from the formal dual of preregular forms.(1 (3) For any n ≥ 1, define We now define a superpotential in a categorical context by making use of the above Hom-space operators.
Definition 4.1.4.Let C be a tensor category with pivotal structure j.Let m be an integer ≥ 2 and V ∈ ob(C).
(1) A morphism s : with embedding ι : * V ֒→ V ⊗(m−1) such that the diagram V (s) = qs, for some q ∈ k × .In particular, s is said to be a superpotential if s is non-degenerate and q = 1.
Remark that the above definition extends our earlier notion of superpotential in the category of finitedimensional vector sapces Vec k : Let V be an n-dimensional k-vector space and let s : k → V ⊗m be a superpotential as in Remark 2.1.2.This is a superpotential in the sense of Definition 4.1.4.
Hom D ((V ⊗m ), 1) In particular, pivotal equivalence preserves preregular forms of the same characteristic.
Proof.Note that the operator D m V can be interpreted as follows: for any V, W ∈ ob(C), we define D V,W : for any f ∈ Hom C (V ⊗ W, 1).We recall the operator E V,W : Hom A straightforward computation shows that the diagram 1) .So our result follows from [29,Proposition 4.3] and the commutativity of (4.2).By Lemma 4.1.5,for the remainder of the paper we assume that the categories we consider are strict.4.2.Preregular forms in comodule categories over copivotal Hopf algebras.In this subsection, we discuss preregular forms in an arbitrary comodule category that admits a pivotal structure [19,39] and 2-cocycle twists of such preregular forms.
Let (H, Φ) be a copivotal Hopf algebra with character Φ.It follows that the antipode S of H is bijective with inverse S −1 = Φ * S * Φ −1 .Moreover, by [3,Proposition 3.10], the category comod fd (H) admits a pivotal structure ϕ : * (−) ∼ − → (−) * .Here ϕ is an isomorphism of monoidal functors between the right and left duality functors.For any finite-dimensional right H-comodule V , the linear map ϕ V : * V → V * is defined as follows: ϕ V = (id ⊗Φ −1 ) • ρ * V (we use the fact that * V = V * as vector spaces).To describe ϕ V explicitly, let {v 1 , . . ., v n } be a basis of V with ρ with dual basis {v 1 , . . ., v n } of * V and V * .We now generalize the notion of a copivotal (or cosovereign) Hopf algebra to a cogroupoid.Definition 4.2.1.A cogroupoid C is called copivotal if for any X ∈ ob(C), there exists some character Φ X : C(X, X) → k such that for any X, Y ∈ ob(C), where the right hand is defined as is the convolution inverse of Φ X .Note that if the cogroupoid C only has one object X, then C is copivotal if and only if the Hopf algebra C(X, X) is a copivotal Hopf algebra (c.f.[3, Definition 3.7 and Remark 3.8]).Example 4.2.3.Note that when m = 2, SL 2 coincides with the cogroupoid B(E, F ) considered in [4].While GL 2 is connected by Corollary 3.2.4,a straightforward calculation shows that for e and f as in Example 3.2.5,where l = 2 and k = 3, we have SL 2 (e, f ) = 0, and so SL 2 is not connected.
Let V be a finite-dimensional k-vector space with a fixed basis {v 1 , . . ., v n } and e : V ⊗m → k be an m-preregular form with associated matrix P ∈ GL n (k) subject to (2.1).We point out that our SL m (e) is the universal Hopf algebra H(e) described in [7, §5].Recall that SL m (e) is generated over k by A = (a ij ) 1≤i,j≤n and B = (b ij ) 1≤i,j≤n subject to relations: 1≤i1,...,im≤n with Hopf algebra structure In particular, we know that SL m (e) is copivotal with character Φ e : SL m (e) → k such that Φ e (A) = P and Φ e (B) = P −1 .We view V as a right SL m (e)-comodule via ρ : Thus the SL m (e)-comodule map e : V ⊗m → k has the following universal property: Then there is a unique Hopf algebra map θ : SL m (e) → H such that the following diagram commutes: Finally, by Lemma 3.1.5,in SL m (e, f ) we have ).It follows that the cogroupoid SL m is copivotal.Lemma 4.2.6.Let e be an m-preregular form on some n-dimensional k-vector space V .In the pivotal tensor category comod fd (SL m (e)), the SL m (e)-comodule map e : V ⊗m → k is preregular.
Proof.We show that e satisfies both conditions in Definition 4.1.1 to be preregular.
).Then we obtain a commutative diagram: It is clear that the H-colinear map e : V ⊗ * V → k is nondegenerate.For (2): let P be the invertible matrix associated to e.By definition, we have for any 1 ≤ i 1 , . . ., i m ≤ n.Hence, e : V ⊗m → k is preregular.
We point out from the above proof that if an H-comodule map e : V ⊗m → k is preregular in comod fd (H), then it is an m-preregular form on V with matrix P ∈ GL n (k) given by ϕ V : * V → V * : v i → 1≤j≤n P ij v j , where {v 1 , . . ., v n } is the dual basis of {v 1 , . . ., v n } of V .
Let H be an arbitrary Hopf algebra and σ be a left 2-cocycle on H.We denote by The following definition introduces the notion of twisting of a preregular form via the monoidal equivalence between comodule categories.Definition 4.2.7.Let m ≥ 2 be an integer and V be a finite-dimensional right comodule over a Hopf algebra H.For any H-comodule map f : V ⊗m → k, we define an H σ -comodule map f σ : F (V ) ⊗σm → k via Proof.To show that f σ is nondegenerate, we consider the commutative diagram ) / / / / ξ V,V ⊗(m−1) where By [36, Lemma 1.5], we know f σ is again a nondegenerate pairing in D together with morphism k F (g) Moreover, we have D m F (V ) (f σ ) = f σ from Lemma 4.1.5since D m V (f ) = f .This proves our result.We now state our main result in this section to describe how a preregular form and its associated cogroupoid behave under 2-cocycle twists.Theorem 4.2.9.Let m ≥ 2 be an integer and V be a finite-dimensional k-vector space.Let e be an mpreregular form on V and σ be a left 2-cocycle on SL m (e).Then e σ is also a m-preregular form on V and SL m (e σ ) ∼ = SL m (e) σ Similarly, we have (h σ −1 ) * σ = h * σ is a left 2-cocycle on SL m (e σ ) h * σ −1 .Note that g * (h σ −1 ) * σ = (h σ −1 • g) * σ = σ on SL m (e).Therefore, we have a Hopf algebra map are isomorphic as Hopf algebras.Acknowledgements.The authors thank Chelsea Walton and Ken Goodearl for useful discussions.Some results in this paper were formulated at the Structured Quartet Research Ensembles (SQuaREs) program in March 2022, and at the BIRS Workshop on Noncommutative Geometry and Noncommutative Invariant Theory in September 2022.The authors thank the American Institute of Mathematics, the Banff International Research Station, and the organizers of the BIRS Workshop for their hospitality and support.Nguyen was partially supported by the Naval Academy Research Council and NSF grant DMS-2201146.Ure was partially supported by an AMS-Simons Travel Grant.Vashaw was partially supported by an Arthur K. Barton Superior Graduate Student Scholarship in Mathematics from Louisiana State University, NSF grants DMS-1901830 and DMS-2131243, and NSF Postdoctoral Fellowship DMS-2103272.Wang was partially supported by Simons Collaboration grant #688403 and AFOSR grant FA9550-22-1-0272.

Example 2 . 1 . 4 .
[32, §3.2] Manin's left universal quantum group aut l (A) associated to the polynomial algebra A = k[x, y] is generated by the entries of the 2 × 2 matrix M = a b c d together with the formal inverse of the determinant δ = ad − cb, subject to the following relations:

) 1 .
Define the action of A on each graded component U d to be given by scalar multiplication U d → U d+1 , according to the matrix M d .Similarly, define the action of B on the graded component U d to be given by scalar multiplication U d → U d−1 , given by the matrix M −1 d−This gives the following diagram, where the action of A moves to the right, and the action of B moves to the left: Section 1.4]) for the ideal of relations for GL 2 (e, f ), under the graded lexicographic ordering with the variables ordered a 11 > a 12 > ... > a 21 > ... > b 11 > b 12 > ... > b lk > D > D −1 (and where δ hi in all formulas represents the Kronecker delta function):

Lemma 4 . 1 . 5 .
Let C and D be any two pivotal tensor categories.For any pivotal equivalence F : C → D, we have the following commutative diagram:

Lemma 4 . 2 . 5 .
The cogroupoid SL m is copivotal, with characters Φ e : SL m (e) → k given by Φ e (A) = P and Φ e (B) = P −1 , that is, S f,e • S e,f = Φ −1 e * id * Φ f in SL m (e, f ), for any two m-preregular forms e : V ⊗m → k and f : W ⊗m → k with associated invertible matrices P and Q, respectively.Proof.It is straightforward to show that SL m is a well-defined cogroupoid.Next, we verify that Φ e is a well-defined character on SL m (e) = SL m (e, e).Suppose V is of dimension k.By (3.1), in SL m (e) we have Φ e
1) We say a Hopf algebra H left coacts on A preserving the grading of A via ρ : A → H ⊗ A if each homogeneous component of A is a left H-comodule via ρ and ρ is an algebra map.In this case, we say A is a left graded comodule algebra over H. (2) Manin's left universal quantum group aut l (A) associated to A is the Hopf algebra that left coacts on A preserving the grading of A via ρ : A → aut l (A) ⊗ A satisfying the following universal property: If H is any Hopf algebra that left coacts on A preserving the grading of A via τ : A → H ⊗ A, then there is a unique Hopf algebra map f : aut l (A) → H such that the diagram Definition 2.3.1.A k-linear left rigid tensor category (C, ⊗, Φ, 1, r, ℓ) is called pivotal if there is a natural isomorphism j : id C → (−) * * of monoidal functors.In this case, j is called a pivotal structure of C. If C and D are two pivotal categories, and (F , ξ) : C → D is a monoidal functor, we say F preserves the pivotal structure, if the diagram As a consequence, the double left dual (−) * * : C → C is a monoidal functor.
[31,ectivity of GL 2 .It is proved in[31, Theorem 7.2.3] that the universal quantum groups of any two Koszul AS-regular algebras are Morita-Takeuchi equivalent as long as the two algebras share the same global dimension.