Abstract
Given a Hopf algebra H and a counital 2-cocycle μ on H, Drinfeld introduced a notion of twist which deforms an H-module algebra A into a new algebra . We show that when A is a quadratic algebra, and H acts on A by degree-preserving endomorphisms, then the twist is also quadratic. Furthermore, if A is a Koszul algebra, then is a Koszul algebra. As an application, we prove that the twist of the q-quantum plane by the quasitriangular structure of the quantum enveloping algebra is a quadratic algebra equal to the -quantum plane.
1 Introduction
In [Citation6], Drinfeld introduced a notion of twist for Hopf algebras and their module algebras. Drinfeld twists have been studied in several different contexts within both mathematics and physics. On the physics side they have, for instance, been used by Vafa and Witten [Citation12] (in the form of cocycle twists; a special case of Drinfeld twists) in work on mirror symmetry, and also in Brouder, Fauser, Frabetti and Oeckl [Citation3] where the operator product and time-ordered products of quantum field theory are shown to be Drinfeld twists of the normal product. Within mathematics, they have also found applications to the study of color Lie algebras [Citation4], non-commutative geometry [Citation1, Citation9] and rational Cherednik algebras and their representation theory [Citation2].
Davies [Citation5] (again in the context of cocycle twists) and Montegomery [Citation11] have found various algebra properties that are preserved by Drinfeld twists, for instance Artin-Schelter regularity. The results of this paper continue this line of work, showing that the properties of being quadratic and Koszul are also preserved under general Drinfeld twists. This offers a new means of showing that an algebra of interest is quadratic (or Koszul), by showing that it is related by twist to another algebra that is known to be quadratic (or Koszul).
In the rest of this section we will briefly recall some basic terminology following Majid [Citation10, Section 2.3]. See also Etingof and Gelaki [Citation7, Section 5.14] for more.
Let k be an arbitrary field, and consider all objects to be linear over k, with . If H is a Hopf algebra over k, then let denote the coproduct and the counit. For an H-module A we denote the action of on as . An H-module algebra A is an algebra for which the product map is an H-module homomorphism, so , and the unit satisfies .
Definition 1.1
([Citation10, Example 2.3.1]). A 2-cocycle of the Hopf algebra H is an invertible satisfying (1.1) (1.1)
A 2-cocycle is said to be counital if (1.2) (1.2)
Counital 2-cocycles are also referred to as bialgebra twists in Etingof and Gelaki [Citation7, Definition 5.14.1], and twisting elements in Giaquinto and Zhang [Citation9, Definition 1.2].
The Drinfeld twist of H by μ is defined to be the Hopf algebra which shares the same algebra structure and counit as H, and has coproduct . We shall refer to as a “Hopf algebra twist” in order to differentiate it from the following, which is also called a Drinfeld twist.
Definition 1.2.
The Drinfeld twist of an H-module algebra (A, m) by μ is the -module algebra where as k-vector spaces, with the product map given by for all .
In Section 2 we consider A to be a quadratic H-module algebra, with the action of H on A being degree-preserving. We show that in this case, the Drinfeld twist of A by a counital 2-cocycle μ is also a quadratic algebra, and we are able to determine the (quadratic) relations of explicity by means of μ acting on the relations of A. Using this, we prove in Section 3 that if A is a Koszul algebra, then the Drinfeld twist must be Koszul. This extends to arbitrary Hopf algebras a result of Davies [Citation5, Proposition 4.25], who proved this for the case when H is the group algebra of a finite abelian group. Finally in Section 4 we give several examples which apply these results. In particular we show that the q-quantum plane may be twisted by the quasitriangular structure of , and the result is the -quantum plane. Additionally, we give new proofs that the quantum symmetric and exterior algebras and are Koszul.
2 Quadratic algebras
Let A be a connected -graded k-algebra, meaning with and . We will assume throughout this paper that A is locally finite-dimensional, meaning that each grading component Ai is finite-dimensional. Now A is also a quadratic algebra if it is generated by its degree 1 elements , and has quadratic relations , so that , where T(V) is the tensor algebra over V and (R) is the 2-sided ideal generated by R inside T(V).
Suppose is both a quadratic algebra, and an H-module algebra, for some Hopf algebra H acting by degree-preserving endomorphisms (i.e. for all and ). Then, it is easy to see that the degree 1 subspace is an H-submodule of A. Additionally, this action of H on V extends naturally by means of the coproduct of H to make T(V) an H-module algebra.
Now a fact that will be very useful throughout the paper is that the space of relations R is an H-submodule of T(V). This can be seen as follows. Note that a quotient of an H-module algebra by an ideal can only be an H-module algebra too if the ideal is also an H-submodule. Since the H-module algebra structure of A arises from quotienting the H-module algebra T(V) by the ideal (R), we see that (R) must also be an H-submodule of T(V). Additionally, since R is precisely the subspace of degree 2 elements in (R), it must be fixed under the degree-preserving endomorphisms of H. And so R is an H-submodule of T(V).
We now give our first main result.
Theorem 2.1.
Let be a quadratic H-module algebra, where H is a Hopf algebra acting by degree-preserving endomorphisms. If μ is a counital 2-cocycle of H, then the Drinfeld twist is a quadratic algebra of the form where .
Proof.
Firstly let us show is a connected -graded algebra under the same grading as on A. Since as vector spaces, forms an -graded vector space under the grading of A. Let m and denote the products on A and respectively. Now is a graded algebra if for all and . But this follows since , and the action of H is degree-preserving, so . is also connected as .
Note that, as the action of H is degree-preserving, is a subspace of . Also, since as -graded vector spaces, may be viewed as the subspace of degree 1 elements of . Therefore, by showing , we will have proven that is a quadratic algebra, as required.
First we prove is generated by V by induction on the degree of elements of . Consider a degree 2 element . Since A is generated by V, for some . Therefore,
Since the action of H is degree preserving, , and so x can be expressed as a linear combination of -products of elements of V, as required.
Now let , and suppose every element of degree k in can be expressed as a linear combination of -products of elements of V. We show that this implies the elements of degree k + 1 have a similar such decomposition. Consider . Since A is generated by V, we may decompose x as a linear combination of terms which are the m-products of k + 1-elements in V, i.e. is of the form , for some . Suppose for some . Then (2.1) (2.1)
Now , and so by the induction hypothesis this can be decomposed into a linear combination of -products of k-elements of V. Therefore can be decomposed using -products of k + 1-elements, and this implies the same holds for x too. This concludes the induction argument proving is generated by V.
There now exists a natural surjective algebra homomorphism . We show to complete the proof that is a quadratic algebra. Note that for the natural embedding , we have . As i is an H-module homomorphism, is an -module homomorphism, and we see . So if and only if , and this holds if and only if . So , and therefore is seen to be a quotient of .
To show that is equal to , we will apply a dimensional argument. Note that all dimensions in the following are finite due to our assumption that A is locally finite-dimensional. Now, since is a quotient of , the dimension of each grading component of must be less than or equal to the dimension of the same degree component of , i.e. (2.2) (2.2)
Next we will consider the Drinfeld twist of by the cocycle , which leads us to another dimensional inequality (see (2.3)). First we must check that we can indeed take such a twist. Note that this twist is over the Hopf algebra , rather than the Hopf algebra H that we have used so far. It is a simple exercise to check is a counital 2-cocycle of the Hopf algebra twist , and we show next that is an -module algebra.
Since as algebras, the H-action on V defines an -action on V. This extends, by means of the twisted coproduct , to make T(V) an -module algebra. Now is an -submodule of T(V) since, if and , where for some , we have where we use the fact that , since we proved above that R is an H-submodule of the H-module algebra T(V). So we have established T(V) is an -module algebra, and is an -submodule, and therefore is an -module algebra as we required.
We can now consider the Drinfeld twist of by . In direct analogy to the argument used for , one may show that V generates , i.e. use the fact is generated by V to express elements of as linear combinations of -products of elements of V. Then one rewrites each -product as a linear combination of -products.
We therefore have a surjective map . It is easy to show that , and so we find is a quotient of A. This implies the following inequality in the dimensions of the grading components of degree i, (2.3) (2.3)
But as discussed at the start of the proof, twisting preserves the grading on algebras, so (2.4) (2.4)
Combining inequalities (2.2), (2.3), and (2.4) we find . But since was a quotient of , we deduce the algebras must be equal. □
3 Koszul algebras
3.1 Statement of the main theorem
Let A be a connected, -graded, and locally finite-dimensional, k-algebra. Define , so , and we call the corresponding quotient map the augmentation map. Now k is an A-module via , and A is called a Koszul algebra if there is a linear graded free resolution of k as an A-module (see Witherspoon [Citation14, Definition 3.4.3]). If A is Koszul, then it is a standard fact that it is also quadratic, so for V = A1 and R a subspace of .
The next result establishes that Koszulity is preserved under Drinfeld twists. This generalizes to arbitrary Hopf algebras a result of Davies [Citation5, Proposition 4.25] who proved this for the case when the Hopf algebra H is the group algebra of a finite abelian group. The rest of the section is dedicated to proving this result.
Theorem 3.1.
Let be a Koszul H-module algebra, where H is a Hopf algebra acting by degree-preserving endomorphisms. If μ is a counital 2-cocycle of H, then the Drinfeld twist is a Koszul algebra given by where .
3.2 Plan for the proof
It follows immediately from Theorem 2.1 that is a quadratic algebra of the form . Using this we can construct a complex , which, by Witherspoon [Citation14, Theorem 3.4.6], is a resolution of k as an -module precisely when is a Koszul algebra. We therefore seek to show is a resolution to complete the proof. We start by considering the Koszul resolution of A, and twist this resolution using a functor of Giaquinto and Zhang. This produces a new resolution of k as an -module, which we denote . We then construct an isomorphism of complexes between and , the existence of which implies is a resolution k as an -module, as required.
3.3 The Koszul resolution of A
Let us start by defining the Koszul resolution of k as an A-module. It is given by , where (3.1) (3.1)
The differentials dn are induced by the canonical embedding of into the Bar resolution of k, where and (3.2) (3.2) for and is the augmentation map of A. is a left A-module under multiplication on the leftmost tensor leg. It is also a left H-module by restricting the natural action of H on (using the coproduct of H) onto . It is clear for the n = 0 and n = 1 cases that is closed under this H-action. For , we showed above that V and R are H-submodules of T(V), and so each can be seen as an H-submodule of A and respectively. Therefore, in this case, is an intersection of H-submodules of , so is an H-submodule itself.
3.4 The Giaquinto and Zhang twisting functor
Let be the category of all left A-modules. Take the category to be the subcategory of whose objects M are also left H-modules such that the following holds for all , (3.3) (3.3) where . The morphisms are those of that are also H-module homomorphisms.
So far, Drinfeld twists have provided a mechanism for deforming the Hopf algebra H and the H-module algebra A. Giaquinto and Zhang [Citation9, Theorem 1.7] extends this by defining a way to twist a module in into a module in . This new twist defines an equivalence of categories . We show next that the Koszul resolution can be defined within , and describe its image under this twisting functor of Giaquinto and Zhang.
Let us first check that (3.3) holds on for all . For , let , then as required. For , let . It is sufficient to show (3.3) holds when for some , since it will then follow for a general element of by linearity. So, where we use coassociativity of H in the third equality. So (3.3) is satisfied and is an object in . It is standard that the differentials dn of the Koszul resolution are A-module homomorphisms, but we note that they are also H-module homomorphisms. Indeed, on inspecting (3.2), we see that as a map, (3.4) (3.4) where m denotes the product map of A and is the augmentation map of A. Since m and are H-module homomorphisms, it follows that dn also is. Therefore the whole Koszul resolution can be defined within the category .
We now apply the functor of Giaquinto and Zhang [Citation9, Theorem 1.7]. Under this functor the A-modules are twisted into -modules in the following way: let as k-vector spaces, and equip with the action
Here we use the fact that A and are H-modules, so is naturally an -module. Therefore defines a k-linear endomorphism of , and also of , since these are equal as k-vector spaces.
We must also apply the twisting functor to the differentials in the Koszul resolution. Giaquinto and Zhang do not explicitly describe how their functor behaves on morphisms, so for completeness, we prove here that we can take it as mapping each morphism in to itself. Suppose with the actions of A on V and W denoted by and respectively. Let be an A-module homomorphism, so . Then is also an -module homomorphism . Indeed where we use the fact is an H-module homomorphism in the first equality. is also an -module homomorphism , since acts on and in exactly the same way that H acts on V and W respectively. Therefore can also be viewed as a morphism in from to , and so it makes sense to let the functor of Giaquinto and Zhang act identically on morphisms. Therefore each differential dn of the Koszul resolution will be sent by the functor of Giaquinto and Zhang to itself.
To summarize, the result of applying the twisting functor to is a complex of -modules which share the same underlying vector spaces, and differentials, as those on . Since is a resolution of k as an A-module, we see that the new “twisted” complex must be a resolution of k as an -module.
3.5 The Koszul complex of
Using Theorem 2.1 we know that is a quadratic algebra of the form . We can therefore construct a complex completely analogously to the Koszul resolution for A given in (3.1), and we describe this explicitly next. Let where
This is a complex of -modules, where the differentials are inherited from the Bar resolution , where and (3.5) (3.5) where we use the same augmentation map as on A, since and A have the same grading.
Note that by Witherspoon [Citation14, Theorem 3.4.6], is a resolution of k as an -module precisely when is a Koszul algebra. We use this to prove our result, showing that is indeed a resolution. To do so we construct an isomorphism of complexes from to , and use the fact established in the last section that is a resolution of k as an -module. To construct this isomorphism of complexes we first define a sequence of elements which turn out to generalize counital 2-cocycles.
3.6 Higher counital 2-cocycles
First we introduce some notation: for , let , where appears in the i-th position. By convention let and .
Recall that a counital 2-cocycle μ must satisfy the 2-cocycle Equationequation (1.1)(1.1) (1.1) . This equation may be rewritten in the above notation as: (3.6) (3.6)
It turns out that μ can be viewed as just one step in a sequence of “higher counital 2-cocycles”, where the next two lemmas justify this terminology. For , let be defined as and for , (3.7) (3.7) where denotes composition, and all products are taken in the algebra . Notice that , which is just the left hand side of the 2-cocycle Equationequation (3.6)(3.6) (3.6) . The following lemma can be viewed as saying that each of the elements fn satisfies a “higher” version of the 2-cocycle Equationequation (3.6)(3.6) (3.6) :
Lemma 3.2.
For all and .
Proof.
Let . Firstly it is clear that . Now let us check for . By definition, , and using (3.7),
Therefore as required. To finish proving the lemma we show that, for each for all . We do so by performing induction on n. The base case n = 2 holds since is precisely the 2-cocycle Equationequation (3.6)(3.6) (3.6) . Now suppose the hypothesis holds for , and let us show it holds for n = k. Take some . We know that , and now by the induction hypothesis we have (3.8) (3.8)
Therefore, where in the first equality we use the definition of , and insert the expression (3.8) for . Next we use the fact that is an algebra homomorphism. In the 3rd equality we take the product of the first two terms in the previous line, and in the 4-th equality we apply (3.6). Finally, by coassociativity of H, , and therefore the final expression above reduces to , as required. □
Recall that μ also satisfies the counital Equationequation (1.2)(1.2) (1.2) , i.e. , where ϵ is the counit of H. The following lemma proves that the elements fn satisfy a generalized notion of counitality,
Lemma 3.3.
For all and .
Proof.
The n = 1 case is precisely (1.2). For arbitrary and , we can apply Lemma 3.2 to express fn as . Then
Now by the counitality of μ. Also, and this is equal to since by the counit axiom for H, . □
3.7 Defining the isomorphism of complexes
Using the higher counital 2-cocycles constructed in the last section we will now define an isomorphism of complexes between and . From this we will deduce that is a resolution, and therefore that is a Koszul algebra.
Recall that the complex embeds into the Bar resolution , so in particular is a subspace of , for all . Additionally, as k-vector spaces, and therefore is also a subspace of . Since A is an H-module algebra, is naturally an -module. Therefore each given in (3.7) has a well-defined action on the space .
Define to be the corresponding sequence of k-linear maps, i.e. for each , Fn is the map given by fn acting on . The first few maps of are depicted in the diagram below,
We check in Section 3.8 that the image of Fn is indeed inside , and in Section 3.9 that the diagram above commutes, i.e. for all . Finally in Section 3.10 we show each Fn has a k-linear inverse. These facts are sufficient to deduce that is a resolution of k as an -module, which is discussed in Section 3.11.
Remark 3.4.
It is possible to show that the k-linear maps Fn are also -module homomorphisms, and therefore that is an isomorphism of complexes. However it is not required that Fn be an -module homomorphism in order to prove is a resolution, so we won’t include proof of this fact here. For brevity though, we will still refer to as being a chain map, or, isomorphism of complexes.
3.8 Checking
For the n = 0 case, , and as vector spaces, , so the image of F0 is equal to , as required.
When n = 1, we have as vector spaces . Now V was defined to be the degree 1 component A1 of A, and since the H-action on A is degree-preserving, V is an H-submodule of A. Therefore is an -submodule of , so the action of μ on will be closed. Due to the equality of vector spaces , we can view the image of as being in .
For , we inspect how fn in (3.7) acts on an element , and show the result lies in . Recall that,
If , then for all . Similarly, we can show by showing for all , where we have used here the fact as vector spaces. Fix an arbitrary . By Lemma 3.2, , and so,
Let us inspect where we land after acts on a. First note , and since A and V are H-modules, the legs of hitting A or V will remain in these spaces. Recall also that R is an H-submodule of T(V), so for all and . Since the in hits the R in , we may apply the fact that R is an H-submodule of T(V), to deduce
Now when acts on an element of we clearly land in . Since i was arbitrary in , we find , and therefore the image of Fn is indeed inside as required.
3.9 commutes with differentials
Let us now check that the maps Fn make the above diagram commute, i.e. for all . We inspect the result of each side of this condition on an element : where in the first equality we apply the definition of the map dn given in (3.4). In the second equality we use the fact that is an -module homomorphism to pull the inside, resulting in the term. Now, using the definition of in (3.5), and notice that
By Lemma 3.2, , and therefore . This nearly concludes the proof that . It only remains to show
As A is an H-module algebra, we have where ϵ is the counit of H. Therefore the degree 0 component of A has the structure of a trivial H-module, i.e. for all and . Since the augmentation map maps into k, we find . Using this, and the fact is an H-module homomorphism, we find where we apply the “higher counitality” of fn (Lemma 3.3) in the final equality to say .
3.10 The inverse chain map
Finally it is clear from the fact that μ is invertible that each map Fn has a k-linear inverse. In particular, let , where and for , (3.9) (3.9)
Although it is not required in this proof, it can be checked that are -module homomorphisms, and it is clear that they will also commute with differentials. Therefore is an isomorphism of complexes, as claimed.
3.11 ) is a resolution of k as an -module
Recall that the complex , which arose from applying the functor of Giaquinto and Zhang to the Koszul resolution , is a resolution of k as an -module. This means that the following complex is exact,
(3.10)
We wish to show that is also a resolution of k as an -module, as this is equivalent to being a Koszul algebra. Hence we must show the following is exact,
In particular we wish to show for and .
We showed in Section 3.9 that commutes with differentials, so for all . Additionally every map Fn is a k-linear isomorphism, and so we find . Also, for , and we see that . Therefore, for , where in the second equality we apply the fact (3.10) is exact so for . Finally , but , so , again using the fact (3.10) is exact. Therefore is a resolution of k as an -module, and is a Koszul algebra.
4 Examples
Next we demonstrate Theorems 2.1 and 3.1 with several examples. In particular, in Section 4.1, we use Theorem 2.1 to determine the result of twisting the quantum plane by the quasitriangular structure of the quantum enveloping algebra . Then in Section 4.2 we apply Theorem 3.1 to provide a new proof of the Koszulity of the quantum symmetric and exterior algebras and .
4.1 The quantum plane and
For a non-zero , the q-quantum plane is given by . In the following we explain why we can twist A by the quasitriangular structure of the quantum enveloping algebra , and we find via Theorem 2.1 that is the quadratic algebra equal to the -quantum plane .
First recall that a quasitriangular structure on a Hopf algebra H is an invertible satisfying: (4.1) (4.1) where τ is the transposition map and has 1 inserted in the middle leg. Quasitriangular structures are known to satisfy the quantum Yang-Baxter equation [10, Lemma 2.1.4]: , and from this, one can check that a quasitriangular structure is also a counital 2-cocycle, i.e. satisfies (1.1) and (1.2). For this reason it makes sense to use quasitriangular structures to perform Drinfeld twists.
Let us introduce the quantum enveloping algebra . We now suppose q also satisfies . Then is defined to be the k-algebra generated by , and satisfying the relations
may additionally be given the structure of a Hopf algebra via the following:
Now the quantum plane is, by construction, a quadratic algebra. Additionally, we may define a representation of on the degree 1 homogeneous subspace of A as follows:
It is well-known (see, for instance, [Citation10, Exercise 9.1.13]) that this action extends to make A a -module algebra. Note that, by construction, this action is degree-preserving.
In the terminology of Vlaar [Citation13, Theorem 6.7], has a quasitriangular structure “up to completion” - meaning that does not lie in , but rather in a completion of this space. Despite this technicality, still satisfies axioms (1.1) and (1.2) of a counital 2-cocycle. However, since is not in , we must check that there is still a well-defined action of on in order for us to define the twist of A by .
Etingof [Citation8, Remark 3.41] states that given two representations of , say and , which are locally nilpotent (i.e. for all or W, there exists some such that ), we have that is a well-defined operator on . Therefore, if the action of on A is locally nilpotent then it will follow that has a well-defined action on . The action on A is indeed locally nilpotent, and this can be seen from how E acts on a general basis element of A: (4.2) (4.2)
It is a simple exercise to check (4.2), first by showing , and then using induction (in the degree b), to show . From (4.2), we see , and so indeed acts locally nilpotently on A. Hence has a well-defined action on and we may construct the Drinfeld twist .
Finally we can apply our first main result, Theorem 2.1. The conditions of the theorem are met since the action of on A is degree-preserving. We deduce that is a quadratic algebra, and is given by . Vlaar [Citation13, Equation 6.37] tells us how acts on explicity, and for the basis vectors and it is, n Therefore and so is equal to the ideal . Hence is the -quantum plane.
4.2 Symmetric and exterior algebras
Here we will consider , and in each example we will take the Hopf algebra H to be the group algebra , where T is the finite abelian group given by for some . The counit and coproduct of H are given by and for all . Let,
By [Citation2, Lemma 4.5], μ is a counital 2-cocycle of . Note also that .
Consider an n-dimensional -vector space V with a fixed basis . For an n × n-matrix satisfying qii = 1 and , the corresponding quantum symmetric algebra is defined as
Likewise the quantum exterior algebra is given as . In the following examples we will be interested in the case when , where
The quantum symmetric and exterior algebras are known to be Koszul, but we show that this can also be deduced as an application of Theorem 3.1.
Example 4.1
(Twisting the symmetric algebra.). Recall that the symmetric algebra is a Koszul algebra (see Witherspoon [Citation14, Example 3.4.11]), and by definition is equal to for . Additionally acts on V via , and this extends to make S(V) an H-module algebra. In particular the action on a monomial is given by
Note that this action of H on S(V) is degree-preserving, and therefore we may apply Theorem 3.1 to deduce that the Drinfeld twist is a Koszul algebra isomorphic to , where
By [Citation2, Corollary 5.8], for , so . Therefore we find that Theorem 3.1 provides a new proof that the quantum symmetric algebra is Koszul.
Example 4.2
(Twisting the exterior algebra). The exterior algebra is also Koszul, and is equal to for . The action of on V given by also extends by algebra automorphisms to the exterior algebra , making an H-module algebra. By construction this action is degree-preserving, and so, by Theorem 3.1, we find is a Koszul algebra. Now for where we apply [Citation2, Corollary 5.8] in the second equality. Therefore , and we have found another proof that is Koszul.
Acknowledgments
I wish to thank Yuri Bazlov for many helpful discussions.
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References
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