aa r X i v : . [ m a t h . QA ] J u l GEOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS
EHUD MEIR
Abstract.
We formulate the generation of finite dimensional pointed Hopf algebrasby group-like elements and skew-primitives in geometric terms. This is done through amore general study of connected and coconnected Hopf algebras inside a braided fusioncategory C . We describe such Hopf algebras as orbits for the action of a reductive groupon an affine variety. We then show that the closed orbits are precisely the orbits of Nicholsalgebras, and that all other algebras are therefore deformations of Nichols algebras. Forthe case where the category C is the category GG YD of Yetter-Drinfeld modules overa finite group G , this reduces the question of generation by group-like elements andskew-primitives to a geometric question about rigidity of orbits. Comparing the resultsof Angiono Kochetov and Mastnak, this gives a new proof for the generation of finitedimensional pointed Hopf algebras with abelian groups of group-like elements by skew-primitives and group-like elements. We show that if V is a simple object in C and B ( V ) is finite dimensional, then B ( V ) must be rigid. We also show that a non-rigid Nicholsalgebra can always be deformed to a pre-Nichols algebra or a post-Nichols algebra whichis isomorphic to the Nichols algebra as an object of the category C . Introduction
One of the fundamental problems in the theory of finite dimensional pointed Hopfalgebras is to determine if such algebras are generated by group-like elements and skew-primitives. This aims to generalize the following classical classification result of Cartier,Milnor, Moore, and Kostant from the 1960s:
Theorem 1.1 (Cartier-Milnor-Moore-Kostant,60s) . Let H be a cocommutative Hopf al-gebra over an algebraically closed field K of characteristic zero. Then H is the crosseddirect product of a group algebra with the universal enveloping algebra of a Lie algebra.In particular, H is generated by group-like elements and primitive elements. This problem was studied thoroughly in case the group of group-like elements in theHopf algebra is abelian and the ground field has characteristic zero. In [AS10] An-druskiewitsch and Schneider proved that such a Hopf algebra must be generated bygroup-like elements and skew-primitives, and gave a complete classification of such al-gebras in case the group of group-like elements does not have prime divisors which are ≤ . This was done by the lifting method and by a deep study of the structure ofthe possible Nichols algebras arising in the category of Yetter-Drinfeld modules over anabelian group. The Nichols algebras correspond to the universal enveloping algebras inthe above theorem. In [He09] Heckenberger classified all Yetter-Drinfeld modules V forwhich the Nichols algebra B ( V ) is finite dimensional. In [Ang13] Angiono described theseNichols algebras explicitly in terms of generators and relations and proved that all finitedimensional connected graded Hopf algebras in the category of Yetter-Drinfeld modules over a finite abelian groups are Nichols algebras. Using the above results Angiono andGarcia-Igelsias gave in [AG19] a complete classification of finite dimensional pointed Hopfalgebras with abelian groups of group-like elements. For more classification results, in-cluding the case where the group of group-like elements is non-abelian, see [AnSa19] andthe survey [And14].The starting point of these classification results is the following: let G be a finitegroup and let H be a finite dimensional pointed Hopf algebra whose group of group-like elements is isomorphic to G . The fact that H is pointed implies that the coradicalfiltration H = KG ⊆ H ⊆ · · · ⊆ H n = H of H is a Hopf algebra filtration. Thisimplies that H gr := ⊕ ni =1 H i /H i − is a graded Hopf algebra. Moreover, if H gr is generatedby group-like elements and skew-primitives then H is generated by group-likes and skew-primitives as well. The inclusion H → H gr splits, and by a result of Radford we canwrite H gr ∼ = R KG where R is a graded Hopf algebra in the category of Yetter-Drinfeldmodules over G , GG YD . The comultiplication of R as a Hopf algebra in GG YD is differentfrom the comultiplication of elements of R in the Hopf algebra H . The skew-primitiveelements become primitive elements in R . The original question then boils down towhether or not R is generated by primitive elements, and not just skew-primitives.In [AS10, AS00] Andruskiewitsch and Schneider studied both the Hopf algebra R andthe dual Hopf algebra R ∗ , proved that finite dimensionality implies that certain relationsamong the elements of these Hopf algebras must hold, and concluded that both R and R ∗ are generated by primitive elements. This means that the algebra R is in fact the Nicholsalgebra B ( V ) where V = P ( R ) is the set of primitive elements in R .Andruskiewitsch and Schneider then also address the questions of the reconstructionof the original algebra A out of A gr , and for what objects V of GG YD the algebra B ( V ) is finite dimensional. The key-point in proving that any finite dimensional pointed Hopfalgebra is generated by group-like elements and skew-primitives is to prove that all Hopfalgebras R in GG YD arising from the above construction are Nichols algebras.In this paper we study a more general problem by using a different, geometric, method.Instead of looking at the category GG YD we look at a general braided fusion category C .For every object B in C we will construct an affine variety X B , whose points representthe structure constants of connected coconnected Hopf algebras (these notations will beexplained in Sections 2 and 5). The group Γ B := Aut C ( B ) acts on X B , and the orbitscorrespond to isomorphism types of Hopf algebras. We will prove the following: Theorem 1.2.
Let R be a connected coconnected finite dimensional Hopf algebra in C such that R ∼ = B as objects of C . The orbit O R of R ∈ X B is closed if and only if thealgebra R is isomorphic to a Nichols algebras. In particular, all the orbits of Γ B in X B are closed if and only if all the connected and coconnected Hopf algebras in C which areisomorphic to B as objects of C are Nichols algebras. If R and R are two algebras in X B , we say that R specializes to R if O R ⊆ O R .We also say in this case that R is a deformation of R . The theorem above thus impliesthat every algebra in X B is a deformation of a Nichols algebra. EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 3
We thus focus our attention on studying deformations of finite dimensional Nicholsalgebras B ( V ) , as such deformations are the possible obstructions to the generation byskew-primitives and the coradical (see Theorem 1.7). Definition 1.3.
The Hopf algebra R ∈ X B is called rigid if R ∈ O R ′ for some R ′ ∈ X B implies that R ∼ = R ′ .The ultimate goal will thus be to prove that B ( V ) is rigid whenever it is finite dimen-sional, as this will imply that all the algebras in X B are Nichols algebras and are thereforegenerated by primitive elements. We will prove the following result: Theorem 1.4.
Assume that V is simple in C and that B ( V ) is finite dimensional. ThenB ( V ) is rigid. Since our aim is to prove that all orbits in X B are closed, it is worthwhile asking howdo hypothetical non-closed orbits in X B look like. To state the next result, recall that a pre-Nichols algebra in a braided monoidal category C is a graded Hopf algebra in C whichis generated by primitive elements (though not all the primitive elements are necessarilyof degree 1). Thus, a Pre-Nichols algebra R is a quotient of the Hopf algebra T ( V ) forsome V ∈ C which also projects onto the Nichols algebra B ( V ) . Dually, a post-Nicholsalgebra is a Hopf subalgebra of the graded-dual Hopf algebra of T ( V ) which containsB ( V ∗ ) . Post- and pre-Nichols algebras are graded-dual to each other (see Section 2.3 of[AARB17]). Theorem 1.5.
Assume that B ( V ) is finite dimensional and not rigid. Then there iseither a finite dimensional pre-Nichols algebra R such that P ( R ) = V ′ ( V and such thatB ( V ) ∈ O R , or a finite dimensional pre-Nichols algebra R such that P ( R ) = V ′ ( V ∗ andsuch that B ( V ∗ ) ∈ O R . Summarizing Theorem 1.2 and 1.5, we get the following result:
Theorem 1.6.
Let C be a braided fusion category. The following conditions are equivalent:(1) For every object B ∈ C , all the orbits of the action of Γ B = Aut C ( B ) on X B areclosed.(2) All finite dimensional Nichols algebras in C are rigid.(3) All finite dimensional pre-Nichols algebras in C are Nichols algebras.(4) Every connected and coconnected Hopf algebra R in C is isomorphic to B ( P ( R )) . In case the category C is the category AA YD if Yetter-Drinfeld modules over a finitedimensional semisimple Hopf algebra A , the Bosonization process which produces from aHopf algebra R in C a Hopf algebra R A in V ec K gives the following result: Theorem 1.7.
Let A be a finite dimensional semisimple Hopf algebra. The following areequivalent:(1) For every B ∈ AA YD the orbits of Γ B in X B are closed.(2) All finite dimensional Nichols algebras in AA YD are rigid.(3) Every Hopf algebra H in which the coradical is a Hopf algebra isomorphic to A isgenerated by the zeroth and first levels of its coradical filtration. EHUD MEIR (4) Every connected and coconnected Hopf algebra R in AA YD is isomorphic to a Nicholsalgebra.For A = KG where G is a finite group the second statement says that every finite dimen-sional pointed Hopf algebra H with G ( H ) = G is generated by group-like elements andskew-primitives. The study of deformations of Hopf algebras was initiated by Gerstenhaber and Schackin [GS90]. Du, Chen and Ye studied deformations of graded Hopf algebras in [DCY07].Angiono, Kochetov and Mastnak studied deformations of Nichols algebras in [AKM15].Deformations were also studied by Makhlouf in [M05]. The deformations in the abovepapers are deformations by a parameter λ . We will show that our notion of rigidity, atleast for Nichols algebras, is equivalent to the rigidity of Angiono, Kochetov and Mastnak.In [AKM15] the authors gave a proof that all Nichols algebras of diagonal type are rigid.Theorem 1.7 above provides a new proof for the generation of pointed Hopf algebras withan abelian group of group-like elements by skew-primitives and group-likes. In Section4 of [Ang13] Angiono proved this result by ruling out the existence of finite dimensionalpre-Nichols algebras which are not Nichols algebras. The proof in this paper follows fromthe rigidity result of [AKM15] which is based on the description of Nichols algebras from[Ang13] by generators and relations, but not on the analysis done in Section 4 of [Ang13].This paper is organized as follows: in Section 2 we will give preliminaries about braidedfusion categories, Hopf algebras, and the results from the theory of algebraic groups andgeometric invariant theory which we will use here. In section 3 we will discuss in moredetail Hopf algebras in braided fusion categories, and prove the equivalence of the secondand third conditions of Theorem 1.7. In Section 4 we will give a description of braidedfusion categories using vector spaces and linear algebra. This will be used in Section 5to show that the collection of all connected and coconnected Hopf algebras which areisomorphic to a given object B of C form an affine variety X B . We will also constructan action of Γ B := Aut C ( B ) on this variety, and show that the orbits correspond toisomorphism classes of Hopf algebras. In the end of Section 5 we will also give a proof ofTheorem 1.4. In Section 6 we will discuss filtrations of Hopf algebras and their relation togeometric invariant theory. In Section 7 we will give a proof of Theorem 1.2 and 1.5. InSection 8 we will explain the relations between the different notions of rigidity and givea new proof, using the results of Angiono Kochotev and Mastnak, to the generation ofpointed Hopf algebras with abelian group of group-like elements by group-like elementsand skew-primitives. 2. Preliminaries
The categories we will consider in this paper are braided fusion categories. We recallhere briefly the relevant definitions. For more on fusion categories and braided fusioncategories see [ENO05]. We assume throughout the paper that our ground field K isalgebraically closed and of characteristic zero. Definition 2.1.
A fusion category C over K is an abelian category which satisfies thefollowing properties: EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 5 (1) The category C is enriched over V ec K . This means that all hom-spaces in C arefinite dimensional K -linear vector spaces.(2) The category C is semisimple. This means that every object in C can be writtenuniquely as a direct sum of simple objects.(3) The category C is monoidal. This means that we have a functor O : C × C → C together with associativity isomorphisms α X,Y,Z : ( X ⊗ Y ) ⊗ Z → X ⊗ ( Y ⊗ Z ) for every three objects X, Y, Z of C satisfying the usual pentagon axiom, and thereis a unique object, up to isomorphism, , such that the functors X ⊗ X and X X ⊗ are both isomorphic to the identity functor.(4) The number of isomorphism classes of simple objects in C is finite.(5) The tensor unit is a simple object in C .(6) The category C is rigid. This means that every object X has a right dual X ∗ anda left dual ∗ X . The right dual is defined uniquely up to an isomorphism by thecondition that there are maps ev X : X ∗ ⊗ X → and coev X : → X ⊗ X ∗ satisfyingsome coherence conditions. The left dual is defined similarly. The semisimplicityof a fusion category implies that left and right duals are isomorphic.A fusion category is called braided if it is equipped with a natural isomorphism σ X,Y : X ⊗ Y → Y ⊗ X for every two objects X, Y ∈ C such that for every X, Y, Z ∈ C the morphism X ⊗ Y ⊗ Z σ X,Y ⊗ Z → Y ⊗ X ⊗ Z Y ⊗ σ X,Z → Y ⊗ Z ⊗ X (2.1)is equal to the morphism X ⊗ Y ⊗ Z σ X,Y ⊗ Z → Y ⊗ Z ⊗ X (2.2)and the morphism X ⊗ Y ⊗ Z X ⊗ σ Y,Z → X ⊗ Z ⊗ Y σ X,Z ⊗ Y → Z ⊗ X ⊗ Y (2.3)is equal to the morphism X ⊗ Y ⊗ Z σ X ⊗ Y,Z → Z ⊗ X ⊗ Y (2.4)(to ease notations, we do not write here the associativity constraints). Notice that we donot assume that σ X,Y σ Y,X = 1 Y ⊗ X . A category satisfying this extra assumption is called symmetric .One important example of a braided fusion category is the Drinfeld center of V ec G .The objects in this category are vector spaces which admit a G -action and a G -grading.The action and the grading should be compatible in the following sense: for g, h ∈ G we EHUD MEIR have g · V h ⊆ V ghg − . This category is braided. The braiding is given by the followingformula: V ⊗ W → W ⊗ Vv ⊗ w g · w ⊗ v for v ∈ V g . This is an example of a braided monoidal category which is also modular .An associative unital algebra inside a monoidal category C is defined as an object A of the category together with morphisms m : A ⊗ A → A and u : → A satisfying theassociativity relation m ( m ⊗ A ) = m (1 A ⊗ m ) and the unit axiom A = m (1 A ⊗ u ) = m ( u ⊗ A ) . (2.5)A co-associative counital coalgebra is defined similarly, by changing the domain andcodomain of all the relevant morphisms. A Hopf algebra R inside a braided monoidalcategory C is an object R of C equipped with the following maps m : R ⊗ R → Ru : → R ∆ : R → R ⊗ R (2.6) ǫ : R → and S : R → R such that the following conditions hold:(1) ( R, m, u ) is an associative unital algebra.(2) ( R, ∆ , ǫ ) is a coassociative counital coalgebra.(3) ∆ and ǫ are algebra maps. This means that the diagrams R ⊗ R ⊗ R ⊗ R Id R ⊗ σ R,R ⊗ Id R / / R ⊗ R ⊗ R ⊗ R m ⊗ m ( ( PPPPPPPPPPPP R ⊗ R ∆ ⊗ ∆ ♥♥♥♥♥♥♥♥♥♥♥♥ m + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ R ⊗ RR ∆ ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ (2.7)and R ⊗ R ǫ ⊗ ǫ (cid:15) (cid:15) m / / R ǫ (cid:15) (cid:15) ⊗ m / / (2.8)are commutative.(4) The map S is an antipode. This means that the two compositions m ( S ⊗ Id R )∆ : R → R and m ( Id R ⊗ S )∆ : R → R are equal to uǫ . EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 7
Notice that algebras and coalgebras can be defined in any monoidal category, whereas thedefinition of a Hopf algebra requires the braiding in the category. If R is a Hopf algebrainside a braided fusion category C , then the dual object R ∗ is again a Hopf algebra, wherethe multiplication is given by ∆ ∗ : R ∗ → ( R ⊗ R ) ∗ ∼ = R ∗ ⊗ R ∗ and the other structuremaps are defined similarly as the dual of the structure maps of R , see Section 5 of [AS00]and the preliminaries in [Z99]. Remark 2.2.
Some of the Hopf algebras in this paper will actually be graded Hopfalgebras in the bigger category Ind ( C ) which contains infinite direct limits of diagrams in C . To make it clear that a certain algebra is already contained in C we will say it is finitedimensional . This is consistent with the notion of finite dimensionality when the categoryis the category of Yetter-Drinfeld modules over some finite dimensional Hopf algebra.We say that an associative algebra A inside a fusion category is connected if A/J ( A ) ∼ = , the trivial algebra in C . Here J ( A ) is the Jacobson radical of A , and is defined as thebiggest nilpotent ideal in A . This definition makes sense in a general fusion category, andnot only for finite dimensional algebras over a field. Indeed, an ideal I of A is a subobjectof A for which the image of the restriction of the multiplication maps I ⊗ A → A and A ⊗ I → A is contained in I . Nilpotency of the ideal means that for a big enough N , the multiplicationmap I ⊗ N = I ⊗ I ⊗ I ⊗ · · · ⊗ I → A is the zero map. In a similar way, we define a coalgebra C to be coconnected, if its dualalgebra C ∗ is connected. This is equivalent to the coradical of C , which is the largestcosemisimple subcoalgebra of C , being one-dimensional. Definition 2.3.
A Hopf algebra is called (co)connected if it is (co)connected as a (co)algebra.A Hopf algebra is called connected coconnected (or ccc) if it is both connected and co-connected.Among the ccc Hopf algebras the Nichols algebras play a prominent role (see [AS02]).We recall now their definition.
Definition 2.4.
For a given object V ∈ C the Nichols algebra B ( V ) is the unique Hopfalgebra in Ind ( C ) which satisfies the following conditions:(1) The Hopf algebra B ( V ) is graded by the non-negative integers as a Hopf algebra.(2) The zeroth component of the grading satisfies B ( V ) = .(3) The first component of the grading satisfies B ( V ) = V , and B ( V ) is generated by V .(4) The subobject of primitive elements of B ( V ) is V . This subobject is defined forany Hopf algebra as P ( R ) = Ker (∆ − u ⊗ − ⊗ u ) . Remark 2.5.
One of the fundamental and very difficult questions in the study of Nicholsalgebras is to determine for which objects V ∈ C the Nichols algebra is finite dimensional. EHUD MEIR
The definition above gives us a concrete way to construct the Nichols algebra. SinceB ( V ) is generated by V , and the elements of V are primitive in B ( V ) we have a surjectiveHopf algebra map π : T ( V ) → B ( V ) . The fact that B ( V ) is graded and the elementsof V are of degree 1 implies that the map π is a graded map. The Hopf algebra B ( V ) can then be constructed from T ( V ) in the following way: we divide T ( V ) first by theHopf ideal I of T ( V ) generated by the primitive elements of T ( V ) in degrees > . Thenin the quotient graded Hopf algebra T ( V ) /I we divide by the ideal I generated by theprimitive elements of degree > in this algebra, and continue inductively. Notice that itmight happen that by dividing out I we get new primitive elements in T ( V ) /I . This isthe reason we need to repeat this process. See also the introduction in [AG19].The Nichols algebra of V and of V ∗ are related in the following way: Recall that fora graded Hopf algebra R = ⊕ n ≥ R n in C , in which all the homogeneous components arefinite dimensional, the graded dual S = M n ≥ ( R n ) ∗ is also a graded Hopf algebra. In case R itself is finite dimensional, this is the same as thedual R ∗ . We claim the following (see also Lemma 5.5 in [AS00] and Proposition 3.2.20in [AnGr99] for the case the category is the category of Yetter-Drinfeld modules over aHopf algebra): Lemma 2.6.
The graded dual of B ( V ) is B ( V ∗ ) . In particular B ( V ) is finite dimensionalif and only if B ( V ∗ ) is finite dimensional.Proof. Let S be the graded dual of B ( V ) . It holds that S = ( B ( V ) ) ∗ = and S =( B ( V ) ) ∗ = V ∗ . We first claim that P ( S ) = S . Notice first that P ( S ) is a gradedsubobject of S . Assume that V ∗ ( P ( S ) . Let n be the minimal integer > such that W = P ( S ) n = 0 . Then h W, V · n i = h ∆ n − W, V ⊗ V ⊗ · · · ⊗ V i = n − X i =0 h ǫ i ⊗ W ⊗ ǫ n − − i , V ⊗ n i = 0 . We have used here the primitivity of W to express ∆ n − using ǫ , the fact that the mul-tiplication in R is dual to the comultiplication in S and the fact that ǫ ( V ) = 0 . But theabove equation implies that h W, B ( V ) n i = 0 since B ( V ) is generated by V . This impliesthat W = 0 , a contradiction.We prove now that S is generated by V ∗ . Assume that this is not the case. Let n be theminimal integer > such that ( V ∗ ) · n ( S n . Using semisimplicity, we can find a subobject = W ⊆ V ⊗ n such that h ( V ∗ ) · n , W i = 0 . A dual argument to the argument above showsthat ∆( W ) ∈ S ⊗ S n ⊕ S n ⊗ S . This follows from the fact that by the minimality of n ,it holds that S m = ( V ∗ ) · m for every m < n . This means that h ∆( W ) , S m ⊗ S n − m i = h ∆( W ) , ( V ∗ ) · m ⊗ ( V ∗ ) · ( n − m ) i = h W, ( V ∗ ) · m + n − m i = h W, ( V ∗ ) · n i = 0 . Since S = this already implies that W is primitive, which is a contradiction toP ( B ( V )) = V . (cid:3) EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 9
Actions of algebraic groups on affine varieties.
We recall the following frame-work and basic facts about actions of algebraic groups. Let Γ be a reductive algebraicgroup acting on an affine variety X . The following holds (see Section 8.3 in [Hu75] andLemma 3.3 in [N78]) Proposition 2.7.
All the orbits of Γ in X are locally closed. For a Γ -orbit O in X , itholds that O \ O is the union of orbits of smaller dimension. Proposition 2.8. If W and W are two closed disjoint Γ -stable subsets of X , then thereis an element f ∈ K [ X ] Γ such that f ( W ) = 1 and f ( W ) = 0 . In other words- we canseparate the subsets W and W by an invariant polynomial. Finite dimensional Hopf algebras in braided fusion categories
Let H be a finite dimensional Hopf algebra in a braided fusion category C . The coradicalfiltration of H is defined as follows (see Chapter IX of [S69] and [AS98]): • H ⊆ H is the coradical. It is the sum of all simple subcoalgebras of H . This isdual to the operation of taking the quotient of H by its Jacobson radical. • For every n > we define H n = Ker ( H ∆ → H ⊗ H → ( H/H n − ) ⊗ ( H/H ) .Equivalently, H n = Ker (cid:0) H ∆ n − → H ⊗ n → ( H/H ) ⊗ n (cid:1) .If H satisfy the dual Chevalley property, i.e. the tensor product of semisimple H -comodules is again a semisimple H -comodule, then H is also a subalgebra of H , andnot only a subcoalgebra (see [AGM17]). In this case the filtration is a filtration of alge-bras as well, that is H i · H j ⊆ H i + j . As a result, the associated graded object H gr := ⊕ H n /H n − (3.1)is a graded Hopf algebra. We will say that H is coardically graded if H ∼ = H gr as Hopfalgebras. The grading gives a split surjection π : H gr → H of Hopf algebras. Using theprocess of Bosonization (or Radford-Majid biproduct) one can also write this algebra inthe form H gr = H R where R is a graded Hopf algebra in the category of Yetter-Drinfeldmodules over H . As a vector space R = { r ∈ H gr | (1 ⊗ π )∆( r ) = r ⊗ ∈ H gr ⊗ H } (3.2)and R = . See [AS10] for the description of R as a Hopf algebra in H H YD .The following lemma is the first step in proving Theorem 1.7: Lemma 3.1.
Let A be a finite dimensional semisimple Hopf algebra. The following con-ditions are equivalent:(1) Every finite dimensional Hopf algebra H in which the coradical H is isomorphicto A is generated by the zeroth and first levels of its coradical filtration.(2) Every coardically graded finite dimensional Hopf algebra H in which the coradical H is isomorphic to A is generated by the zeroth and first levels of its coradicalfiltration.(3) Every coradically graded finite dimensional Hopf algebra R ∈ AA YD in which R = is generated by its primitive elements. Proof.
The first condition clearly implies the second one. On the other hand, if thesecond condition holds and H is a Hopf algebra such that H ∼ = A , we can pass to theassociated graded Hopf algebra H gr . Since this Hopf algebra is coardically graded thesecond condition implies that it is generated by its zeroth and first terms of the coradicalfiltration, and the same thus holds also for H (see Lemma 2.2. in [AS98]).We next prove that the second and third conditions are equivalent. Indeed, if H iscoradically graded then the above discussion implies that H ∼ = H R where R is acoradically graded Hopf algebra in AA YD . It then holds that H is generated by its firstand zeroth terms of its coradical filtration if and only if the same holds for R . But this isequivalent to R being generated by its primitive elements. (cid:3) Lemma 3.2. (see also Lemma 5.5 in [AS00] ) Assume that R is a Hopf algebra inside abraided fusion category C . If R is generated by P ( R ) , and R ∗ is generated by P ( R ∗ ) , then R is isomorphic to B ( P ( R )) (that is: R is a Nichols algebra). Remark 3.3.
This proposition also holds if one replaces the braided fusion category witha finite braided tensor category.
Proof.
Write P ( R ) = V and P ( R ∗ ) = W . Write π : T ( V ) → R for the resulting surjectiveHopf-algebra map in Ind ( C ) . Let p : T ( V ) → B ( V ) be the canonical surjection. We willshow that π splits via p .Assume that U ⊆ T ( V ) is a primitive subobject of degree n > (that is: U ⊆ P ( T ( V )) ).We will show that π ( U ) = 0 . The primitivity of W implies that h W, V · n i = h n − X i =0 ǫ i ⊗ W ⊗ ǫ n − − i , V ⊗ n i = 0 . So in particular h W, π ( U ) i = 0 . Using now the primitivity of U (which also implies theprimitivity of π ( U ) , since π is a Hopf-algebra morphism) we get h W · m , π ( U ) i = h W ⊗ m , m − X i =0 i ⊗ π ( U ) ⊗ m − i − i = 0 for every m . But this implies that π ( U ) is perpendicular to the subalgebra of R ∗ generatedby W , which is R ∗ itself. This means that π ( U ) = 0 , so U ⊆ Ker ( π ) .This implies that the ideal I ⊆ T ( V ) generated by primitive elements of degree > is contained in Ker ( π ) . We thus get a surjective Hopf algebra map π : T ( V ) /I → R .Denote by I the ideal of T ( V ) /I generated by primitive elements of degree > in thisalgebra. By the same argument, I ⊆ Ker ( π ) . We define now inductively ideals I n and Hopf algebra surjections π n : T ( V ) /I n → R such that I n ⊆ T ( V ) /I n − is the idealgenerated by all primitive elements of degree >1. This is the same as the chain of idealswhich appears after Definition 2.4. The union of the inverse images of the ideals I n inside T ( V ) is exactly the kernel of the surjection p : T ( V ) → B ( V ) . We thus get a surjectivemap π : B ( V ) → R which is injective on V . This map must be injective as well, because if n is the minimalnumber such that Ker ( π ) n = 0 then Ker ( π ) n must be a primitive object by gradingconsideration, and by the definition of the Nichols algebra this is impossible. (cid:3) EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 11
Lemma 3.4.
The pairing P ( R ) ⊗ P ( R ∗ ) → is non-degenerate if and only if R ∼ = B ( P ( R )) .Proof. One direction follows from the fact that the graded dual of B ( V ) is B ( V ∗ ) , see2.6. Assume, on the other hand, that the pairing is non-degenerate. By the previousproposition, it will be enough to prove that R ∗ is generated by W = P ( R ∗ ) . By a dualargument, R is then generated by V = P ( R ) , and we can use the previous proposition tofinish the proof.Now for the coradical filtration ( R ∗ i ) of R ∗ it holds that R ∗ = K and R ∗ = K ⊕ W .Since the coradical filtration is dual to the radical filtration, It holds that R ∗ ∼ = ( R/J ) ∗ where J is the Jacobson radical of R . The fact that the pairing between V and W isnon-degenerate then means that the image of V inside R/J must be onto. But thismeans that V generates R as a unital algebra. By a similar argument W generates R ∗ ,and by the previous proposition R is a Nichols algebra. (cid:3) We are now ready to prove the equivalence of the third and fourth conditions of 1.7.
Proof.
Lemma 3.1 and Lemma 3.2shows that the third condition of Theorem 1.7 is equiva-lent to the statement that all finite dimensional coardically graded Hopf algebras R ∈ AA YD with R = are Nichols algebras. If the fourth condition of Theorem 1.7 holds and allthe ccc Hopf algebras in AA YD are Nichols algebras, this is in particular true for coardi-cally graded Hopf algebras with R = , and the fourth condition thus implies the thirdcondition.If on the other hand the third condition holds and R is a ccc Hopf algebra in AA YD then the Hopf algebra R gr arising from the coradical filtration of R is a Nichols algebra.This implies that R is generated by its primitive elements. Similarly, the dual R ∗ is alsogenerated by its primitive elements, and by Lemma 3.2 R is a Nichols algebra. (cid:3) Braided fusion categories by linear algebra
Let C be a braided fusion category. Write X , . . . X s for a set of representatives of theisomorphism classes of simple objects of C . We would like to describe all the data encodedin the structure of C as a braided fusion category using vector spaces and linear maps.We begin with the hom-spaces, which are very easy to describe. Indeed, it holds thatHom C ( X i , X j ) = ( if i = jK · Id X i if i = j (4.1)since C is a semisimple category, and the X i are non-isomorphic simple objects. Everyobject of C is isomorphic to a direct sum of simple objects. Instead of writing an objectof C as L i X ⊕ a i i we will use the isomorphic object M i U i ⊗ X i where U i are plain vector spaces. Notice that the fact that C is a K -linear category meansthat taking tensor products of objects in C with vector spaces makes sense. The hom-spaces in C are then given byHom C ( M i V i ⊗ X i , M j U j ⊗ X j ) = M i Hom K ( V i , U i ) (4.2)We describe next the tensor product and the associativity constraints. Assume that [ X i ] · [ X j ] = X k N ki,j [ X k ] in the Grothendieck ring of C . For every three indices i, j, k ∈ { , . . . s } fix a vector space V ki,j of dimension N ki,j . We can then write X i ⊗ X j ∼ = M k V ki,j ⊗ X k . Notice that (cid:0) M i U i ⊗ X i (cid:1) ⊗ (cid:0) M j W j ⊗ X j (cid:1) ∼ = M i,j,k U i ⊗ W j ⊗ V ki,j ⊗ X k . For i, j, k ∈ { , . . . s } we then have ( X i ⊗ X j ) ⊗ X k ∼ = M a V ai,j ⊗ X a ⊗ X k ∼ = M a,b V ai,j ⊗ V ba,k ⊗ X b while on the other hand X i ⊗ ( X j ⊗ X k ) ∼ = M c X i ⊗ V cj,k ⊗ X c ∼ = M c,b V bi,c ⊗ V cj,k ⊗ X b . The associativity constraints α i,j,k : ( X i ⊗ X j ) ⊗ X k → X i ⊗ ( X j ⊗ X k ) are then given by a family of linear maps α a,b,ci,j,k : V ai,j ⊗ V ba,k → V bi,c ⊗ V cj,k . (4.3)The Pentagon axiom then translates into a list of axioms which say that certain sumsof compositions of the linear maps α a,b,ci,j,k are equal. More precisely, for every i, j, k, l ∈{ , . . . s } writing the pentagon diagram for the tensor product of X i ⊗ X j ⊗ X k ⊗ X l givesus that for every a, b, c ∈ { , . . . s } the composition V ai,j ⊗ V ba,k ⊗ V cb,l α a,b,di,j,k −→ M d V bi,d ⊗ V dj,k ⊗ V cb,l α b,c,ei,d,l −→ M d,e ⊗ V ci,e ⊗ V dj,k V ed,l α d,e,fj,k,l −→ M e,f V ci,e ⊗ V cj,f ⊗ V fk,l (4.4)is equal to the composition V ai,j ⊗ V ba,k ⊗ V cb,l α b,c,fa,k,l −→ M f V ai,j ⊗ V ca,f ⊗ V fk,l α a,c,ei,j,f −→ q M f,e V ci,e ⊗ V tj,f ⊗ V fk,l . (4.5)Assuming that X = is the tensor unit, the unit axioms for the monoidal category C can be translated as saying that V j ,i and V ji, are zero if i = j , are one dimensional in case EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 13 i = j , and that there are distinguished bases l i ∈ V i ,i and r i ∈ V ii, such that for every i, j, k ∈ { , . . . s } it holds that α i,k,ji, ,j : V ii, ⊗ V ki,j → V ki,j ⊗ V j ,j (4.6)sends r i ⊗ v to v ⊗ l i for v ∈ V ki,j .The rigidity of the category can be described in this language in the following way:for every i ∈ { , . . . s } there is a unique ¯ i ∈ { , . . . s } such that V i, ¯ i and V i,i are one-dimensional, and V i,j = 0 for j = ¯ i . The evaluation ev i : X ¯ i ⊗ X i → X (4.7)is then given by a linear isomorphism which we denote by the same symbol ev i : V ¯ i,i → K and the coevaluation X → X i ⊗ X ¯ i is then given by a linear isomorphism coev i : K → V i, ¯ i .The rigidity axioms translate again to equality between compositions of linear maps. Theequality between the composition X i → ( X i ⊗ X ¯ i ) ⊗ X i → X i ⊗ ( X ¯ i ⊗ X i ) → X i (4.8)and Id X i translates to the equality K → V i, ¯ i ⊗ V i ,i → V ii, ⊗ V i,i → K = K Id K → K (4.9)where the first map sends 1 to coev i ⊗ l i , the second map is α , ,ii, ¯ i,i and the third map sends r i ⊗ v to ev i ( v ) ∈ K .The braided structure is given by maps σ X i ,X j : X i ⊗ X j → X j ⊗ X i . This is the sameas a collection of linear isomorphisms σ ki,j : V ki,j → V kj,i , (4.10)which should satisfy the axioms arising from the braid relations.The introduction of the vector spaces V ki,j here and the linear maps α a,b,ci,j,k and σ ki,j can beseen as a way to “introduce coordinates” on the category C . We use here the fact that asan abelian category, C is very simple to understand on the level of objects and morphisms.The additional braided monoidal structure is described using linear algebra. This will beused later on in the construction of the variety X B . This should be seen as more of anauxiliary result, and will not play a dominant role in the sequel.5. The variety X B and the action of the group Usually, when one speaks of “a Hopf algebra R inside the braided fusion category C ”it is understood that R is an object of C which is equipped with structure maps whichare not written explicitly. We will take here a different approach. We will fix an object B inside our braided fusion category C , and ask what are all the possible Hopf algebrastructures one can give on that object.A Hopf algebra is given by morphisms m : B ⊗ B → B,u : K → B, ∆ : B → B ⊗ B (5.1) ǫ : B → K and S : B → B which satisfy some axioms. We can thus think of a Hopf algebra as a point in the affinespace A N = Hom C ( B ⊗ B, B ) ⊕ Hom C ( K, B ) ⊕ Hom C ( B, B ⊗ B ) ⊕ Hom C ( B, K ) ⊕ Hom C ( B, B ) . (5.2)We will write a point in this spaces as ( m, u, ∆ , ǫ, S ) . Notice that not all points in thisaffine space will define Hopf algebra structure, and not all Hopf algebras will be ccc. Wewrite X B ⊆ A N for the subset of all points ( m, u, ∆ , ǫ, S ) which define a ccc Hopf algebra structure on B .For t = ( m, u, ∆ , ǫ, S ) ∈ X B we write ( B, t ) for the Hopf algebra B with structure givenby t . When we will say “ R is a Hopf algebra in X B ” we will mean that R = ( B, t ) and t ∈ X B . We will also write R ∈ X B .Write Γ B = Aut C ( B ) . When no confusion will arise we will write Γ = Γ B . The group Γ acts on the different direct summands in A n by conjugation. The action of Γ on B ⊗ B is the diagonal one, and on K is the trivial one. This induces a linear action of Γ on A N .We claim the following: Lemma 5.1.
The action of Γ on A N stabilizes the subset X B . Two points ( m , u , ∆ , ǫ , S ) and ( m , u , ∆ , ǫ , S ) in X B define isomorphic Hopf algebras if and only if they lie inthe same Γ -orbit. The stabilizer of ( m, u, ∆ , ǫ, S ) in Γ can be identified with the group ofautomorphisms of the Hopf algebra that this tuple defines.Proof. Write t i = ( m i , u i , ∆ i , ǫ i , S i ) for i = 1 , . The Hopf algebra axioms can be phrasedas equalities between certain linear maps. Associativity of the multiplication, for example,is the equality m (1 B ⊗ m ) α B,B,B = m ( m ⊗ B ) (5.3)as morphisms in Hom C (( B ⊗ B ) ⊗ B, B ) . If γ : B → B is an automorphism in C , and if γ ( t ) = ( t ) then we have that m (1 B ⊗ m ) α B,B,B = γm ( γ − ⊗ γ − )(1 B ⊗ γm ( γ − ⊗ γ − )) α B,B,B = γm (1 B ⊗ m ) α B,B,B ( γ − ⊗ γ − ⊗ γ − ) (5.4)where we used the naturality of α and the definition of the action of γ . On the otherhand a similar calculation gives us m ( m ⊗ B ) = γm ( m ⊗ B )( γ − ⊗ γ − ⊗ γ − ) . (5.5)This shows that if γ ( t ) = t then m is associative if and only if m is associative.Similarly, all the other Hopf algebra axioms are valid for t if and only if they are valid for t . So if γ ( t ) = t then t defines a Hopf algebra if and only if t does. Moreover, t willdefine a ccc Hopf algebra if and only if Ker ( ǫ ) is a nilpotent ideal in B with respect to themultiplication m and Ker ( u ∗ ) is a nilpotent ideal in B ∗ with respect to the multiplication ∆ ∗ . For the same reason as above, this happens if and only if Ker ( ǫ ) is a nilpotent idealwith respect to the multiplication m and Ker ( u ∗ ) is a nilpotent ideal in B ∗ with respectto the multiplication ∆ ∗ . In other words, Γ stabilizes the subset X B of A N . EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 15
We will think of the equation γ ( t ) = t as saying that γ defines an isomorphismbetween ( B, t ) and ( B, t ) . Indeed, γm ( γ − ⊗ γ − ) = m can be rephrased as sayingthat the diagram B ⊗ B γ ⊗ γ / / m (cid:15) (cid:15) B ⊗ B m (cid:15) (cid:15) B γ / / B (5.6)is commutative. Similar statements hold for u, ∆ , ǫ and S . This implies that if γ ( t ) = t then t and t define isomorphic Hopf algebras. On the other hand, if t and t defineisomorphic Hopf algebras on B , take an isomorphism γ : ( B, t ) → ( B, t ) between theseHopf algebras. Then by the same calculations as above we get that γ ( t ) = t . So theorbits of Γ in X B are in one to one correspondence with isomorphism classes of ccc Hopfalgebras which are isomorphic to B as an object of C . Finally, the equality γ ( t ) = t for t ∈ X B just means that γ : ( B, t ) → ( B, t ) is an automorphism. (cid:3) The rest of this section will be devoted to prove the following claim:
Lemma 5.2.
The subset X B is an affine sub-variety of A N . The group Γ is isomorphicto a direct product of general linear groups, and the action of Γ on A N is rational.Proof. The proof of the lemma will be based on analyzing objects and morphisms in thecategory C . We begin by writing B as B = M i B i ⊗ X i (5.7)where X i are representatives of the isomorphism classes of simple objects of C and B i arevector spaces. By Equation 4.2 this already gives us an isomorphism Γ ∼ = Q i GL ( B i ) . Wechoose a basis { e i , . . . e id i } for B i .A Hopf algebra structure on B will be given by maps m : B ⊗ B → B , u : → B , ∆ : B → B ⊗ B , ǫ : B → and S : B → B . By writing tensor products using the spaces V ki,j from Section 4 we see that the morphism m is given by M i,j B i ⊗ B j ⊗ X i ⊗ X j → M k B k ⊗ X k . (5.8)Rewriting the first object using the vector spaces V ki,j gives us M i,j,k B i ⊗ B j ⊗ V ki,j ⊗ X k → M k B k ⊗ X k . (5.9)This means that m is equivalent to a collection of linear maps m ki,j : B i ⊗ B j ⊗ V ki,j → B k . (5.10)Similarly, the morphism u is equivalent to a map u : K → B , (5.11)the morphism ∆ is equivalent to a collection of maps ∆ i,jk : B k → B i ⊗ B j ⊗ V ki,j , (5.12) the morphism ǫ is equivalent to a map ǫ : B → K (5.13)and the antipode S is equivalent to a collection of linear maps S i : B i → B i . (5.14)We rewrite now the affine variety A N as A N = M i,j,k Hom K ( B i ⊗ B j ⊗ V ki,j , B k ) ⊕ Hom K ( K, B ) ⊕ M i,j,k Hom K ( B k , B i ⊗ B j ⊗ V ki,j ) ⊕ Hom K ( B , K ) ⊕ M i Hom K ( B i , B i ) . (5.15)This description of A N shows us that the action of Aut C ( B ) = Q i GL ( B i ) on it isrational. Indeed, it is simply given by pre- and post-composing of linear maps.A choice of bases for B i and for V ki,j for all i, j, k will give us a basis for A N . Thisenables us to describe the structure we have at hand, the tuple t = ( m, u, ∆ , ǫ, S ) , as acollection of numbers, the structure constants of t . Indeed, using the bases for B i and V ki,j we can describe the different structure maps as linear maps between vector spaces withgiven bases, and these are just given by matrices of scalars.We explain now why the subset X B is in fact an affine variety. The idea is to show thatall the Hopf algebra axioms can be expressed using polynomial equations. We will alsoshow that the property of being ccc can be described using polynomial equations.We begin with proving this for the associativity. The proof for the other Hopf algebraaxioms is similar. The associativity axioms says that m ( m ⊗ B ) = m (1 B ⊗ m ) α B,B,B .Writing this using the maps α a,b,ci,j,k : V ai,j ⊗ V ba,k → V bi,c ⊗ V cj,k and m ki,j we get that associativityis equivalent to the commutativity of the diagram B i ⊗ B j ⊗ B k ⊗ V ai,j ⊗ V ba,k P c α a,b,ci,j,k / / m ai,j (cid:15) (cid:15) L c B i ⊗ B j ⊗ B k ⊗ V bi,c ⊗ V cj,km cj,k (cid:15) (cid:15) B a ⊗ B k ⊗ V ba,km ba,k (cid:15) (cid:15) L c B i ⊗ B c ⊗ V bi,cm bi,c (cid:15) (cid:15) B b = / / B b (5.16)where we simplified the morphisms by writing α a,b,ci,j,k instead of B i ⊗ B j ⊗ B k ⊗ α a,b,ci,j,k andsimilarly for the other morphisms. The linear map m ai,j goes from B i ⊗ B j ⊗ V ai,j to B a ,but it is clear how to get a linear map from it as shown in the diagram. If we writenow the linear maps m ki,j and α a,b,ci,j,k in terms of the bases of B i and V ki,j we get a set ofquadratic polynomials on the structure constants whose vanishing is equivalent to theassociativity of m . For similar reasons the other Hopf algebra axioms can be written aswell as polynomials in the structure constants.It is left to show that being ccc is a closed condition for Hopf algebras. For this, write n = P i dim K ( B i ) . We claim the following: Lemma 5.3.
An ideal J of ( B, t ) is nilpotent if and only if J n = 0 . EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 17
Proof.
One direction is clear. For the other direction, if J is nilpotent, the sequence ofideals J ) J ) J · · · is strictly monotonic decreasing until it stabilizes at zero. Thus,the sequence d a := dim Hom C ( ⊕ i X i , J a ) satisfy n > d > d > . . . (5.17)and this sequence of numbers stabilizes at zero. This implies that if J is nilpotent its n -thpower must already be zero. (cid:3) By definition, a Hopf algebra ( B, t ) is ccc if it is connected and coconnected. We willshow that being connected is a closed condition. The fact that coconnectedness is aclosed condition follows from a dual argument. By definition of connectedness, ( B, t ) isconnected if the ideal Ker ( ǫ ) is nilpotent. This is equivalent to Ker ( ǫ ) n = 0 by the abovelemma. It holds that the map P = 1 B − uǫ : B → B (5.18)is a projection on Ker ( ǫ ) (this holds in any Hopf algebra, and follows from the fact that ǫ ◦ u = Id ). The nilpotency of Ker ( ǫ ) is thus equivalent to the fact that the map B ⊗ n P ⊗ n −→ B ⊗ n m n − −→ B is the zero map. But again, this can be written as a polynomial equation using thestructure constants of u , ǫ and m . The subset X B is thus a subvariety of A N , and theaction of Γ on it is rational. The theory of algebraic groups and geometric invariant theorycan thus be applied in our setting. (cid:3) Definition 5.4. If R and R are two Hopf algebras in X B , we say that R is a special-ization of R and write R R if O R ⊆ O R . Remark 5.5.
We have used here a slightly heavy categorical language, in order to con-struct the variety X B in the most general way possible. If, for example, the category C is GG YD there is a way around this: we can fix only the dimension of B as a vector space,and consider also the action and coaction of KG as part of the structure of B , instead ofsomething that is given a-priori, as we have done here. Describing the isomorphism typeof B as an object in C can then be done by declaring what the trace of the operations ofthe elements of D ( G ) , the Drinfeld double of G , on B , should be. The construction hererelies heavily on the fact that the category we are working in is semisimple. Indeed, thesemisimplicity gives us an easy classification of the object of the category and their au-tomorphism groups. See also [AA18] for the study of Nichols algebras in non-semisimplebraided monoidal categories. Remark 5.6.
The Hopf algebras in GG YD which one encounters in the study of pointedHopf algebras are usually graded. We study here ccc Hopf algebras and not gradedalgebras for two reasons. Firstly, being connected and coconnected is a conditions whichcan be described by polynomial equations. If we consider instead the variety of all gradedHopf algebras we will get something which is too rigid, and we will not be able to see thespecializations in the orbits of X B . Secondly, all finite dimensional graded Hopf algebraswith R = are automatically ccc. Indeed, this follows from the fact that if R = L i ≥ R i then the Jacobson radical is L i ≥ R i , and the quotient is isomorphic to . The sameholds for the dual. Remark 5.7.
The map R R ∗ gives us an isomorphism of varieties X B ∼ = X B ∗ whichcommutes with the action of Aut C ( B ) ∼ = Aut C ( B ∗ ) . In particular, if R , R ∈ X B then R R if and only if R ∗ R ∗ in X B ∗ .The following lemma will be useful for the proof of Theorem 1.4. Lemma 5.8.
Assume that R R ′ . Then P ( R ) is isomorphic to a subobject of P ( R ′ ) .Proof. The object of primitive elements in R is the same as the kernel of the map T R = ∆ − u ⊗ R − R ⊗ u : R → R ⊗ R We can write this map as the direct sum of maps B i ⊗ X i → ⊕ j,k B j ⊗ B k ⊗ V ki,j ⊗ X i . Sucha map is thus equivalent to a collection of maps ( T R ) i : B i → B j ⊗ B k ⊗ V ki,j . For everylinear map L and any natural number m the condition that rank ( L ) ≤ m is a Zariskiclosed condition. This implies that for every i it holds that rank (( T R ′ ) i ) ≤ rank (( T R ) i ) and therefore dim( Ker ( T R ′ ) i ) ≥ dim( Ker (( T R ) i ) . This gives us the desired result. (cid:3)
Proof of Theorem 1.4. If R B ( V ) then P ( R ) is isomorphic to a subobject of P ( B ( V )) = V . Similarly R ∗ B ( V ) ∗ ∼ = B ( V ∗ ) by Lemma 2.6 and Remark 5.7 so P ( R ∗ ) is isomorphicto a subobject of P ( B ( V ∗ )) = V ∗ . Since V is simple, V ∗ is simple as well, and it followsthat P ( R ) = 0 or V and P ( R ∗ ) = 0 or V ∗ . The option P ( R ∗ ) = 0 is not possible, since thiswould imply that J/J = 0 where J is the Jacobson radical of R , and R must then be thetrivial algebra. In a similar way, P ( R ) = 0 is impossible. We are left with the situationwhere P ( R ) = V and P ( R ∗ ) = V ∗ . But this already implies that P ( R ) ⊗ P ( R ∗ ) → isa non-degenerate pairing, which implies by Lemma 3.4 that R is a Nichols algebra, asdesired. (cid:3) Filtrations of Hopf algebras
As we have seen in previous sections, understanding specializations is fundamental tounderstand ccc Hopf algebras. Using the Hilbert-Mumford criterion, we will show that if O R ⊆ O R and O R is closed, the specialization R R follows from a filtration on R ,in a way which we will describe now. Let R = ( B, t ) be a ccc Hopf algebra in C . Definition 6.1 (see also [M05]) . A Hopf algebra filtration on R is a chain of C -subobjectsof R · · · R − ⊆ R − ⊆ R ⊆ R ⊆ · · · such that the following properties hold: R i = R for i >> ,R i = 0 for i << , EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 19 ∀ i, j : m ( R i ⊗ R j ) ⊆ R i + j , ∀ k : ∆( R k ) ⊆ X i + j = k R i ⊗ R j , (6.1) ǫ ( R − ) = 0 , Im ( u ) ⊆ R , ∀ i : S ( R i ) ⊆ R i . We will denote the filtration by ( R i ) . Lemma 6.2.
Every ccc Hopf algebra filtration ( R i ) of R determines a Z -graded ccc Hopfalgebra R gr with the same underlying object B , defined as follows:(1) As an object of C , R = ⊕ i ∈ Z R i /R i − . (2) The unit u ∈ R gr is the image of u in R /R − .(3) The counit ǫ : R gr → K is given by ⊕ R i /R i − → R /R − ǫ → K where ǫ here isthe map which is induced from ǫ , since ǫ ( R − ) = 0 .(4) The condition on the antipode S implies that it defines a collection of induced maps S i : R i /R i − → R i /R i − . The antipode S of R gr is P i S i .(5) The condition on m implies that we have an induced map m i,j : R i /R i − ⊗ R j /R j − → R i + j /R i + j − for every i, j ∈ Z . We define m = P i,j m i,j .(6) The condition on ∆ implies that we have an induced map ∆ i,j : R i + j /R i + j − → R i /R i − ⊗ R j /R j − for every i, j ∈ Z . We define ∆ = P i,j ∆ i,j .Proof. All the maps for R gr are well defined due to the condition the structure maps of R satisfy. As we will prove in the next lemma, the condition on the filtration implies that R ∼ = ⊕ i R i /R i − as an object of C . This implies that R gr ∼ = R ∼ = B as objects in C . Thefact that R is a ccc Hopf algebra can be verified directly, or using the next lemma. (cid:3) Lemma 6.3.
Let ( R i ) be a Hopf algebra filtration of the ccc Hopf algebra R . Then O R gr ⊆ O R . In particular, O R gr is also contained in the closed subset X B , and as a result R gr is also a ccc Hopf algebra.Proof. We first claim that we can write R as R = ⊕ i ∈ Z T i such that R i = ⊕ j ≤ i T j where T i are subobjects of R in C . This follows from the semisimplicity of C together withthe conditions on the filtration R i . Indeed, take i << for which R i = 0 . Define T j = 0 for all j ≤ i . Then choose T i + k inductively to be a direct sum complement of T i + k − in R i + k . Notice that this implies that T j ∼ = R j /R j − for every j ∈ Z , and that B ∼ = R ∼ = ⊕ T j ∼ = ⊕ R j /R j − ∼ = R gr as objects of C .Next, we write all the structure maps of R in terms of the direct sum decomposition R = ⊕ i T i . The conditions on the filtration gives us m = X i + j ≥ k m ki,j , ∆ = X i + j ≤ k ∆ i,jk ,u = X i ≤ u i , ǫ = X i ≥ ǫ i (6.2) S = X i ≥ j S ji where m ki,j : T i ⊗ T j → T k , ∆ i,jk : T k → T i ⊗ T j ,u i : K → T i , ǫ i : T i → K, and (6.3) S ji : T i → T j . Using the identification R ∼ = ⊕ i T i ∼ = ⊕ i R i /R i − ∼ = R gr we see that the multiplication in R gr is given by P i,j m i + ji,j , the comultiplication is given by P i,j ∆ i,ji + j , the unit by u = u ,the counit is ǫ , and the antipode is P i S ii . We thus see that in passing from R to R gr we “deleted” all the parts of the structure maps which are of positive degree and stayedwith maps of degree zero (maps of negative degree do not appear here at all), where thedegree of a map T i ⊗ · · · ⊗ T i r → T j ⊗ · · · ⊗ T j m is i + i + · · · + i r − j − j − · · · − j m .We use this idea to prove that R gr ∈ A N is in the closure of the orbit of R . Thiswill already imply that R gr is a ccc Hopf algebra, because O R ⊆ X B , and X B is closed.To prove this, we will use a so called one-parameter subgroup of Γ . That is: a grouphomomorphism φ : G m → Γ =
Aut C ( B ) . We define φ as follows φ ( λ ) = X i ∈ Z λ − i Id T i . (6.4)We claim that φ ( G m )( R ) contains R gr in its closure. Since φ ( G m ) is a subgroup of Γ thiswill be enough. To prove this write φ ( λ )( R ) = ( B, t λ ) = ( B, m λ , u λ , ∆ λ , ǫ λ , S λ ) . (6.5)We get m λ = X i + j ≥ k λ i + j − k m ki,j , ∆ λ = X i + j ≤ k λ k − i − j ∆ i,jk u λ = X i ≤ λ − i u i , ǫ λ = X i ≥ λ i ǫ i (6.6) S λ = X i ≥ j λ i − j S ji . This description shows us that lim λ → φ ( λ )( R ) exists, since in all the above expression λ appears only with non-negative powers. Taking the limit λ → gives us lim λ → φ ( λ )( B, t ) =(
B, m ′ , u ′ , ∆ ′ , ǫ ′ , S ′ ) where m ′ = X i,j m i + ji,j = m, ∆ ′ = X i,j ∆ i,ji + j = ∆ u ′ = u = u, ǫ ′ = ǫ = ǫ (6.7) S ′ = X i S ii = S This shows us that the limit point is exactly the structure constants of R gr . We aredone. (cid:3) EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 21
Remark 6.4. If R ∈ X B and φ : G m → Γ is any one-parameter subgroup for which lim λ → φ ( λ )( R ) exists, then we get a filtration on R by setting T i = Ker ( φ ( λ ) − λ − i ) ⊆ R for a generic λR i = ⊕ j ≤ i T j . The fact that lim λ → φ ( λ )( R ) exists implies, by the same argument as above, that ( R i ) isa Hopf algebra filtration. We thus see that the isomorphism classes of ccc Hopf algebraswhich appear on the boundary of R by the action of a 1-parameter subgroup are exactlythe ccc Hopf algebra arising from R by a Hopf algebra filtration. Moreover, by the Hilbert-Mumford criterion if O R ′ ⊆ O R and O R ′ is closed, then there exists a one-parametersubgroup φ : G m → Γ such that R ′ = lim λ → φ ( λ )( R ) (see Theorem 1.4. in [K78]). Thisimplies that in order to understand specializations, and especially specializations to cccHopf algebras with closed orbits, we need to study Hopf algebra filtrations.We finish this section with two filtrations which are canonically associated to any cccHopf algebra: the radical and the coradical filtration. They are dual to one another, ina way which we shall explain below. As was explained in the introduction, the coradicalfiltration of a Hopf algebra (not necessarily a ccc one) is used in a fundamental way inthe classification of non-semisimple Hopf algebras. The use of both filtrations togetherwill play an important role in studying closure of orbits in this paper.For the radical filtration, let J = Ker ( ǫ ) be the Jacobson radical of R . We claim thefollowing: Lemma 6.5.
The filtration R i = J − i for i < and R i = R for i ≥ is a Hopf algebrafiltration.Proof. Since J is a nilpotent ideal, the condition R i = 0 for i << holds. It is clear thatthe condition R i = R for i >> holds as well (it holds, in fact, for i = 0 ). The fact that J i ⊆ J j when j ≤ i implies that R i · R j ⊆ R i + j . The condition u ∈ R is immediate, andthe condition ǫ ( R − ) = 0 follows from the fact that J = Ker ( ǫ ) . For the condition on ∆ ,notice that ∆( J ) ⊆ J ⊗ R + R ⊗ J . For any i, j ≥ the braiding σ : R ⊗ R → R ⊗ R satisfies σ ( J i ⊗ J j ) = J j ⊗ J i . A direct calculation implies that ∆( J k ) ⊆ P i + j = k J i ⊗ J j ,which is what we wanted to prove. (cid:3) We write R gra for the graded Hopf algebra arising from R via the radical filtration.We recall here also the definition of the dual filtriation, the coradical filtration, fromSection 3: We define R i = 0 for i < , R = Im ( u ) and R n = Ker (cid:16) R ∆ n − → R ⊗ n → ( R/R ) ⊗ n (cid:17) for n > . Again, a direct verification, using the fact that the dual algebra is connected, reveals thefact that this is a filtration of Hopf algebras as well. We write R grc for the graded Hopfalgebra associated to this filtration. We thus see that R R gra and R R grc for everyccc Hopf algebra R in X B . The two filtrations are dual to one another in the followingsense: For i ∈ Z let R i be the i -th level of the coradical filtration, and let X i ⊆ R ∗ bedefined as X i = Ker ( R ∗ → R ∗− i ) . (6.8) Then it holds that ( X i ) is the radical filtration on R ∗ . This duality induces isomorphisms ( R gra ) ∗ ∼ = ( R ∗ ) grc and ( R grc ) ∗ ∼ = ( R ∗ ) gra .7. A proof of Theorem 1.2 and 1.5
In this section we prove that a ccc Hopf algebra in X B has a closed orbit if and onlyif it is isomorphic to a Nichols algebra. Let R be a Hopf algebra in X B . Recall theassociated graded Hopf algebras R gra and R grc from Section 6. We know that R R gra and R R grc . In particular, if the orbit of R is closed we get that R ∼ = R grc and R ∼ = R gra .The following proposition shows that if O R is closed then R is a Nichols algebra. Proposition 7.1.
Assume that the Hopf algebras R gra and R grc are isomorphic as Hopfalgebras. Then R is a Nichols algebra.Proof. Assume that the two Hopf algebras are isomorphic. Write R gra = L ni =0 A i where A i = J i /J i +1 (for the sake of simplicity, we use here positive grdaing instead of thenegative grading from Section 6 for the radical filtrations). Then the algebra R gra isgenerated by the elements in A . These elements are also primitive, since ∆( A ) ⊆ A ⊗ A + A ⊗ A and A ∼ = R/J ∼ = . This implies, in particular, that R gra is generatedby primitive elements. The algebra R grc = ⊕ mi =0 C i has all primitives in degree 1. Sinceit is isomorphic to the algebra R gra , it is also generated by these elements. This impliesthat R grc is a graded Hopf algebra which is generated in degree 1 and has all its primitiveelements in degree 1. By Definition 2.4 this implies that R grc is a Nichols algebra. Itfollows that R is generated by primitive elements. By a dual argument, and by using thefact that ( R gra ) ∗ ∼ = ( R ∗ ) grc and ( R grc ) ∗ ∼ = ( R ∗ ) gra we get that R ∗ is also generated by itsprimitive elements. Lemma 3.2 gives us the desired result. (cid:3) This finishes the proof that if O R is closed then R is a Nichols algebra, because the factthat R R grc and R R gra together with the closure of O R implies that R grc ∼ = R ∼ = R gra . Next, we will show that if R is a Nichols algebra then O R is closed. Assume thatthis is not the case and let R = B ( V ) be a Nichols algebra with a non-closed orbit. Theclosure O R is the union of the orbit of R with orbits of smaller dimension. An orbit ofminimal dimension in O R is closed. It follows that R R ′ for some R ′ ∈ X B with O R ′ closed. But we already know that this implies that R ′ is a Nichols algebra.Write R ′ = B ( V ′ ) . Then we have B ( V ) B ( V ) ′ . By 5.8 we know that this impliesthat P ( B ( V )) = V is isomorphic to a subobject of P ( B ( V ′ )) = V ′ . Since R ≇ R ′ theobject V must be isomorphic to a proper subobject of V ′ . Write V ′ = V ⊕ V ′′ . Thesplit inclusion of objects in C , V → V ′ → V , induces a split inclusion of Nichols algebrasB ( V ) → B ( V ′ ) → B ( V ) . This implies that B ( V ) ≇ B ( V ′ ) because then B ( V ′ ) properlycontains B ( V ) , and this contradicts the fact that R and R ′ are isomorphic to the sameobject of C . This finishes the proof of Theorem 1.2The proof of Theorem 1.2 gives us a description of the irreducible components of X B : Theorem 7.2.
Let B ∈ C . For every V ∈ C such that B ( V ) ∈ X B write X V = { R | R ∈ X B and R B ( V ) } . Then the subsets X V are stable under the action of Γ and areexactly the connected components of X B . EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 23
Proof.
The fact that X V is stable under Γ is immediate. Notice that for dimension consid-erations the number of objects V such that B ( V ) ∈ X B is finite. We denote these objectsby V , V , . . . V m . We claim now that the dimension of the invariant subalgebra K [ X B ] Γ is finite. Indeed, if R B ( V ) then continuity considerations imply that f ( R ) = f ( B ( V )) for every f ∈ K [ X B ] Γ . This implies that φ : K [ X B ] Γ → K m f ( f ( B ( V )) , . . . , f ( B ( V m ))) is an injective algebra homomorphism. Since B ( V i ) ≇ B ( V j ) when i = j and O B ( V i ) and O B ( V j ) are closed and disjoint Proposition 2.8 implies that for every i = j there is afunction f ij ∈ K [ X B ] Γ such that f ij ( B ( V i )) = f ij ( B ( V j )) . Since φ is an algebra map thisimplies that φ is also surjective and thus an isomorphism. We thus see that all the X V i areclosed, since X V i is the zero set of the polynomial − φ − ( e i ) (where { e i } is the standardbasis of K m ).We next claim that X V i is connected for every i . Indeed, assume that X V i = Y ⊔ Y with Y and Y closed and nonempty. Take y ∈ Y . Then O y = Γ · y is connected,contained in X V i and intersects Y , so O y ⊆ Y . But then O B ( V i ) ⊆ O y ⊆ Y . By a similarargument O B ( V i ) ⊆ Y and this is a contradiction. (cid:3) Due to the last theorem, we can focus our attention on the different subvarieties X V .These subvarieties are stable under the action of Γ . The conditions in Theorem 1.6 thenboil down to the statement that if B ( V ) is finite dimensional, then the variety X V has asingle orbit under the action of Γ . We finish with a proof of Theorem 1.5: Proof of Theorem 1.5.
Assume that B ( V ) is not rigid. In other words, assume that thereare non-closed orbits in X V . Take a non-closed orbit O R of minimal dimension in X V .We will prove that such an orbit is the orbit of a pre-Nichols algebra or a post-Nicholsalgebra. For this consider the Hopf algebras R gra and R grc . If both these Hopf algebrasare isomorphic to R , then R ∼ = R gra ∼ = R grc , which implies that R itself is a Nichols algebraby Proposition 7.1. This is a contradiction. If both these algebras are non-isomorphic to R , then from the fact that R grc , R gra ∈ O R it follows that the dimensions of the orbits O R gra and O R grc are smaller than the dimension of O R . By the minimality condition on R , this implies that R grc ∼ = R gra ∼ = B ( V ) . But By Proposition 7.1 again, this implies that R itself is a Nichols algebra, which leads again to a contradiction.We thus see that either R ∼ = R gra and R ≇ R grc , or R ≇ R gra and R ∼ = R grc . Assumefirst that R ∼ = R gra and R ≇ R grc . Since R ∼ = R gra , R has a grading R = ⊕ R i such thatthe Jacobson radical J satisfy J i = ⊕ j ≥ i R j for every i ≥ . The grading implies that R ,which generate R as an algebra, is a primitive object. But this already implies that R isa pre-Nichols algebra, as it lies between T ( R ) and B ( R ) . Notice that it is impossiblethat R ∼ = V . Indeed, if this was the case then from dimension considerations the factthat R projects onto B ( R ) would imply that R ∼ = B ( R ) = B ( V ) , contradicting the factthat the orbit of R is not closed. By Lemma 5.8 it follows that R is isomorphic to asubobject of V . We thus see that it must be a proper subobject. This shows that if R ∼ = R gra then R is a pre-Nichols algebra. If R ∼ = R grc then byduality of the radical and coradical filtrations we get that R ∗ is a pre-Nichols algebra.This finishes the proof of Theorem 1.5 and also of 1.6 (cid:3) Different notions of rigidity
In this paper we call a ccc Hopf algebra R rigid if any Hopf algebra which speciailzesto it is isomorphic to it. There are other notions of rigidity, using deformations by aone-parameter family. We will explain here the relations between them.In [AKM15] and [DCY07] a deformation of a graded bialgebra (not necessarily a finitedimensional one) B = ⊕ i ≥ B ( i ) by a parameter λ is defined as a pair ( m λ , ∆ λ ) such that m λ = ∞ X i =0 m ( i ) λ i and ∆ λ = ∞ X i =0 ∆ ( i ) λ i (8.1)where m ( i ) , ∆ ( i ) are maps of degree − i , m (0) = m and ∆ (0) = ∆ , and ( m λ , ∆ λ , u, ǫ ) definesa bialgebra structure on B for every λ . (Bialgebra deformations of Hopf algebras areautomatically Hopf algebras as well. Due to the uniqueness of the antipode, we do notneed to consider it as part of the deformation data). It is shown that this is the same asa filtered Hopf algebra U which satisfies grU ∼ = B . In [AKM15] and [DCY07] a gradedHopf algebra B is called rigid if it has no non-trivial deformations. We will call it heredeformation rigid. We claim the following: Lemma 8.1.
A finite dimensional Nichols algebra B ( V ) is rigid with respect to Definition1.3 if and only if both B ( V ) and B ( V ∗ ) are deformation rigid.Proof. Remark 5.7 implies that B ( V ) is rigid if and only if B ( V ∗ ) is rigid (with respect toDefinition 1.3). To prove the first direction it will thus be enough to show that if B ( V ) isrigid with respect to Definition 1.3 then it is deformation rigid.Assume that ( m λ , ∆ λ ) is a one-parameter deformation of B ( V ) . Using the gradingB ( V ) = ⊕ B ( V ) i we have a one-parameter family φ : G m → Aut C ( B ) which sends λ tothe automorphism which acts on B ( V ) i by the scalar λ − i . We then have that ( m λ , ∆ λ ) = φ ( λ ) · ( m , ∆ ) and ( m , ∆ ) = lim λ → λ · ( m , ∆ ) . Since B ( V ) is rigid, any algebra specializing to it is isomorphic to it. So we get a Hopfalgebra isomorphism Ψ : B ( V ) ∼ = ( B, m , ∆ , u, ǫ, S ) where S is the uniquely definedantipode.Next, we claim that B ( V ) consists of primitive elements with respect to the comul-tiplication ∆ (and therefore, with respect to ∆ λ for every λ ∈ K × as well). Indeed,by grading consideration we have that ∆ | B ( V ) = ∆(0) + ∆ (1) , where ∆ (0) = ∆ . Themap ∆ (1) : B ( V ) → B ( V ) ⊗ B ( V ) = ⊗ = must be zero since otherwise this willcontradict the fact that ǫ ( B ( V ) ) = 0 . This implies that the isomorphism Ψ will mapB ( V ) to B ( V ) and B ( V ) to B ( V ) . Without loss of generality we can assume that Ψ | B ( V ) = Id B ( V ) (recall that both B ( V ) and the deformed algebra have the same under-lying object B ). Since B ( V ) n is the image of B ( V ) ⊗ n → B ( V ) with respect to the iteratedmultiplication m , and since L i ≤ n B ( V ) i contains the image of B ( V ) ⊗ n → B ( V ) under theiterated multiplication m , we get that Ψ preserves the filtration B (0) ⊆ B (1) ⊆ · · · of B EOMETRIC PERSPECTIVE ON NICHOLS ALGEBRAS 25 given by B ( n ) = L i ≤ n B ( V ) i . But this already implies that the deformation ( m λ , ∆ λ ) istrivial with respect to the definition in Section 2.3 of [AKM15].In the other direction, assume that B ( V ) and B ( V ∗ ) are deformation rigid. By Theorem1.6 we know that if B ( V ) is not rigid then there is a pre-Nichols Hopf algebra R such that R B ( V ) or R B ( V ∗ ) . Since both B ( V ) and B ( V ∗ ) are deformation rigid, we canassume without loss of generality that the first case holds. The algebra R is generated byits primitive elements, and therefore admits a filtration R (0) ⊆ R (1) ⊆ · · · where R ( n ) = P ni =0 P ( R ) i . This can easily be seen to be a Hopf algebra filtration of R .The associated graded object is then B ( V ) with its usual grading. Since we assumed thatB ( V ) is deformation rigid we get that R ∼ = B ( V ) , which is a contradiction. (cid:3) We recall here Theorem 6.2 from [AKM15]. This result relies on the classification resultfrom [Ang13].
Theorem 8.2.
Assume that B ( V ) is finite dimensional and that the braiding on V is ofdiagonal type. Then B ( V ) is deformation rigid. This theorem, combined with Theorem 1.6 gives a new proof for the generation ofpointed Hopf algebras with abelian groups of group-like elements by group-like elementsand skew-primitives. Indeed, the above theorem implies that all the finite dimensionalNichols algebra in GG YD for G abelian are rigid, and therefore by Theorem 1.6 all finitedimensional ccc Hopf algebras in GG YD are Nichols algebras, and every Hopf algebras H such that H = KG with G abelian is generated by group-like elements and skew-primitives. Acknowledgments
I would like to thank Nicolás Andruskiewitsch, Iván Angiono and Istvan Heckenbergerfor their useful comments.
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