On defining ideals and differential algebras of Nichols algebras
aa r X i v : . [ m a t h . QA ] S e p ON DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OFNICHOLS ALGEBRAS
XIN FANG
Abstract.
This paper is devoted to understanding the defining ideal of a Nicholsalgebra from the decomposition of specific elements in the group algebra of braidgroups. A family of primitive elements are found and algorithms are proposed. Toprove the main result, the differential algebra of a Nichols algebra is constructed.Moreover, another point of view on Serre relations is provided. Introduction
Nichols algebras, as their name says, are constructed by W.D. Nichols in [18]with the name "bialgebra of type one" for classifying finite dimensional gradedHopf algebras generated by elements in degree and . No much attention waspaid to his work at that time until quantized enveloping algebras are constructedby Drinfel’d and Jimbo in the middle of eighties.The construction of Nichols, after having slept for about 15 years, is highlightedby M. Rosso in his article [21] to give a functorial and coordinate free construction ofquantized enveloping algebras, which meanwhile gives the motivation and anotherpoint of view to researches on pointed Hopf algebras.The construction in [21] starts with a Hopf algebra H and an H -Hopf bimodule M . Then the set of right coinvariants M R admits a braiding σ M : M R ⊗ M R → M R ⊗ M R . Once the usual flip is replaced by this braiding, the classical constructionof shuffle algebra will give a new Hopf algebra whose structure is controlled bythis braiding, and is called quantum shuffle algebra. This construction gives manyinteresting examples: the positive part of a quantized enveloping algebra associatedto a symmetrizable Cartan matrix can be found as a Hopf sub-algebra generatedby H and M R in the quantum shuffle algebra with some particular Hopf algebra H and H -Hopf bimodule M .A well-known result affirms that there is an equivalence of braided categorybetween the category of Hopf bimodules over H and that of left H -Yetter-Drinfel’dmodules, given by sending a Hopf bimodule M to the set of its right coinvariants M R ; so it is possible to work in the context of Yetter-Drinfel’d modules at thevery beginning. This gives a translation of language between Nichols-Rosso andAndruskiewitsch-Schneider.The dual construction of quantum shuffle algebra is easier to understand: it isthe braided tensor algebra T ( V ) for V = M R and the Nichols algebra N ( V ) isthe quotient of T ( V ) by some Hopf ideal I ( V ) . As an example, it gives the strictpositive part of quantized enveloping algebras when H and V are properly chosen.
1N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 2
In the latter case, as a quotient of the braided Hopf algebra T ( V ) , N ( V ) can beviewed as imposing some relations in T ( V ) which have good behavior under thecoproduct. It is natural to ask for the structure of such ideals, but unfortunately,this Hopf ideal is defined as the maximal coideal contained in the set of elementsin T ( V ) of degree no less than and it is very difficult to read out these relationsdirectly from such an abstract definition.Similar problems arise in some other places in mathematics. For example, theGabber-Kac theorem in the theory of Kac-Moody Lie algebras is of the same philos-ophy: starting with a symmetrizable Cartan matrix and some Chevalley generators,it is possible to construct a Lie algebra with Chevalley relations; but to get a Kac-Moody Lie algebra, one is forced to do the quotient by an ideal with some maximalproperties and of course with mysteries. It is in Gabber-Kac [7] that they provedthat this ideal is generated by Serre relations, which completes the whole story. Itshould be remarked that this result is not simple at all: the proof clarifies somestructures and uses several tools in Kac-Moody Lie algebras, such as generalizedCasimir elements and Verma modules.As we know, for Nichols algebras, the problem of deciding generating relationsin the ideal I ( V ) is still open and the best general result can be found in theliterature is due to M. Rosso [21], P. Schauenburg [22] and others, which affirmsthat elements of degree n in I ( V ) are those annihilated by the symmetrizationoperator S n . However, it is still difficult to make these relations explicit because ofthe appearance of large amount of equations when a basis is chosen.The main objective of this paper is to give some restrictions on elements in I ( V ) to obtain a number of important relations imposed, and we conjecture that theserelations generate I ( V ) as an ideal.To be more precise, the restriction is at first given by some operators P n , calledDynkin operators, on T ( V ) ; these operators can be viewed as analogues of theDynkin projection operators in the characterization of free Lie algebras. Our firstrestriction is passing from ker( S n ) to ker( S n ) ∩ Im ( P n ) : though the latter is some-how subtle at a first glimpse, in some important cases, they will generate the Hopfideal I ( V ) ; in general, if a conjecture of Andruskiewitsch-Schneider is true, thestatement above holds for any Nichols algebras. This is proved by showing that allprimitive elements of degree n are eigenvectors of P n with eigenvalue n .Another restriction is given by concentrating on some levels in T ( V ) having theirorigin in the decomposition of the element S n in the group algebra C [ B n ] , where B n is the braid group on n strands. The main idea here is building a bridge toconnect some solutions of equation S n x = 0 in V ⊗ n with the invariant space forthe central element θ n of B n , which are much easier to understand and compute.Moreover, it throws some light on understanding the structure of Nichols algebrasfrom the representation theory of braid groups, though the latter is difficult indeed.When constructing this bridge, we captured the appearance of the Dynkin opera-tor P n as an important ingredient. Moreover, it is a central tool when constructingsolutions of S n x = 0 from solutions of the equation θ n x = x . As an example, weobtain quantized Serre relations from a pedestrian point of view: that is to say, the N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 3 machinery will tell us what these relations are by assuming almost no knowledgeon Lie theory and quantized enveloping algebra.A natural question is posed when observing the calculation in the exterior alge-bra and quantized enveloping algebras: whether elements with levels in T ( V ) areprimitive?The second part of this paper is devoted to give a positive answer in the generalcase: that is to say, for any braiding coming from a Yetter-Drinfel’d structure.The proof is based on the construction of the differential algebra of a Nicholsalgebra, which can be viewed as a generalization of the construction of the quantizedWeyl algebra over a quantized enveloping algebra given in [6]. This construction,when restricted to Nichols algebras of diagonal type, gives a generalization for theconstruction of skew-derivations dates back to N.D. Nichols [18], Section 3.3 andexplicitly used in M. Graña [8] and I. Heckenberger [9], [10]. The advantage of ourapproach is: at the very beginning, we never make hypotheses on the type of thebraiding, so this is a general construction shared by all kinds of Nichols algebrascoming from a Yetter-Drinfel’d module.Once restricting ourselves to the diagonal case, with the help of these differen-tial operators, we proved the classical Taylor lemma, which generalizes a result inHeckenberger [10]. Moreover, when the derivation is given by a primitive elementof degree , a decomposition theorem of T ( V ) as a braided algebra is obtained,which can be viewed as a generalization of a result in classical Weyl algebra givenby A. Joseph [12] in solving the Gelfan’d-Kirillov conjecture.At last, we show that elements with levels are all primitive with the help of thesedifferential operators.The organization of this paper is as follows:In Section 2, some notions in Hopf algebras are recalled and notations are intro-duced. Section 3 is devoted to defining Dynkin operators and proving the "con-volution invariance" of these operators. In Section 4, we are concerned with thedecomposition of specific elements in the group algebra of braid groups, which is analgebraic preparation for solving equation S n x = 0 . The construction of the bridgementioned above is given in Section 5 and some properties of ker( S n ) ∩ Im ( P n ) areobtained. Section 6 contains examples in the diagonal case; a concrete calculationof quantized Serre relations in the case U q ( sl ) is among examples. In Section 7, thedifferential algebra of a Nichols algebra is constructed and then the Taylor Lemmaand a decomposition theorem are proved as an application in Section 8. Finally,the main theorem on primitive elements is demonstrated with the help of the dif-ferential algebra in Section 9. Acknowledgements.
I am grateful to my advisor Marc Rosso for the discus-sion and treasurable remarks on this work. I would like to thank Victoria Lebedfor her comments on an early version of this paper. I want to thank the referee forpreventing me from mistakes and simplifying the proof of Theorem 1.
N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 4 Recollections on Hopf algebras
From now on, suppose that we are working in the complex field C . All algebrasand modules concerned are over C . Results in this section will hold for any fieldwith characteristic . All tensor products are over C if not specified otherwise.This section is devoted to giving a recollection on some constructions in Hopfalgebras and fixing notations.2.1. Yetter-Drinfel’d modules.
Let H be a Hopf algebra. A vector space V iscalled a (left) H -Yetter-Drinfel’d module if:(1) It is simultaneously an H -module and an H -comodule;(2) These two structures satisfy the Yetter-Drinfel’d compatibility condition:for any h ∈ H and v ∈ V , X h (1) v ( − ⊗ h (2) .v (0) = X ( h (1) .v ) ( − h (2) ⊗ ( h (1) .v ) (0) , where ∆( h ) = P h (1) ⊗ h (2) and ρ ( v ) = P v ( − ⊗ v (0) are Sweedler notationsfor coproduct and comodule structure map.Morphisms between two H -Yetter-Drinfel’d modules are linear maps preserving H -module and H -comodule structures.We denote the category of H -Yetter-Drinfel’d modules by HH Y D ; it is a tensorcategory.The advantage of working in the category of Yetter-Drinfel’d module is: for
V, W ∈ HH Y D , there exists a braiding σ V,W : V ⊗ W → W ⊗ V , given by σ V,W ( v ⊗ w ) = P v ( − .w ⊗ v (0) . This braiding gives HH Y D a braided tensor category structure.Let A and B be two H -module algebras in HH Y D . Then the vector space A ⊗ B admits an algebra structure with the following multiplication map: A ⊗ B ⊗ A ⊗ B id ⊗ σ B,A ⊗ id −−−−−−−→ A ⊗ A ⊗ B ⊗ B m A ⊗ m B −−−−−→ A ⊗ B, where m A and m B are multiplications in A and B , respectively. We let A ⊗ B denotethis algebra.2.2. Braided Hopf algebras in HH Y D .Definition 1 ([3], Section 1.3) . A braided Hopf algebra in the category HH Y D is acollection ( A, m, η, ∆ , ε, S ) such that: (1) ( A, m, η ) is an algebra in HH Y D ; ( A, ∆ , ε ) is a coalgebra in HH Y D . That is tosay, m, η, ∆ , ε are morphisms in HH Y D ; (2) ∆ : A → A ⊗ A is an algebra morphism; (3) ε : A → C is an algebra morphism; (4) S is the convolution inverse of id A ∈ End ( A ) . The most important example of a braided Hopf algebra is the braided tensorHopf algebra defined as follows.
N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 5
Example 1 ([3]) . Let V ∈ HH Y D be an H -Yetter-Drinfel’d module. There exists abraided Hopf algebra structure on the tensor algebra T ( V ) = ∞ M n =0 V ⊗ n . (1) The multiplication is the original one on T ( V ) given by the concatenation. (2) The coalgebra structure is defined on V by: for any v ∈ V , ∆( v ) = v ⊗ ⊗ v , ε ( v ) = 0 . Then it can be extended to the whole T ( V ) by the universalproperty of T ( V ) as an algebra. Nichols algebras.
Let V ∈ HH Y D be a finite dimensional C -vector space and T ( V ) be the braided tensor Hopf algebra over V as defined in Example 1; it is N -graded.We will recall briefly the definition and the explicit construction of Nichols al-gebras, which dates back to [18] and is given in [3]. Another definition in a dualpoint of view is given in [21] under the name quantum shuffle algebra and is de-noted by S σ ( V ) . The difference between them is: the construction in [21] is inthe graded dual of T ( V ) , so instead of being a quotient object, it will be a sub-object in the graded dual. But they are isomorphic as braided Hopf algebra up toa symmetrization morphism. Definition 2 ([3]) . A graded braided Hopf algebra R = L ∞ n =0 R ( n ) is called aNichols algebra of V if (1) R (0) ∼ = C , R (1) ∼ = V ∈ HH Y D ; (2) R is generated as an algebra by R (1) ; (3) R (1) is the set of all primitive elements in R .We let N ( V ) denote this braided Hopf algebra. Remark 1.
It is conjectured by Andruskiewitsch and Schneider in [3] that when R is finite dimensional, (3) implies (2). There is a construction of N ( V ) from T ( V ) as shown in [3]: let T ≥ ( V ) = M n ≥ V ⊗ n and I ( V ) be the maximal coideal of T ( V ) contained in T ≥ ( V ) . Then I ( V ) is alsoa two-sided ideal; the Nichols algebra N ( V ) of V can be constructed as T ( V ) / I ( V ) .We let S denote the convolution inverse of id : N ( V ) → N ( V ) .For k ∈ N , let N ( V ) k denote the subspace of degree k elements in N ( V ) . Fromdefinition, N ( V ) = C and N ( V ) = V is the set of all primitive elements in N ( V ) .2.4. Nichols algebras of diagonal type.
In this subsection, we recall a partic-ular type of Nichols algebra, which will be a good source of examples in our laterdiscussions. A concrete approach can be found in [3].Let G be an abelian group and H = C [ G ] be its group algebra: it is a commuta-tive and cocommutative Hopf algebra. We let b G denote the character group of G . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 6
Let V ∈ HH Y D be a finite dimensional H -Yetter-Drinfel’d module and dimV = n .Let T ( V ) denote the braided tensor Hopf algebra and N ( V ) denote the associatedNichols algebra.As shown in [18] or Remark 1.5 in [3], the category HH Y D is made of a G -gradedvector space V = L g ∈ G V g such that for any h ∈ G and v ∈ V g , h.v ∈ V g . Thecomodule structure is given by: for V in HH Y D and v ∈ V g in the decompositionabove, the comodule structure map δ : V → H ⊗ V is δ ( v ) = g ⊗ v . Definition 3.
Let V ∈ HH Y D be a finite dimensional H -Yetter-Drinfel’d module. V is called of diagonal type if there exists a basis { v , · · · , v n } of V , g , · · · , g n ∈ G and χ , · · · , χ n ∈ b G such that for any g ∈ G and v i ∈ V g i , g.v i = χ i ( g ) v i . Sometimes, we call T ( V ) of diagonal type if V is of diagonal type.In this case, the braiding in HH Y D is given by: for
V, W ∈ HH Y D , σ V,W : V ⊗ W → W ⊗ V, σ
V,W ( v ⊗ w ) = ( g.w ) ⊗ v for any g ∈ G , v ∈ V g and w ∈ W .In particular, if we choose V = W and v , · · · , v n be a basis of V as in thedefinition above, the braiding, when acting on basis elements, is given by: for ≤ i, j ≤ n , σ V,V ( v i ⊗ v j ) = χ j ( g i ) v j ⊗ v i . So σ V,V is completely determined by the matrix ( χ j ( g i )) ≤ i,j ≤ n . We denote q ij = χ j ( g i ) and call ( q ij ) ≤ i,j ≤ n the braiding matrix associated to σ V,V .It is convenient to define a bicharacter χ over G when G = Z n to rewrite thebraiding above. Definition 4.
A bicharacter on an abelian group A is a map χ : A × A → C ∗ suchthat: χ ( a + b, c ) = χ ( a, c ) χ ( b, c ) , χ ( a, b + c ) = χ ( a, b ) χ ( a, c ) , for any a, b, c ∈ A . Suppose that G = Z n and V ∈ HH Y D . Then V , T ( V ) and N ( V ) are all Z n -graded.Let v , · · · , v n be a basis of V as in Definition 3, α , · · · , α n be a free basis of Z n and deg ( v i ) = α i be their grade degrees in Z n .If this is the case, a bicharacter over Z n can be defined using the braiding matrix:for any ≤ i, j ≤ n , χ : Z n × Z n → C ∗ is determined by χ ( α i , α j ) = q ij .2.5. Radford’s biproduct.
Let A ∈ HH Y D be a braided Hopf algebra. Then A ⊗ H admits a Hopf algebra structure from a construction in Radford [19], which is calledthe biproduct of A and H .These structures are defined by:(1) The multiplication is given by the crossed product: for a, a ′ ∈ A , h, h ′ ∈ H , ( a ⊗ h )( a ′ ⊗ h ′ ) = X a ( h (1) .a ′ ) ⊗ h (2) h ′ ; N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 7 (2) The comultiplication is given by: for a ∈ A and h ∈ H , ∆( a ⊗ h ) = X ( a (1) ⊗ ( a (2) ) ( − h (1) ) ⊗ (( a (2) ) (0) ⊗ h (2) ); (3) The antipode is completely determined by: for a ∈ A and h ∈ H , S ( a ⊗ h ) = X (1 ⊗ S H ( h ) S H ( a ( − ))( S A ( a (0) ) ⊗ . Proposition 1 (Radford) . With structures defined above, A ⊗ H is a Hopf algebra.We let A♯H denote it.
Quantum doubles and Heisenberg doubles.
We recall first the definitionof a generalized Hopf pairing, which gives duality between Hopf algebras.
Definition 5 ([14]) . Let A and B be two Hopf algebras with invertible antipodes.A generalized Hopf pairing between A and B is a bilinear form ϕ : A × B → C suchthat: (1) For any a ∈ A , b, b ′ ∈ B , ϕ ( a, bb ′ ) = P ϕ ( a (1) , b ) ϕ ( a (2) , b ′ ) ; (2) For any a, a ′ ∈ A , b ∈ B , ϕ ( aa ′ , b ) = P ϕ ( a, b (2) ) ϕ ( a ′ , b (1) ) ; (3) For any a ∈ A , b ∈ B , ϕ ( a,
1) = ε ( a ) , ϕ (1 , b ) = ε ( b ) . Remark 2.
From the uniqueness of the antipode and conditions (1)-(3) above, wehave: for any a ∈ A , b ∈ B , ϕ ( S ( a ) , b ) = ϕ ( a, S − ( b )) . Starting with a generalized Hopf pairing between two Hopf algebras, we can de-fine the quantum double and the Heisenberg double of them, which will be essentialin our later construction of differential algebras of Nichols algebras.
Definition 6 ([14]) . Let A , B be two Hopf algebras with invertible antipodes and ϕ be a generalized Hopf pairing between them. The quantum double D ϕ ( A, B ) isdefined by: (1) As a vector space, it is A ⊗ B ; (2) As a coalgebra, it is the tensor product of coalgebras A and B ; (3) As an algebra, the multiplication is given by: ( a ⊗ b )( a ′ ⊗ b ′ ) = X ϕ ( S − ( a ′ (1) ) , b (1) ) ϕ ( a ′ (3) , b (3) ) aa ′ (2) ⊗ b (2) b ′ . Then we construct the Heisenberg double of A and B : it is a crossed product ofthem where the module algebra type action of A on B is given by the Hopf pairing. Definition 7 ([16]) . The Heisenberg double H ϕ ( A, B ) of A and B is an algebradefined as follows: (1) As a vector space, it is B ⊗ A and we let b♯a denote a pure tensor; (2) The product is given by: for a, a ′ ∈ A , b, b ′ ∈ B , ( b♯a )( b ′ ♯a ′ ) = X ϕ ( a (1) , b ′ (1) ) bb ′ (2) ♯a (2) a ′ . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 8 Dynkin operators and their properties
In this section, we will define Dynkin operators in the group algebras of braidgroups. The definition of these operators is motivated by the iterated brackets inLie algebras which are used by Dynkin in the proof of the Dynkin-Wever-Spechettheorem for characterizing elements in free Lie algebras (for example, see [20]).As will be shown in this section, Dynkin operators have good properties underthe convolution product, which generalizes the corresponding result in free Liealgebras. This will be used to detect primitive elements later.3.1.
Definition of Dynkin operators.
We suppose that n ≥ is an integer.Let S n denote the symmetric group: it acts on an alphabet with n letters bypermuting their positions. It can be generated by the set of transpositions { s i =( i, i + 1) | ≤ i ≤ n − } .Let B n be the braid group of n strands, it is defined by generators σ i for ≤ i ≤ n − and relations: σ i σ j = σ j σ i , for | i − j | ≥ σ i σ i +1 σ i = σ i +1 σ i σ i +1 , for ≤ i ≤ n − . Let π : B n → S n be the canonical surjection which maps σ i ∈ B n to s i = ( i, i +1) ∈ S n .We consider the group S ± n = Z / Z × S n , where Z / Z = {± } is the signature.We are going to define a subset P i,j ⊂ S ± n by induction on | i − j | for ≤ i ≤ j ≤ n .We omit the signature . Define P i,i = { (1) } , P i,i +1 = { (1) , − ( i, i + 1) } , and P i,j = P i +1 ,j ∪ ( P i,j − ◦ − ( i, j, j − , · · · , i + 1)) , where ◦ is the product in C [ S n ] .Moreover, we define P i,j = X ( ε,ω ) ∈P i,j εω ∈ C [ S n ] . Let σ ∈ S n and σ = s i · · · s i r be a reduced expression of σ . It is possible todefine a corresponding lifted element T σ = σ i · · · σ i r ∈ B n . This gives a linear map T : C [ S n ] → C [ B n ] called Matsumoto section. For ≤ k ≤ n , let S k,n − k ⊂ S n denote the set of ( k, n − k ) -shuffles in S n defined by: S k,n − k = { σ ∈ S n | σ − (1) < · · · < σ − ( k ) , σ − ( k + 1) < · · · < σ − ( n ) } . Example 2.
We explain the definition of these elements P i,j in an example for S : P , = { (1) , − (12) } , P , = P , ∪ ( P , ◦ − (132)) = { (1) , − (23) , − (132) , (13) } , P , = { (1) , − (34) , − (243) , (24) } , P , = P , ∪ ( P , ◦ − (1432))= { (1) , − (34) , − (243) , (24) , − (1432) , (142) , (1423) , − (14)(23) } . These elements P i,j and P i,j come from iterated brackets: [ a, [ b, [ c, d ]]] = abcd − abdc − acdb + adcb − bcda + bdca + cdba − dcba. N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 9
When S ± acts on letters abcd by permuting their position and then multiplying bythe signature, an easy computation gives: [ a, [ b, [ c, d ]]] = T P , ( abcd ) . Definition 8.
We call these P i,j Dynkin operators in C [ S n ] and correspondingelements T P i,j Dynkin operators in C [ B n ] . For σ ∈ S n , let l ( σ ) denote the length of σ . It is exactly the number of generatorsappearing in any reduced expression of σ . Remark 3.
In general, the Matsumoto section T : S n → B n is not a grouphomomorphism, but we have the following property: for w, w ′ ∈ S n , if l ( ww ′ ) = l ( w ) + l ( w ′ ) , then T w T w ′ = T ww ′ . Lemma 1.
Let w ∈ P ,k and σ ∈ S k,n − k . Then T wσ = T w T σ .Proof. Recall that the length of an element in S n equals to the number of inversionsof its action on { , · · · , n } . As w permutes only the first k positions, and the ( k, n − k ) -shuffle σ preserves the order of first k positions, the number of inversionsof wσ is the sum of those for w and σ , which means that l ( wσ ) = l ( w ) + l ( σ ) andthen the lemma comes from the remark above. (cid:3) The following lemma is helpful for the understanding of the operator P i,j and forour further applications. Lemma 2.
For n ≥ and ≤ i < j ≤ n , the following identity holds in C [ B n ] : T P i,j = (1 − σ j − σ j − · · · σ i )(1 − σ j − σ j − · · · σ i +1 ) · · · (1 − σ j − ) . Proof.
It suffices to show it for i = 1 and j = n . We prove it by induction on n .The case n = 2 is clear.Suppose that the lemma holds for n − . From the definition of P ,n , P ,n = P ,n − P ,n − ◦ (1 , n, · · · , , so P ,n = P ,n − (1 , n, · · · , ◦ P ,n and then T P ,n = (1 − σ n − σ n − · · · σ ) T P ,n . From the induction hypothesis, T P ,n = (1 − σ n − σ n − · · · σ ) · · · (1 − σ n − σ n − )(1 − σ n − ) , which finishes the proof. (cid:3) Properties of Dynkin operators.
We treat T ( V ) as a braided Hopf algebraas in Section 2.At first, we define Dynkin operators on T ( V ) . Definition 9.
We define a graded endomorphism Φ ∈ L ∞ n =0 End ( V ⊗ n ) by: Φ(1) =0 and for x ∈ V ⊗ n with n ≥ , Φ( x ) = T P ,n ( x ) ∈ V ⊗ n . It can be viewed as a linear map
Φ : T ( V ) → T ( V ) and is called a Dynkin operatoron T ( V ) . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 10
Using this notation, we can deduce from Lemma 2 the following inductive char-acterization of Φ : for v ∈ V and w ∈ V ⊗ n , we have:(1) Φ( vw ) = (cid:26) vw, if w ∈ C , (1 − T (1 ,n +1 , ··· , )( v Φ( w )) , if n ≥ . Moreover, the following identity is clear:(2) T (1 ,n +1 , ··· , ( v Φ( w )) = (Φ | V ⊗ n ⊗ id )( T (1 ,n +1 , ··· , ( vw )) . The following proposition can be viewed as a generalization of a classical resultfor free Lie algebras in [20]. As T ( V ) is a braided Hopf algebra, we let ∗ denotethe convolution product in End ( T ( V )) . Theorem 1.
Let x ∈ V ⊗ n . Then (Φ ∗ id )( x ) = nx. Proof.
The proof is given by induction on the degree n . The case n = 1 is trivial.Let n ≥ . Suppose that the theorem holds for all elements of degree n − . Itsuffices to show that for any v ∈ V and v ∈ V ⊗ n − , (Φ ∗ id )( vw ) = nvw. We write ∆( w ) = 1 ⊗ w + P w ′ ⊗ w ′′ where w ′ ∈ ker ε = L ∞ k =1 V ⊗ k . For ahomogeneous element x ∈ T ( V ) , we let l ( x ) denote its degree. As ∆ is an algebramorphism, with these notations, ∆( vw ) = v ⊗ w + 1 ⊗ vw + X vw ′ ⊗ w ′′ + (1 ⊗ v )( X w ′ ⊗ w ′′ ) , and then (Φ ∗ id )( vw ) = Φ( v ) w + X Φ( vw ′ ) w ′′ + (Φ | V ⊗ l ( w ′ ) ⊗ id )( T (1 ,l ( w ′ )+1 , ··· , ( vw ′ )) w ′′ . By the induction hypothesis, ( n − w = (Φ ∗ id )( w ) = X Φ( w ′ ) w ′′ , then after (1), X Φ( vw ′ ) w ′′ = X v Φ( w ′ ) w ′′ − X T (1 ,l ( w ′ )+1 , ··· , ( v Φ( w ′ )) w ′′ . In this formula, the first term is ( n − vw and the second one, after (2), equals to (Φ | V ⊗ l ( w ′ ) ⊗ id )( T (1 ,l ( w ′ )+1 , ··· , ( vw ′ )) w ′′ . Combining these formulas terminates the proof of the theorem. (cid:3)
To write down the formula in a more compact form, we define the number oper-ator:
Definition 10.
The number operator N : T ( V ) → T ( V ) is the linear map definedby: N (1) = 0 and for any x ∈ V ⊗ n with n ≥ , N ( x ) = nx. N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 11
So the formula in Theorem 1 can be written as (Φ ∗ id )( x ) = N ( x ) . As S is the convolution inverse of the identity map, we have: Corollary 1.
Let x ∈ T ( V ) . Then: ( N ∗ S )( x ) = Φ( x ) . As an application of Corollary 1, we may descend Φ from braided tensor Hopfalgebra T ( V ) to Nichols algebra N ( V ) . Proposition 2. Φ( I ( V )) ⊂ I ( V ) , so Φ induces a linear map Φ : N ( V ) → N ( V ) .Proof. From the definition of I ( V ) , it is both a coideal and a two-sided ideal of T ( V ) . So the coproduct on it satisfies: ∆( I ( V )) ⊂ I ( V ) ⊗ T ( V ) + T ( V ) ⊗ I ( V ) . From Corollary 1, Φ( I ( V )) = ( N ∗ S )( I ( V )) ⊂ I ( V ) because S ( I ( V )) ⊂ I ( V ) and N respects I ( V ) (note that I ( V ) is a homogeneous ideal). (cid:3) Decompositions in braid groups
The objective of this section is to give a preparation for results that will be usedin our investigations on the Dynkin operators and their relations with the structureof Nichols algebras. As such operators live in the group algebra C [ B n ] , we wouldlike to give first some decomposition results for some specific elements in C [ B n ] .4.1. Central element.
Let n ≥ be an integer and Z ( B n ) denote the center of B n .In the braid group B n , there is a Garside element ∆ n = ( σ · · · σ n − )( σ · · · σ n − ) · · · ( σ σ ) σ . The following characterization of Z ( B n ) is well known. Proposition 3 ([15], Theorem 1.24) . For n ≥ , let θ n = ∆ n . Then Z ( B n ) isgenerated by θ n . For the particular case n = 2 , we have θ = ∆ = σ .Between lines of the proof of the proposition above in [15], the following lemmais obtained. Lemma 3 ([15]) . For any ≤ i ≤ n − , σ i ∆ n = ∆ n σ n − i . Lemma 4.
The following identities hold: (1)
For any ≤ s ≤ n − , σ s ( σ n − σ n − · · · σ ) = ( σ n − σ n − · · · σ ) σ s +1 . (2) The element ∆ n has another presentation: ∆ n = σ n − ( σ n − σ n − ) · · · ( σ · · · σ n − σ n − ) . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 12 (3)
The element θ n has another presentation: θ n = ∆ n = ( σ n − σ n − · · · σ ) n . Proof. (1) This can be proved by a direct verification.(2) ∆ n is the image under the Matsumoto section of the element σ ∈ S n suchthat for any ≤ i ≤ n , σ ( i ) = n − i + 1 . It is easy to check that onceprojected to S n , the element on the right hand side is exactly σ . Moreover,this decomposition is reduced because both sides have the same length,which finishes the proof.(3) This identity comes from a direct computation using Lemma 3: ∆ n = ∆ n ∆ n = ∆ n ( σ σ · · · σ n − )( σ σ · · · σ n − ) · · · ( σ σ ) σ = ( σ n − σ n − · · · σ )( σ n − σ n − · · · σ ) · · · ( σ n − σ n − ) σ n − ∆ n = ( σ n − · · · σ ) · · · ( σ n − σ n − ) σ n − ( σ · · · σ n − ) · · · ( σ σ ) σ = ( σ n − σ n − · · · σ ) n . (cid:3) The following proposition is the main result of this subsection.
Proposition 4.
The element θ n has another presentation: θ n = ∆ n = ( σ n − σ n − · · · σ ) n − . Proof.
From Lemma 4, ∆ n = ( σ n − σ n − · · · σ )( σ n − σ n − · · · σ ) · · · ( σ n − σ n − · · · σ ) . At first, using Lemma 4, it is possible to move the first σ towards right until it cannot move anymore. We exchange it with ( σ n − σ n − · · · σ ) for n − times, whichgives σ n − finally and so: ∆ n = ( σ n − σ n − · · · σ )( σ n − σ n − · · · σ ) · · · ( σ n − σ n − · · · σ ) . Repeating this procedure with the help of Lemma 4 for the first σ , · · · , σ n − , wewill obtain the presentation as announced in the proposition. (cid:3) Decompositions in the group algebra.
In this subsection, we will work inthe group algebra C [ B n ] for some n ≥ .The symmetrization operator in C [ B n ] is defined by: S n = X σ ∈ S n T σ ∈ C [ B n ] . Because V is a braided vector space and B n acts naturally on V ⊗ n , we may treat S n as a linear operator in End ( V ⊗ n ) .For ≤ i ≤ n − , let ( i, i + 1) ∈ S n be a transposition. We have seen that T ( i,i +1) = σ i . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 13
The element S n ∈ C [ B n ] has a remarkable decomposition as shown in [5]. Forany ≤ m ≤ n , we define T m = 1 + σ m − + σ m − σ m − + · · · + σ m − σ m − · · · σ ∈ C [ B n ] . Proposition 5 ([5]) . For any n ≥ , S n = T T · · · T n ∈ C [ B n ] . In fact, this proposition is true when being projected to C [ S n ] ; then notice thatthe expansion of the product on the right hand side contains only reduced terms.Recall the definition of P ,n in Section 3.1. To simplify the notation, we denote P n = T P ,n ∈ C [ B n ] . This element will be an important ingredient in our further discussion.For n ≥ , recall the decomposition of P n given in Lemma 2: P n = (1 − σ n − σ n − · · · σ )(1 − σ n − σ n − · · · σ ) · · · (1 − σ n − ) . This element P n permits us to give a much more refined structure of T n . Weintroduce another member T ′ n = (1 − σ n − σ n − · · · σ )(1 − σ n − σ n − · · · σ ) · · · (1 − σ n − ) ∈ C [ B n ] . Proposition 6.
For n ≥ , the decomposition T n P n = T ′ n holds in C [ B n ] .Proof. The Proposition 6.11 in [5] affirms that if all inverses appearing are welldefined, then T n = (1 − σ n − σ n − · · · σ ) · · · (1 − σ n − )(1 − σ n − ) − · · · (1 − σ n − · · · σ ) − . So the proposition follows from Lemma 2. (cid:3)
Corollary 2.
The following identity holds in C [ B n ] : n − X k =0 ( σ n − σ n − · · · σ ) k ! (1 − σ n − σ n − · · · σ ) = 1 − ∆ n = 1 − θ n . Moreover, for ≤ s ≤ n − , let ι s : B s ֒ → B n be the canonical embedding ofbraid groups on the last s strands. If θ s is the central element in B s , we denote θ ι s s = ι s ( θ s ) and θ ι = σ n − , then there exists an element L n = X k =0 ( σ n − σ n − ) k ! · · · n − X k =0 ( σ n − σ n − · · · σ ) k ! ∈ C [ B n ] , such that in C [ B n ] , L n T ′ n = (1 − θ n )(1 − θ ι n − n − ) · · · (1 − θ ι ) . The study of the ideal I ( V ) We keep assumptions and notations in previous sections.
N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 14
A result on the quotient ideal.
As we have seen, the Nichols algebraassociated to an H -Yetter-Drinfel’d module V is a quotient of the braided tensorHopf algebra T ( V ) by a maximal coideal I ( V ) contained in T ≥ ( V ) . This definitiontells us almost nothing about the concrete structure of I ( V ) : as I ( V ) is an ideal,the Nichols algebra N ( V ) can be viewed as imposing some relations in T ( V ) , butsuch relations can never be read directly from the definition.As we know, the best result for the structure of I ( V ) is obtained by M. Rosso in[21] in a dual point of view and by P. Schaurenburg in [22]. We recall this resultin this subsection.Let S n : V ⊗ n → V ⊗ n be the element in C [ B n ] defined in Section 4.2. Proposition 7 ([21], [22]) . Let V be an H -Yetter-Drinfel’d module. Then N ( V ) = M n ≥ (cid:0) V ⊗ n / ker( S n ) (cid:1) . So to make more precise the structure of I ( V ) , it suffices to study each subspace ker( S n ) . In the following part of this section, we want to characterize a part ofelements in ker( S n ) and show that in cases of great interest, this part is the essentialone for understanding the structure of ker( S n ) .5.2. General assumption.
From now on, assume that n ≥ is an integer. Tostudy the structure of ker( S n ) , we want first to concentrate on some essential "lev-els" in it. Definition 11.
Let < s < n be an integer and i : B s → B n be an injection ofgroups. We call i a positional embedding if there exists some integer ≤ r ≤ n − s such that for any ≤ t ≤ s − , i ( σ t ) = σ t + r . For an element v ∈ V ⊗ n , if v ∈ ker( S n ) , there are two possibilities:(1) There exists some ≤ s < n and some positional embedding of groups ι : B s ֒ → B n such that v is annihilated by ι ( S s ) ;(2) For any s and positional embedding ι as above, v is not in ker( ι ( S s )) .Elements falling in the case (2) are much more interesting in our framework. Sowe would like to give a more concrete assumption for the purpose of concentratingon such elements; here, we want to impose a somehow stronger restriction.Let v ∈ V ⊗ n be a non-zero element and C [ X v ] denote the C [ B n ] -submodule of V ⊗ n generated by v , that is to say, C [ X v ] = C [ B n ] .v . Because C [ X v ] is a C [ B n ] -module, S n : C [ X v ] → C [ X v ] is well defined.We fix this v ∈ V ⊗ n as above, the restriction on v we want to impose is as follows: Definition 12.
An element v ∈ V ⊗ n is called of level n if S n v = 0 and for any ≤ s ≤ n − and any positional embedding ι : B s ֒ → B n , the equation ι ( θ s ) x = x has no solution in C [ X v ] . Proposition 8. If v ∈ V ⊗ n is a non-zero solution of equation (1 − σ n − σ n − · · · σ ) x =0 , then θ n .v = v . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 15
Proof. If (1 − σ n − σ n − · · · σ ) v = 0 , from Corollary 2, n − X k =0 ( σ n − σ n − · · · σ ) k ! (1 − σ n − σ n − · · · σ ) v = (1 − θ n ) v. (cid:3) Remark 4. As V ⊗ n is a C [ B n ] -module, we can define H ⊂ V ⊗ n as the subspaceof V ⊗ n formed by eigenvectors of θ n with eigenvalue (that is to say, H = { w ∈ V ⊗ n | θ n w = w } ).As θ n ∈ Z ( B n ) , for any w ∈ H and Y ∈ C [ B n ] , we have θ n Y w = Y w , thus H is a C [ B n ] -submodule of V ⊗ n . It means that if v ∈ H , then C [ X v ] ⊂ H . Lemma 5.
Let v ∈ V ⊗ n be a non-zero element of level n . Then for any ≤ i ≤ n − and any positional embedding ι i : B i ֒ → B n , ι i ( S i ) x = 0 has no solution on C [ X v ] .Proof. Let ι i : B i ֒ → B n be a positional embedding such that ι i ( S i ) x = 0 has asolution in C [ X v ] . Then ι i ( S i ) = ι i ( T ) · · · ι i ( T i ) ∈ C [ B n ] . The equation ι i ( S i ) x = 0 has a solution in C [ X v ] means that det ( ι i ( S i )) = 0 , so there exists some ≤ j ≤ i such that det ( ι i ( T j )) = 0 .Because ι i ( T j ) ι i ( P j ) = ι i ( T ′ j ) , we obtain that det ( ι i ( T ′ j )) = 0 . From the definitionof T ′ j , there exists some ≤ k ≤ j − such that det (1 − ι i ( σ j − σ j − · · · σ k )) = 0 . So we can choose another positional embedding ι : B j − k +1 → B n such that for theaction of C [ B n ] on C [ X v ] , det (1 − ι ( σ j − k σ j − k − · · · σ )) = 0 . So, from Proposition 8, ι ( θ j − k +1 ) x = x has a non-zero solution on C [ X v ] , whichcontradicts to the assumption that v is of level n . (cid:3) Solutions.
Fix some n ≥ , we want to solve the equation S n x = 0 on C [ X v ] for some non-zero element v ∈ V ⊗ n .We define an element in C [ B n ] : X = (1 − σ n − σ n − · · · σ ) · · · (1 − σ n − σ n − · · · σ ) · · · (1 − σ n − σ n − )(1 − σ n − ) . Proposition 9. If v ∈ V ⊗ n is a non-zero element of level n , then X is invertibleon C [ X v ] .Proof. We may view X as an element in End ( C [ X v ]) . If X is not invertible, det ( X ) = 0 . From the definition, there must exist some term, say (1 − σ n − σ n − · · · σ i ) ,for some ≤ i ≤ n − , having determinant . So there exists some nonzero element w ∈ C [ X v ] such that (1 − σ n − σ n − · · · σ i ) w = 0 . But from Proposition 8, we mayfind some positional embedding ι : B n − i +1 ֒ → B n such that ι ( θ n − i +1 ) w = w , whichcontradicts to the assumption that v is of level n . (cid:3) N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 16
The level n assumption we are working with will give more information on solu-tions of equation S n x = 0 . Proposition 10.
Let v ∈ V ⊗ n be a non-zero element of level n . (1) There exists a bijection between nonzero solutions of equation T ′ n x = 0 andof the equation (1 − σ n − σ n − · · · σ ) x = 0 in C [ X v ] . (2) Equations S n x = 0 and T n x = 0 have the same solutions in C [ X v ] .Proof. (1) From Proposition 9, X − : C [ X v ] → C [ X v ] is well defined. So thisproposition comes from the identity: T ′ n = (1 − σ n − σ n − · · · σ ) X .(2) Let w be a non-zero solution of T n x = 0 , then from Proposition 5, S n w = 0 .Conversely, let u be a non-zero solution of S n x = 0 . If T n u = 0 , again fromProposition 5, T n u will be a non-zero solution of equation T · · · T n − x =0 on C [ X v ] , so S n − x = 0 has a non-zero solution on C [ X v ] , contradictsLemma 5 above. (cid:3) Recall that P n = T P ,n ∈ C [ B n ] as defined in the last section, P n ∈ End ( C [ X v ]) .Now, let w ∈ ker( S n ) ∩ Im ( P n ) be a non-zero element of level n . Then fromLemma 5, w satisfies T n w = 0 . Moreover, because it is in Im ( P n ) , we can choosesome w ′ such that P n ( w ′ ) = w , then T ′ n w ′ = T n w = 0 . From the identity T ′ n = (1 − σ n − σ n − · · · σ ) X , Xw ′ is a solution of the equation (1 − σ n − σ n − · · · σ ) x = 0 , so from Proposition 8, θ n Xw ′ = Xw ′ . This discussiongives the following proposition. Proposition 11.
Let w ∈ ker( S n ) ∩ Im ( P n ) be an element of level n . Then θ n w = w .Proof. From the definition of w ′ , if θ n fixes w ′ , then it fixes w . So if θ n w = w , thenit does not fix w ′ and then Xw ′ (see Remark 4), which is a contradiction. (cid:3) Remark 5.
Let H denote the eigenspace of θ n corresponding to the eigenvalue as in the Remark 4 above. If we let E n denote the set of elements in Im ( P n ) withlevel n in V ⊗ n , then the proposition above implies that E n ⊂ P n ( H ) . We have constructed solutions of equation θ n x = x on C [ X v ] from some kinds ofelements in ker( S n ) ∩ Im ( P n ) . Now we proceed to consider the construction in theopposite direction.Let w ∈ C [ X v ] be a solution of θ n x = x . If u = n − X k =0 ( σ n − σ n − · · · σ ) k ! w = 0 , it will be a solution of the equation (1 − σ n − σ n − · · · σ ) x = 0 , then X − u is asolution of the equation T ′ n x = 0 and P n X − u will be a non-trivial solution of N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 17 S n x = 0 if it is not zero; moreover, it is in Im ( P n ) , from which we obtain anelement in ker( S n ) ∩ Im ( P n ) .There are some possibilities for the appearance of zero elements when passingfrom the solutions of θ n x = x to those of S n x = 0 . The appearance of zeros mostlycomes from the fact that an element satisfying θ n x = x may be the solution of ι s ( θ s ) x = x for some ≤ s ≤ n − and some positional embedding ι s : B s ֒ → B n .The subspace ker( S n ) ∩ Im ( P n ) is sufficiently important, as will be shown in thenext subsection.5.4. Properties of ker( S n ) ∩ Im ( P n ) . In this subsection, suppose that n ≥ isan integer.Instead of End ( C [ X v ]) , it is better in this subsection to view S n , P n as elements in End ( V ⊗ n ) . We want to show that ker( S n ) ∩ Im ( P n ) contains all primitive elementsand in some special cases (for example, the diagonal case), it generates ker( S n ) . Proposition 12.
Let v ∈ V ⊗ n be a homogeneous primitive element of degree n .Then v ∈ ker( S n ) ∩ Im ( P n ) .Proof. The fact v ∈ ker( S n ) is a corollary of the definition of Nichols algebra andProposition 7. So it suffices to show that v ∈ Im ( P n ) .The element v is primitive means that ∆( v ) = v ⊗ ⊗ v . From Theorem 1, Φ ∗ id = N , so nv = Φ ∗ id ( v ) = m ◦ (Φ ⊗ id )∆( v ) = Φ( v ) , and then v = 1 n Φ( v ) = 1 n P n ( v ) ∈ Im ( P n ) . (cid:3) The second property we want to establish is that in the diagonal case, thesesubspaces ker( S n ) ∩ Im ( P n ) will generate the ideal (also coideal) I ( V ) .Recall that from the definition of Nichols algebra and Proposition 7, the subspace I ( V ) = M n ≥ ker( S n ) ⊂ T ( V ) is a maximal coideal contained in T ≥ ( V ) . Moreover, it is a homogeneous ideal.Let J ⊂ T ≥ ( V ) be a coideal in HH Y D containing the subspace M n ≥ (ker( S n ) ∩ Im ( P n )) . Such a coideal does exist as I ( V ) satisfies these conditions. Proposition 13.
Let T ( V ) be of diagonal type. Then the ideal generated by J in T ( V ) is I ( V ) .Proof. Let K be the two-sided ideal generated by J ⊂ T ≥ ( V ) in T ( V ) . Then K is also an ideal in T ≥ ( V ) . As a two-sided ideal generated by a coideal, K is alsoa coideal. From the maximality of I ( V ) , K ⊂ I ( V ) . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 18
We proceed to prove that T ( V ) /K ∼ = N ( V ) . For this purpose, the followinglemma is needed. Lemma 6 ([11]) . Suppose that the Nichols algebra is of diagonal type. Let K ⊂ T ≥ ( V ) be simultaneously an ideal, a coideal and an H -Yetter-Drinfel’d module. Ifall primitive elements in T ( V ) /K are concentrated in V , then T ( V ) /K ∼ = N ( V ) . From this lemma, it suffices to show that there is no non-zero primitive elementof degree greater than in T ( V ) /K .Suppose that v is such a non-zero element which is moreover homogeneous ofdegree n , so in T ( V ) , ∆( v ) ∈ v ⊗ ⊗ v + K ⊗ T ( V ) + T ( V ) ⊗ K. As K ⊂ I ( V ) , ∆( v ) ∈ v ⊗ ⊗ v + I ( V ) ⊗ T ( V ) + T ( V ) ⊗ I ( V ) . But in T ( V ) / I ( V ) , from the definition of Nichols algebra, there is no such element,which forces v ∈ I ( V ) and then S n v = 0 .We need to show that in fact v ∈ K . From Corollary 1, P n ( v ) = N ∗ S ( v ) ∈ nv + K, then v − k ∈ Im ( P n ) for some k ∈ K . As S n v = 0 and K ⊂ ker( S n ) , v − k ∈ ker( S n ) ∩ Im ( P n ) ⊂ K ; this implies v ∈ K . (cid:3) This proposition shows the importance of these subspaces ker( S n ) ∩ Im ( P n ) inthe study of the defining ideal.5.5. Main theorem.
The main result of this paper is:
Theorem 2.
Elements of level n are primitive. From this theorem, level n solutions of S n x = 0 are primitive elements of degree n , so they are in ker( S n ) ∩ Im ( P n ) . Moreover, this introduces a method to findprimitive elements in T ( V ) .The proof of this theorem will be given in the end of this paper, after introducingthe differential algebra of a Nichols algebra.6. Applications
In this section, we give some applications of the machinery constructed above.Though the discussion in the last section is somehow elementary, it may giveremarkable results and good points of view once being applied to some concreteexamples.
N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 19
A general application for the diagonal type.
Let H be the group algebraof an abelian group G , V ∈ HH Y D be of diagonal type, T ( V ) and N ( V ) be thebraided tensor Hopf algebra and Nichols algebra, respectively.Suppose that dimV = m , with basis v , · · · , v m such that σ ( v i ⊗ v j ) = q ij v j ⊗ v i . From the definition of the braiding, the action of C [ B n ] on V ⊗ n has the followingdecomposition V ⊗ n = M i ∈ I C [ B n ] .v i · · · v i m m , where the sum runs over I = { i = ( i , · · · , i m ) | i + · · · + i m = n } .We fix some i = ( i , · · · , i m ) ∈ N m such that i ∈ I and a monomial v i = v i · · · v i m m , then in C [ X i ] = C [ B n ] .v i , if θ n x = x has a solution, we must obtain θ n v i = v i .Indeed, when projected canonically to S n , the element θ n ∈ B n corresponds to , so if θ n x = x , θ n will stablize all components of x . From the decompositionabove, for any component x of x , there exists a nonzero constant c and an element σ ∈ B n such that cx = σ ( v i ) , thus v i = σ − ( cx ) . Thus all level n elements arecontained in the sum of some C [ X i ] for some v i satisfying θ n v i = v i .To exclude those elements which have not level n but are stable under the actionof θ n , some notations are needed.We fix some i and v i . It is more convenient to write v i = e · · · e n , where e i aresome v j ’s. Then let T i = ( t ij ) denote a matrix in M n ( C ) with t ii = 1 and for i = j , t ij are defined by σ ( e i ⊗ e j ) = t ij e j ⊗ e i . For some ≤ s ≤ n , ≤ k < · · · < k s ≤ n , k = ( k , · · · , k s ) , we define: Π ks = s Y i =1 s Y j =1 t k i ,k j . The following proposition is an easy consequence of the definition.
Proposition 14.
With the notations above, we have: (1) θ n v i = v i if and only if Π kn = 1 . (2) If for any ≤ s ≤ n − and any k , Π ks = 1 , then v i satisfies the assumptionin Definition 12. Moreover, all elements in ker( S n ) ∩ C [ X v i ] are of level n . Remark 6.
Under the assumptions (2) in the proposition above, (1)
From Theorem 2, all elements in ker( S n ) ∩ C [ X v i ] are primitive. (2) From Remark 5, all elements in ker( S n ) ∩ C [ X v i ] can be constructed from C [ X v i ] by the method given in the end of Section 5.3. So such a family ofprimitive elements can be easily and directly computed. N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 20
Remark 7. If v ∈ V ⊗ V is an element in ker( S ) ∩ Im ( P ) , then it must beof level and so primitive. Moreover, in the diagonal case, level elements in ker( S ) ∩ Im ( P ) can be obtained from monomials stablized by θ by applying P . Exterior algebras.
In this subsection, as a warm up, we apply results of theprevious section to the construction of exterior algebras.The main ingredient is the Hopf algebra H = C [ G ] , where G = Z / Z = { , ε } .Let V be a finite dimensional vector space with basis v , · · · , v m .(1) The action of H on V is given by: for v ∈ V , ε.v = − v ;(2) The coaction is given by: δ ( v ) = ε ⊗ v , where δ : V → H ⊗ V .This makes V an H -Yetter-Drinfel’d module.We form the braided Hopf algebra T ( V ) and want to calculate relations appearingin the ideal I ( V ) .At first, we consider relations in V ⊗ n of level n . In fact, for n ≥ , there are nosuch relations because if v = v i · · · v i n ∈ V ⊗ n is a pure tensor such that θ n v = v ,from the definition of the braiding, there must exist some ≤ s < t ≤ n such that σ ( v i s ⊗ v i t ) = v i s ⊗ v i t , which contradicts the definition of level n relations.So it suffices to consider relations of level in V ⊗ . We start from consideringall solutions of θ x = x in V ⊗ . These solutions are: v i v j , for ≤ i, j ≤ n .As in the procedure of constructing solutions of S n x = 0 from θ n x = x given inthe last section, the action of P on these elements gives: P ( v i v j ) = v i v j + v j v i , so v i v j + v j v i ∈ ker( S ) ∩ Im ( P ) . Moreover, from Remark 7 in the last subsection,we obtain ker( S ) ∩ Im ( P ) = span { v i v j + v j v i | ≤ i, j ≤ n } . Quantized enveloping algebras.
In this subsection, we will discover thequantized Serre relations in the definition of the quantized enveloping algebra U q ( g ) associated to a symmetrizable Kac-Moody Lie algebra g by assuming almost noknowledge about the existence of such relations.Let q be a nonzero complex number such that for any N ≥ , q N = 1 . Let g be asymmetrizable Kac-Moody Lie algebra of rank n , C = ( C ij ) n × n be its generalizedCartan matrix and A = DC be the symmetrization of the Cartan matrix by somediagonal matrix D = ( d , · · · , d n ) with d i positive integers which are relativelyprime. We denote A = ( a ij ) n × n .At first, we briefly recall the construction of the strict positive part of U q ( g ) inthe framework of Nichols algebras. This construction is due to M. Rosso and canbe found in [21] with a slightly different language.Let H = C [ G ] be the group algebra where G is the abelian group Z n . Let K , · · · , K n denote a basis of Z n . Then H is a commutative and cocommutativeHopf algebra.Let V be a C -vector space of dimension n with basis E , · · · , E n . We define an H -Yetter-Drinfel’d module structure on V by: N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 21 (1) The action of K i on E j is given by: K i .E j = q a ij E j ;(2) The coaction of E i is given by: δ ( E i ) = K i ⊗ E i , where δ : V → H ⊗ V isthe structure map of left H -comodule structure on V .Starting with this V ∈ HH Y D , the braided tensor algebra T ( V ) and the correspond-ing Nichols algebra N ( V ) can be constructed. The defining ideal is denoted by I ( V ) .Assume that we know nothing about this ideal I ( V ) (because from the generaltheory of quantized enveloping algebras, we know that I ( V ) is generated by quan-tized Serre relations). Here, our point of view is much more pedestrian: if we donot know them, how to find?Results in the previous section will offer us a method.At first, we want to concentrate on the case U q ( sl ) , the simplest one which hassuch quantized Serre relations. In this case, H = Z with basis K , K , V is ofdimension with basis E , E .We would like to compute the level relations in I ( V ) .At first, we write down all monomials of degree which are stabilized by theaction of θ but not for all θ with possible embeddings. They are: E E , E E E , E E , E E , E E E , E E . After the action of σ σ , we obtain: E E , E E E + q E E , E E + q − E E E ,E E + q − E E E , E E E + q E E , E E . In this case, X = 1 − σ , so the action of X − on these elements will give: x = 21 − q − E E , x = 11 − q − E E E + q − q E E ,x = 11 − q E E + q − − q − E E E , x = 11 − q E E + q − − q − E E E ,x = 11 − q − E E E + q − q E E , x = 11 − q − E E . It is easy to compute the action of P on all possible monomials: P ( E E ) = E E − ( q + q − ) E E E + E E ,P ( E E E ) = 2 E E E − q − E E − qE E ,P ( E E ) = (1 − q ) E E − ( q − − E E ,P ( E E ) = (1 − q ) E E − ( q − − E E ,P ( E E E ) = 2 E E E − qE E − q − E E ,P ( E E ) = E E − ( q + q − ) E E E + E E . And then P ( x ) = 21 − q − ( E E − ( q + q − ) E E E + E E ) , N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 22 P ( x ) = − q − − q − ( E E − ( q + q − ) E E E + E E ) ,P ( x ) = 21 − q ( E E − ( q + q − ) E E E + E E ) ,P ( x ) = 21 − q ( E E − ( q + q − ) E E E + E E ) ,P ( x ) = − q − − q − ( E E − ( q + q − ) E E E + E E ) ,P ( x ) = 21 − q − ( E E − ( q + q − ) E E E + E E ) . So starting with solutions of θ x = x with level , the solutions in Im ( P ) with level we obtained for the equation S x = 0 are exactly the quantized Serrerelations of degree .Moreover, we show that there are no other relations of level . If w ∈ ker( S ) issuch an element, it will be stable under the action of θ , so it is a linear combinationof monomials above, then it must be a linear combination of degree Serre relations.Finally, we turn to the level n elements for an arbitrary integer n ≥ . Asexplained in Section 6.1, it suffices to consider a monomial of form E s E t for somepositive integers s and t .The action of θ s + t on this monomial gives: θ s + t ( E s E t ) = q s − s + t − t − st E s E t . So this monomial is stablized by θ s + t if and only if s − s + t − t − st = 12 (cid:0) ( s − t ) + ( s − + ( t − − (cid:1) = 0 . The only possible positive integer solutions ( s, t ) of the equation ( s − t ) + ( s − + ( t − = 2 are (2 , , (2 , and (1 , . But ( s, t ) = (2 , is not of level because we can always find a subword which is fixed by θ .As a conclusion, the only possible level in this case is and all possible relationscoming from level elements are quantized Serre relations as shown above.6.4. Primitivity of Serre relations.
As an application of the main theorem, wededuce a short proof for the primitivity of Serre relations with little computation.A direct proof can be found in the appendix of [2].Let A = ( a ij ) n × n = DC be a symmetrized Cartan matrix and V be a Z n -Yetter-Drinfeld module of diagonal type with dimension n . Notations in the previoussubsection are adopted. Moreover, suppose that the braiding matrix ( q ij ) satisfies:(3) q ij q ji = q c ij ii , ≤ i, j ≤ n, and these − c ij are the smallest integers such that the equations (3) hold. Proposition 15.
For any ≤ i, j ≤ n , i = j , we denote N = 1 − c ij . Then P N +1 ( v Ni v j ) is a primitive element, where P N +1 is the Dynkin operator. N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 23
Proof.
At first, it is easy to show that (1 − σ N σ N − · · · σ )( v Ni v j ) = (1 − q − c ij ii q ij q ji ) v Ni v j . From the hypothesis (3) above, the right hand side is , so from Proposition 8, θ N +1 ( v Ni v j ) = v Ni v j .Moreover, it is obvious that for any < s < N + 1 and any positional embedding ι : B s ֒ → B N +1 , ι ( θ s )( v Ni v j ) = v Ni v j . As a consequence, in the algorithm afterRemark 5, X − is well defined and from the definition of X , X − ( v Ni v j ) = λv Ni v j for some non-zero constant λ . Then P N +1 ( v Ni v j ) is a nonzero solution of S N +1 x = 0 of level n , so it is primitive by Theorem 2. (cid:3) Quantized enveloping algebras revisited.
We keep notations in the be-ginning of Section 6.3.As A = DC is a symmetrized Cartan matrix, for any ≤ i, j ≤ n , we have d i c ij = d j c ji . Then from the definition of q ij , the following lemma is clear. Lemma 7.
Let A = DC be a symmetrized Cartan matrix. Then for any ≤ i, j ≤ n , q ij q ji = q c ij ii . Combined with Proposition 15, this lemma gives:
Corollary 3.
Let g be a symmetrizable Kac-Moody algebra. Then degree n quan-tized Serre relations in U q ( g ) are of level n . Moreover, the union of level n elementsfor n ≥ generates I ( V ) as an ideal. Remark 8.
The corollary above explains the reason for the importance of sym-metrizable Kac-Moody algebras: they contain sufficient Serre relations. This givesa strong constraint on the representation theory of such Lie algebras. Differential algebras of Nichols alegbras
The first part of this section is devoted to the generalization of some results in[6], then we recall the construction of a pairing between two Nichols algebras.7.1.
Pairings between Nichols algebras.
In this subsection, we want to recalla result of [4] and [17]. It should be remarked that these two constructions, thoughin different languages (one is dual to the other), are essentially the same.Let H = ∞ M n =0 H n , B = ∞ M n =0 B n be two graded Hopf algebras with finite dimensional graded components. Definition 13.
A generalized Hopf pairing φ : H × B → C is called graded if forany i = j , φ ( H i , B j ) = 0 . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 24
We fix a graded Hopf pairing φ : H × B → C between H and B and assumemoreover that φ is non-degenerate.Let V ∈ HH Y D and W ∈ BB Y D be two Yetter-Drinfel’d modules, φ : V × W → C be a non-degenerate bilinear form such that for any h ∈ H , b ∈ B , v ∈ V and w ∈ W ,(4) φ ( h.v, w ) = X φ ( h, w ( − ) φ ( v, w (0) ) , (5) φ ( v, b.w ) = X φ ( v ( − , b ) φ ( v (0) , w ) , where δ V ( v ) = P v ( − ⊗ v (0) and δ W ( w ) = P w ( − ⊗ w (0) are H -comodule and B -comodule structure maps, respectively.Let T ( V ) , T ( W ) be the corresponding braided tensor Hopf algebras and N ( V ) , N ( W ) be Nichols algebras associated to V and W , respectively. Let B H ( V ) = N ( V ) ♯H and B B ( W ) = N ( W ) ♯B denote crossed biproducts defined in Section 2.5. Theorem 3 ([4],[17]) . There exists a unique graded Hopf pairing φ : B H ( V ) × B B ( W ) → C , extending φ and φ . Moreover, it is non-degenerate. In the following argument, attention will be paid to a particular case of thistheorem. In our framework, we take H = B and V = W in Theorem 3, φ : H × H → C a non-degenerate graded Hopf pairing and φ : V × V → C a non-degeneratebilinear form satisfying the compatibility conditions above. So the machinery inTheorem 3 produces a non-degenerate graded Hopf pairing φ : B H ( V ) × B H ( V ) → C . This will be the main tool in our further construction.7.2.
Double construction and Schrödinger representation.
In this subsec-tion, as a review, we will apply results from [6], Section 2 to the case of Nicholsalgebras.Suppose that H is a graded Hopf algebra, V ∈ HH Y D is an H -Yetter-Drinfel’dmodule and φ : B H ( V ) × B H ( V ) → C is the non-degenerate graded Hopf pairingconstructed in the last section.We recall some results from [6] briefly.To indicate their positions, we denote B + H ( V ) = B H ( V ) , B − H ( V ) = B H ( V ) and D φ ( B H ( V )) = D φ ( B + H ( V ) , B − H ( V )) their quantum double. The Schrödinger representation defined in [6] gives a modulealgebra type action of D φ ( B H ( V )) on these two components.(1) On B + H ( V ) , the action is given by: for a, x ∈ B + H ( V ) and b ∈ B − H ( V ) , ( a ⊗ .x = X a (1) xS ( a (2) ) , (1 ⊗ b ) .x = X ϕ ( x (1) , S ( b )) x (2) . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 25 (2) On B − H ( V ) , the action is given by: for a ∈ B + H ( V ) and b, y ∈ B − H ( V ) , ( a ⊗ .y = X ϕ ( a, y (1) ) y (2) , (1 ⊗ b ) .y = X b (1) yS ( b (2) ) . As has been shown in [6], these actions give both B + H ( V ) and B − H ( V ) a D φ ( B H ( V )) -module algebra structure.Moreover, we can construct the Heisenberg double H φ ( B H ( V )) = H φ ( B + H ( V ) , B − H ( V )) , which, in general, is not a Hopf algebra.In [6], we defined an action of D φ ( B H ( V )) on H φ ( B H ( V )) by: for a, a ′ ∈ B + H ( V ) and b, b ′ ∈ B − H ( V ) , ( a ⊗ b ) . ( b ′ ♯a ′ ) = X ( a (1) ⊗ b (1) ) .b ′ ♯ ( a (2) ⊗ b (2) ) .a ′ , which makes H φ ( B H ( V )) a D φ ( B H ( V )) -module algebra.The following two results are also obtained in [6]. Proposition 16 ([6]) . We define a D φ ( B H ( V )) -comodule structure on B + H ( V ) and B − H ( V ) by: for a ∈ B + H ( V ) and b ∈ B − H ( V ) , B + H ( V ) → D φ ( B H ( V )) ⊗ B + H ( V ) , a X a (1) ⊗ ⊗ a (2) , B − H ( V ) → D φ ( B H ( V )) ⊗ B − H ( V ) , b X ⊗ b (1) ⊗ b (2) . Then with the Schrödinger representation and comodule structures defined above,both B + H ( V ) and B − H ( V ) are in the category D φ D φ Y D . Moreover, the Heisenberg double H φ ( B H ( V )) is in the D φ ( B H ( V )) -Yetter-Drinfel’dmodule category. Theorem 4 ([6]) . We define a D φ ( B H ( V )) -comodule structure on H φ ( B H ( V )) by:for a ∈ B + H ( V ) and b ∈ B − H ( V ) , H φ ( B H ( V )) → D φ ( B H ( V )) ⊗ H φ ( B H ( V )) ,b♯a X (1 ⊗ b (1) ) . ( a (1) ⊗ ⊗ b (2) ♯a (2) . Then with the module structure defined above and this comodule structure, H φ ( B H ( V )) is in the category D φ D φ Y D . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 26
Construction of differential algebras.
In this section, we want to con-struct the differential algebra of a Nichols algebra. It generalizes the constructionof quantized Weyl algebra in [6].But it should be remarked that the construction in [6] concentrates on a specificHopf algebra, say C [ Z n ] and a special action on the Nichols algebra. So to generalizeit, we need some more work.Let N + ( V ) and N − ( V ) be Nichols algebras contained in B + H ( V ) and B − H ( V ) respectively. We would like to give both N + ( V ) and N − ( V ) a D φ ( B H ( V )) -Yetter-Drinfel’d module algebra structure.For N + ( V ) , the D φ ( B H ( V )) -module structure is given by the Schrödinger rep-resentation and the D φ ( B H ( V )) -comodule structure is given by: N + ( V ) → D φ ( B H ( V )) ⊗ N + ( V ) ,b X ( b (1) ♯ ( b (2) ) ( − ⊗ ⊗ ( b (2) ) (0) . This is obtained from the formula in Proposition 16.
Proposition 17.
With the structures defined above, N + ( V ) is a D φ ( B H ( V )) -Yetter-Drinfel’d module algebra.Proof. At first, we need to show that the Schrödinger representation preserves N + ( V ) .Let a ∈ B H ( V ) . It suffices to prove that if x ∈ N + ( V ) , then both ( a ⊗ .x and (1 ⊗ a ) .x are contained in N + ( V ) . For this purpose, because the action is linear, wewrite a = b♯h ∈ N + ( V ) ♯H . From the formula given in Radford’s crossed biproduct, (( b♯h ) ⊗ .x = X ( b♯h ) (1) xS (( b♯h ) (2) )= X ( b (1) ♯ ( b (2) ) ( − h (1) )( x♯ S (( b (2) ) (0) ♯h (2) )= X ( b (1) (( b (2) ) ( − h (1) .x ) ♯ ( b (2) ) ( − h (2) )(1 ♯S ( h (3) ) S (( b (2) ) ( − ))( S (( b (2) ) (0) ) ♯ X ( b (1) (( b (2) ) ( − h (1) .x ) ♯ ( b (2) ) ( − h (2) S ( h (3) ) S (( b (2) ) ( − ))( S (( b (2) ) (0) ) ♯ X b (1) (( b (2) ) ( − h (1) .x ) S (( b (2) ) (0) ) ♯ , which is in N + ( V ) .For the other action, we have: (1 ⊗ ( b♯h )) .x = X φ ( x (1) , S ( b♯h )) x (2) . From the definition of the crossed biproduct, when restricted to N + ( V ) , the co-product gives ∆ : N + ( V ) → B H ( V ) ⊗ N + ( V ) , so the result is in N + ( V ) .Thus the action and coaction of D φ ( B H ( V )) on N + ( V ) are both well definedand as a consequence, N + ( V ) is a D φ ( B H ( V )) -module algebra.These structures are compatible because we have seen that the coaction definedabove is just the restriction of the coproduct in D φ ( B H ( V )) on N + ( V ) . (cid:3) N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 27
The same argument, once applied to N − ( V ) , implies that N − ( V ) is a D φ ( B H ( V )) -Yetter-Drinfel’d module algebra.As remarked after the definition of Yetter-Drinfel’d modules, we may use thenatural braiding in the category D φ D φ Y D to give N − ( V ) ⊗ N + ( V ) an associativealgebra structure, which is denoted by W φ ( V ) and is called the differential algebraof the Nichols algebra N ( V ) = N − ( V ) .This gives a natural action of W φ ( V ) on N − ( V ) , where N + ( V ) ⊂ W φ ( V ) acts by"differential". Remark 9. (1)
It should be pointed out that N − ( V ) ⊗ N + ( V ) is a subalgebraof H φ ( B H ( V )) . This can be obtained from the definition of the braiding σ in the category D φ D φ Y D and the formula for the action of D φ ( B H ( V )) on B − H ( V ) .Moreover, it is exactly the subalgebra of H φ ( B H ( V )) generated by N − ( V ) and N + ( V ) . (2) This action of W φ ( V ) on N − ( V ) can be explained as follows: we considerthe trivial N + ( V ) -module C given by the counit ε , then Ind W φ ( V ) N + ( V ) ( C .
1) = W φ ( V ) ⊗ N + ( V ) C . is isomorphic to N − ( V ) as a vector space and from this, N − ( V ) can beregarded as a W φ ( V ) -module. Non-degeneracy assumption.
We should point out that results in previoussections do not depend on the non-degeneracy of the generalized Hopf pairing. Someresults concerned with this property will be discussed in this subsection.Recall the notation N ( V ) = N − ( V ) . So W φ ( V ) acts on N ( V ) by: for x ∈ N + ( V ) ⊂ W φ ( V ) and y ∈ N ( V ) , x.y = X φ ( x, y (1) ) y (2) . Because the generalized Hopf pairing is graded and non-degenerate, results in [6]can be generalized to the present context.Let v , · · · , v n be a basis of V . Lemma 8.
Let y ∈ N ( V ) , y / ∈ C ∗ such that for any basis element v i ∈ N + ( V ) ⊂ W φ ( V ) , v i .y = 0 . Then y = 0 . Proposition 18.
Let y ∈ N ( V ) such that y = 0 . Then there exists x ∈ N + ( V ) ⊂ W φ ( V ) such that x.y is a non-zero constant. Explicit construction of pairing.
Let V ∈ HH Y D be a finite dimensionalYetter-Drinfel’d module. Then its linear dual V ∗ ∈ HH Y D is also a Yetter-Drinfel’dmodule such that the evaluation map ev : V ∗ ⊗ V → C is in HH Y D (see [3], Section1.2).As both V and V ∗ are braided vector spaces with braidings coming from theYetter-Drinfel’d module structures, both T ( V ) and T ( V ∗ ) are braided Hopf alge-bras. The canonical pairing V ∗ ⊗ V → C given by the evaluation map f ⊗ v f ( v ) N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 28 extends to a generalized pairing φ : T ( V ∗ ) ⊗ T ( V ) → C . Moreover, the radicals of this pairing are I ( V ∗ ) and I ( V ) , respectively ([1], Example3.2.23, Definition 3.2.26), so it descends to a non-degenerate generalized pairing φ : N ( V ∗ ) ⊗ N ( V ) → C . This is the main ingredient we will use to prove the main theorem.8.
Applications to Nichols algebras
At first, we want to show that the differential algebra above generalizes theskew-derivation defined by N.D. Nichols [18], explicitly used in M. Graña [8] and I.Heckenberger in [10] for Nichols algebras of diagonal type.8.1.
Derivations.
We keep notations from previous sections, fix a basis v , · · · , v n of V and a dual basis v ∗ , · · · , v ∗ n with respect to the evaluation map. Definition 14.
For a ∈ T ( V ∗ ) , we define the left derivation ∂ La : T ( V ) → T ( V ) by: for y ∈ N ( V ) , ∂ La ( y ) = X φ ( a, y (1) ) y (2) . If a = v ∗ i , the notation ∂ Li is adopted for ∂ Lv ∗ i . The left derivation ∂ La descends to N ( V ) and gives ∂ La : N ( V ) → N ( V ) .In the proposition below, we suppose that the Nichols algebra is of diagonal type. Proposition 19.
For i = 1 , · · · , n , the definition of ∂ Li above coincides with theone given in [10] .Proof. From the definition of ∂ Li , ∂ Li ( v i · · · v i k ) = X φ ( v ∗ i , ( v i · · · v i k ) (1) )( v i · · · v i k ) (2) . Then after the definition of the coproduct in Nichols algebras, the fact that σ is ofdiagonal type and φ is graded, we obtain that the terms satisfying φ ( v ∗ i , ( v i · · · v i k ) (1) ) = 0 are contained in those given by the shuffle action S ,k − in the coproduct formula.A simple calculation shows that S ,k − = { , σ , σ σ , · · · , σ · · · σ k − } , which gives exactly the formula in the definition after Heckenberger. (cid:3) Remark 10.
The advantage of our definition for differential operators on Nicholsalgebras are twofold: (1)
This is a global and functorial construction, we never need to work in aspecific coordinate system at the beginning;
N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 29 (2)
We make no assumption on the type of the braiding, it has less restrictionand can be applied to more general cases, for example: Hecke type, quantumgroup type, and so on.
Remark 11.
In the same spirit, for a ∈ T ( V ∗ ) , the right differential operator ∂ Ra can be similarly defined by considering the right action: for y ∈ T ( V ) , ∂ Ra ( y ) = X y (1) φ ( a, y (2) ) . The left derivation ∂ Ra descends to N ( V ) and gives ∂ Ra : N ( V ) → N ( V ) . Some results from [10] can be generalized with simple proofs.
Lemma 9.
Let x ∈ T ( V ) and a ∈ T ( V ∗ ) . Then: ∆( ∂ La ( x )) = X ∂ La ( x (1) ) ⊗ x (2) , ∆( ∂ Ra ( x )) = X x (1) ⊗ ∂ Ra ( x (2) ) . Proof.
We prove it for ∂ La : ∆( ∂ La ( x )) = ∆ (cid:16)X φ ( a, x (1) ) x (2) (cid:17) = X φ ( a, x (1) ) x (2) ⊗ x (3) = X ∂ La ( x (1) ) ⊗ x (2) . (cid:3) Moreover, we have following results:
Lemma 10.
For any a, b ∈ T ( V ∗ ) , ∂ La ∂ Rb = ∂ Rb ∂ La . Lemma 11.
For x, y ∈ T ( V ) and ≤ i ≤ n , we have: ∂ Li ( xy ) = ∂ Li ( x ) y + X x (0) φ (( v ∗ i ) ( − , x ( − ) ∂ L ( v ∗ i ) (0) ( y ) . Proof.
At first, from the definition, ∆( xy ) = X x (1) (( x (2) ) ( − .y (1) ) ⊗ ( x (2) ) (0) y (2) . so the action of ∂ Li gives: ∂ Li ( xy ) = X φ ( v ∗ i , x (1) (( x (2) ) ( − .y (1) ))( x (2) ) (0) y (2) = X (cid:0) φ ( v ∗ i , x (1) ) ε (( x (2) ) ( − .y (1) ) + ε ( x (1) ) φ ( v ∗ i , ( x (2) ) ( − .y (1) ) (cid:1) ( x (2) ) (0) y (2) = X φ ( v ∗ i , x ) y + φ ( v ∗ i , x ( − .y (1) ) x (0) y (2) = ∂ Li ( x ) y + X φ (( v ∗ i ) ( − , x ( − ) x (0) ∂ L ( v ∗ i ) (0) ( y ) . (cid:3) N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 30
Taylor Lemma.
This subsection is devoted to generalizing the Taylor Lemmain [13] to Nichols algebras of diagonal type. We keep notations in the last subsectionand suppose that G = Z n .For any x ∈ T ( V ∗ ) , the left derivation ∂ Lx : T ( V ) → T ( V ) will be denoted by ∂ x in this subsection.From now on, we fix a homogeneous primitive element w ∈ T ( V ∗ ) of degree α (the degree given by Z n ), denote q α,α = χ ( α, α ) and T ( V ) ∂ w = { v ∈ T ( V ) | ∂ w ( v ) = 0 } . Remark 12.
It is easy to see that if w ∈ T ( V ∗ ) is a non-constant element, ∂ w isa locally nilpotent linear map as it decreases the degree when acting on an element. Lemma 12 (Taylor Lemma) . Suppose that q is not a root of unity. If there existssome homogeneous element a ∈ T ( V ) such that ∂ w ( a ) = 1 , then a is free over T ( V ) ∂ w and as vector spaces, we have: T ( V ) = T ( V ) ∂ w ⊗ C C [ a ] . Proof.
Recall that α is the degree of w .As T ( V ) is Z n -graded and ∂ w is a linear map of degree − α , we may suppose that a is homogeneous of degree α .It is clear that T ( V ) ∂ w ⊗ C [ a ] ⊂ T ( V ) . Now we prove the other inclusion.As we are working under the diagonal hypothesis, a simple computation givesthat ∂ nw ( a n ) = ( n ) q α,α ! . Then if for some x i ∈ T ( V ) ∂ w , P x i a i = 0 , applying ∂ w sufficiently many times willforce all x i to be zero.Let x ∈ T ( V ) be a homogeneous element. Then so is ∂ w ( x ) . We let µ denotethe degree of x and n ∈ N a positive integer such that ∂ nw ( x ) = 0 but ∂ n +1 w ( x ) = 0 .If x ∈ T ( V ) ∂ w , the lemma is proved. Now we suppose that x / ∈ T ( V ) ∂ w , whichimplies that n > .We let λ denote the degree of ∂ nw ( x ) and q α,λ = χ ( α, λ ) , then ∂ nw ( ∂ nw ( x ) a n ) = q nα,λ ∂ nw ( x ) ∂ nw ( a n ) . If we define X = x − n ) q α,α ! 1 q nα,λ ∂ nw ( x ) a n , then X ≡ x (mod T ( V ) ∂ w ⊗ C [ a ] ) and ∂ nw ( X ) = ∂ nw ( x ) − n ) q α,α ! 1 q nα,λ q nα,λ ∂ nw ( x ) ∂ nw ( a n ) = 0 . So the lemma follows by induction on the nilpotent degree of x . (cid:3) For a homogeneous element b ∈ T ( V ) , we let ∂ ◦ b denote its degree. N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 31
Remark 13.
Let w ∈ T ( V ∗ ) be an element such that it has non-zero image in N ( V ) . As the pairing we are considering is non-degenerate, the element a alwaysexists if q is not a root of unity. We denote q ii = χ ( α i , α i ) and suppose that q is not a root of unity. Theorem 5.
Let w = v ∗ i for some ≤ i ≤ n and a ∈ T ( V ) be a homogeneouselement satisfying ∂ i ( a ) = 1 . We dispose − ∂ ◦ v i the degree of a and define theadjoint action of C [ a ] on T ( V ) ∂ i by: for a homogeneous element b ∈ T ( V ) ∂ i , a · b = ab − χ ( ∂ ◦ a, ∂ ◦ b ) ba. Then we can form the crossed product of C [ a ] and T ( V ) ∂ i with the help of thisaction, which is denoted by T ( V ) ∂ i ♯ C [ a ] . With this construction, the multiplicationgives an isomorphism of algebra: T ( V ) ∼ = T ( V ) ∂ i ♯ C [ a ] . Proof.
At first, we should show that the action defined above preserves T ( V ) ∂ i . Let b ∈ T ( V ) ∂ i and denote q i,b = χ ( ∂ ◦ v i , ∂ ◦ b ) . Then ∂ i ( a · b ) = ∂ i ( ab ) − q − i,b ∂ i ( ba )= ∂ i ( a ) b + χ ( ∂ ◦ v i , ∂ ◦ a ) a∂ i ( b ) − q − i,b ∂ i ( b ) a − q − i,b q i,b b∂ i ( a )= ∂ i ( a ) b − b∂ i ( a ) = 0 , where equations ∂ i ( b ) = 0 and ∂ i ( a ) = 1 are used. So the crossed product is welldefined.Note that a is primitive. We proceed to prove that the multiplication is analgebra morphism: what needs to be demonstrated is that for any m, n ∈ N andhomogeneous elements x, y ∈ T ( V ) ∂ w , ( x ⊗ a m )( y ⊗ a n ) = xa m ya n . From definition, it suffices to show that (1 ⊗ a m )( y ⊗
1) = a m y. At first, it should be pointed out that from the definition, the action of C [ a ] on T ( V ) ∂ i is just the commutator coming from a braiding. If we let Φ denote thelinear map in Theorem 1, then: (1 ⊗ a m )( y ⊗
1) = (Φ ⊗ id )(∆( a m )( y ⊗ . So it suffices to prove that m ◦ (Φ ⊗ id )(∆( a m )( y ⊗ a m y. We proceed to show this by induction.For m = 1 , m ◦ (Φ ⊗ id )(∆( a )( y ⊗ m ◦ (Φ ⊗ id )( ay ⊗ χ ( ∂ ◦ a, ∂ ◦ y ) y ⊗ a )= ay − χ ( ∂ ◦ a, ∂ ◦ y ) ya + χ ( ∂ ◦ a, ∂ ◦ y ) ya = ay. N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 32
For the general case, we denote ∆( a m − )( y ⊗
1) = P x ′ ⊗ x ′′ , then m ◦ (Φ ⊗ id )(∆( a m )( y ⊗ m ◦ (Φ ⊗ id ) (cid:16) ( a ⊗ ⊗ a ) (cid:16)X x ′ ⊗ x ′′ (cid:17)(cid:17) = m ◦ (Φ ⊗ id ) (cid:16)X ax ′ ⊗ x ′′ + X χ ( ∂ ◦ a, ∂ ◦ x ′ ) x ′ ⊗ ax ′′ (cid:17) = X a Φ( x ′ ) x ′′ − X χ ( ∂ ◦ a, ∂ ◦ x ′ )Φ( x ′ ) ax ′′ + X χ ( ∂ ◦ a, ∂ ◦ x ′ )Φ( x ′ ) ax ′′ = X a Φ( x ′ ) x ′′ = a m y, where the last equality comes from the induction hypothesis. (cid:3) Remark 14.
In the theorem, we need to take the opposite of the degree of a be-cause it acts as a differential operator, which has negative degree. But to get anisomorphism, a positive one is needed. Primitive elements
This last subsection is devoted to giving a proof of Theorem 2.At first, as in the construction of Section 7, let φ : T ( V ∗ ) ⊗ T ( V ) → C be ageneralized pairing which descends to a non-degenerate pairing φ : N ( V ∗ ) ⊗ N ( V ) → C between Nichols algebras.For x ∈ T ( V ) , we let ∆ i,j ( x ) denote the component of ∆( x ) of bidegree ( i, j ) . Proposition 20.
Let x ∈ ker( S n ) be a non-zero solution of equation S n x = 0 oflevel n . Then for any i = 1 , · · · , n , ∂ Ri ( x ) = 0 .Proof. From the definition of ∂ Ri and the fact that φ is graded, the possible non-zeroterms in ∂ Ri ( x ) are those belonging to ∆ n − , ( x ) in ∆( x ) .From the definition of the coproduct, ∆ n − , corresponds to the action of theelement X σ ∈ S n − , T σ . It is clear that S n − , = { , σ n − , σ n − σ n − , · · · , σ n − · · · σ } , so in fact, ∆ n − , corresponds to the part T n in the decomposition of S n . Thecondition of x being of level n means that T n x = 0 , thus ( id ⊗ φ ( v ∗ i , · )) ◦ ∆ n − , ( x ) = 0 , and so ∂ Ri ( x ) = 0 . (cid:3) Corollary 4.
With the assumption in the last proposition, for any non-constant a ∈ T ( V ∗ ) , we have ∂ Ra ( x ) = 0 . N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 33
Proposition 21.
Let x ∈ T ( V ) be a homogeneous element which is not a constant.If for any non-constant element a ∈ T ( V ∗ ) , ∂ Ra ( x ) = 0 , then ∆( x ) − x ⊗ ∈ T ( V ) ⊗ I ( V ) and x ∈ I ( V ) .Proof. If for any a ∈ T ( V ∗ ) , ∂ Ra ( x ) = 0 , then P x (1) φ ( a, x (2) ) = 0 for any a . Wechoose x (1) to be linearly independent. If x (2) is not a constant, it must be in theright radical of φ , which is exactly I ( V ) .So ∆( x ) − x ⊗ ∈ T ( V ) ⊗ I ( V ) . We obtain that x ∈ I ( V ) by applying ε ⊗ id onboth sides. (cid:3) It is clear that if x ∈ ker( S ) be a non-zero solution of S x = 0 with level , then x is primitive. Theorem 6.
Let n ≥ and x ∈ ker( S n ) be a non-zero solution of equation S n x = 0 with level n . Then x is primitive and it is in Im ( P n ) .Proof. The case n = 2 is clear.Let n > and x be a solution of equation S n x = 0 with level n . It suffices toprove that ∆ i,n − i ( x ) = 0 for any ≤ i ≤ n − .From the definition, components in ∆ i,n − i ( x ) can be obtained by acting a shuffleelement on x , we want to show that ∆ i,n − i ( x ) = P x ′ ⊗ x ′′ = 0 .From Corollary 4, x is of level n implies that for any non-constant a ∈ T ( V ∗ ) , ∂ Ra ( x ) = 0 . So from Proposition 21, ∆( x ) − x ⊗ ∈ T ( V ) ⊗ I ( V ) and then x ′′ ∈ I ( V ) , S n − i x ′′ = 0 . It is easy to see that there exists a positional embedding ι : B n − i ֒ → B n such that ι ( S n − i ) X σ ∈ S i,n − i T σ ( x ) = 0 , which means that the equation ι ( S n − i ) v = 0 has a non-zero solution in C [ X x ] . Itcontradicts Lemma 5. (cid:3) Corollary 5.
Let n ≥ and E n be the set of level n solutions of equation S n x = 0 in V ⊗ n . Then E n is a subspace of T ( V ) . If we denote P = L n ≥ E n , then P is acoideal and the ideal K generated by P in T ( V ) is contained in I ( V ) . List of notationsNotation(s) Section Notation(s) Section P i,j , P i,j , S k,n − k , T w X, E n Φ , N Π ks θ n , ∆ n B + H ( V ) , B − H ( V ) S n , T n , P n , T ′ n , L n N + ( V ) , N − ( V ) C [ X v ] , H ∂ La , ∂ Ra , ∂ Li , ∂ Ri N DEFINING IDEALS AND DIFFERENTIAL ALGEBRAS OF NICHOLS ALGEBRAS 34
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