Examples of Nichols algebras associated to upper triangular solutions of the Yang-Baxter equation in rank 3
aa r X i v : . [ m a t h . QA ] J u l EXAMPLES OF NICHOLS ALGEBRAS ASSOCIATED TOUPPER TRIANGULAR SOLUTIONS OF THEYANG–BAXTER EQUATION IN RANK 3
JOÃO MATHEUS JURY GIRALDI AND LEONARDO DUARTE SILVA
Abstract.
We determine some Nichols algebras that admit a non-trivial quadratic relation associated to some families of upper triangularsolutions of the Yang–Baxter equation of dimension 3. Introduction
Let k be a field. Nichols algebras are Hopf algebras in braided tensorcategories with very special properties; they appeared first in the work ofNichols [N] and play an important role in the classification of (pointed) Hopfalgebras [A, AS2].Recall that a braided vector space is a pair ( V, c ) where V is a vector spaceand c ∈ GL ( V ⊗ V ) , called a braiding , satisfies the braid equation : ( c ⊗ id)(id ⊗ c )( c ⊗ id) = (id ⊗ c )( c ⊗ id)(id ⊗ c ) . (1.1)We observe that the braid equation (1.1) is equivalent to the quantum Yang-Baxter equation (QYBE) (2.3). To each braided vector space one attachesa graded connected braided Hopf algebra, generated in degree one and withall primitive elements in this degree, called the Nichols algebra of ( V, c ) orsimply of V and denoted by B ( V ) .Despite this simple definition, the actual computation of the defining rela-tions and the dimension of B ( V ) is a hard task, however needed for the clas-sification program of Hopf algebras. A large amount of work on this questionis available, specially when ( V, c ) is of diagonal type [He1, He2, An1, An2]but not exclusively, see references in the survey [A].Assume that V has finite dimension and let ( v i , v i ) be dual bases for V .The braiding c (or even V ) is rigid if the map c ♭ : V ∗ ⊗ V → V ⊗ V ∗ givenby f ⊗ v P i (ev ⊗ id ⊗ id)( f ⊗ c ( v ⊗ v i ) ⊗ v i ) is bijective.If c is rigid, then B ( V ) can be realized as a braided Hopf algebra in thecategory of Yetter-Drinfeld modules HH YD over some Hopf algebra H . Thus,a new Hopf algebra B ( V ) H is obtained via the process of bosonization .For this reason we restrict our attention to rigid braidings. Mathematics Subject Classification.
Assume that k is algebraically closed field of characteristic zero. In [Hi1],Hietarinta classified all solutions of the QYBE in rank 2. Based on thisresult, in [AGi], the authors considered the rigid braidings which are notof diagonal type. They computed the Nichols algebras associated to themassuming that admit at least one quadratic relation.Later Hietarinta also classified the non-trivial (rigid) upper triangular so-lutions of the QYBE for dimension 3 in [Hi2]. The following question arisesnaturally: to study the Nichols algebras related to such solutions. It turnsout that this is considerably more difficult than the case of rank 2.In the present paper, we analyze the Nichols algebras associated to someof the families in [Hi2], namely ( R ,j ) j ∈ I . As in [AGi], we restrict to thosethat have at least one quadratic relation. Technically, there is a quadraticrelation if and only if t = 1 where t is a parameter to be explained in themain part of the paper. We compute most of the Nichols algebras like this.Here is our main result: Theorem 1.1.
Let ( V, c ) be the rigid braided vector space correspondingto some R -matrix R ,j . Then, B ( V ) has quadratic relations if and onlyif t = 1 . Furthermore, the explicit presentation of B ( V ) , a PBW-basis,the dimension and the GK-dimension are given if t = 1 and some extraconditions hold ( when needed ) , see Table 1. Table 1.
Examples of Nichols algebras of rank 3
Name Ref. B ( V ) Extra Conditions R , §3.1 Table 2 R , §3.2 Propositions 3.4, 3.7, Table 3 R , §3.3 Table 4 R , §3.4 Table 5 R , §3.5 Propositions 3.11, 3.12 t = 1 and p = 2 q , or t = − and p = 0 R , §3.6 Table 6 R , §3.7 Table 7 t = 1 and a = p − q ,or t = − and a = − p R , §3.8 Table 8 R , §3.9 Table 9 b = 1 R , §3.10 Table 10 To determine these Nichols algebras, we use the following technique, al-ready used in many papers.First, we look for relations in small degrees. For this, we use the quantumsymmetrizers Q n (2.1) and the following convenient fact: Let x ∈ T ( V ) . If ∂ f ( x ) = 0 for all f ∈ V ∗ , then x ∈ J ( V ) . See §2.1 for details and notation.Thus, we have a set of homogeneous relations. Write I for the ideal generatedby such relations. ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 3
Now, let B = T ( V ) /I be the pre-Nichols algebra and π : B ։ B ( V ) bethe natural projection. Suppose that there is a relation = r ∈ ker π whichwe assume homogeneous of minimal degree. Then we have that ∂ f ( r ) = 0 byminimality of the degree. If this leads to a contradiction, then π is actuallyan isomorphism. Otherwise, we get a new relation and repeat the process.Finally, we also mention that the constructions made in Subsections 3.2.1and 3.2.2 can be generalized for the cases which are excluded by the extraconditions. We obtain analogous relations to the ones presented in Proposi-tions 3.4 and 3.7. However, we were not able to prove that they are enough.The paper is organized as follows. In Section 2 we recall definitions andfacts that are used throughout the paper. Then, in Section 3 we do thecase-by-case analysis of the Nichols algebras with quadratic relations of thebraided vector spaces ( V, c ) corresponding to the families ( R ,j ) j ∈ I . Notation.
Let j, k ∈ Z . If j ≤ k , then we denote I j,k = { j, j + 1 , . . . , k } and I k = I ,k . If j > k , then I j,k = ∅ . Sums and products over an empty setof indices are 0 and 1, respectively.The maps χ o , χ e : Z −→ { , } stand for the odd and even characteristicfunctions. We denote the floor function by ⌊ ⌋ : R → Z . Acknowledgments.
The authors thank Nicolás Andruskiewitsch and IvánAngiono for fruitful conversations at different moments of our study.2.
Preliminaries
Yetter-Drinfeld modules and Nichols algebras.
Fix H a Hopfalgebra. A left Yetter-Drinfeld module M over H is a left H -module ( M, · ) and a left H -comodule ( M, λ ) that satisfies the compatibility condition λ ( h · m ) = h (1) m ( − S ( h (3) ) ⊗ h (2) · m (0) , h ∈ H, m ∈ M. The related category is denoted by HH YD . If S is bijective, then HH YD isbraided and monoidal: the braiding c M,N : M ⊗ N → N ⊗ M is given by c M,N ( m ⊗ n ) = m ( − · n ⊗ m (0) for all m ∈ M, n ∈ N and M, N ∈ HH YD . AHopf algebra in HH YD is called a braided Hopf algebra.Given V ∈ HH YD , we say that a braided N -graded Hopf algebra R = L n ≥ R ( n ) ∈ HH YD is a Nichols algebra for V if k ≃ R (0) , V ≃ R (1) ∈ HH YD , R (1) = P ( R ) and R is generated as algebra by R (1) .Nichols algebra always exists and is unique up to isomorphism. It isusually denoted by B ( V ) = L n ≥ B n ( V ) and is given by the quotient ofthe tensor algebra T ( V ) by the largest homogeneous Hopf ideal J ( V ) = L n ≥ J n ( V ) generated by homogeneous elements of degree ≥ .The ideal J ( V ) has an alternative description: J n ( V ) is the kernel of the n th quantum symmetrizer associated to cQ n = X σ ∈ S n ρ n ( M ( σ )) ∈ End( T n ( V )) , n ≥ , (2.1) GIRALDI AND SILVA where ρ n : B n → GL ( T n ( V )) , n ≥ , is the representation induced by c onthe braid group B n , and M is the Matsumoto section corresponding to thecanonical projection B n ։ S n . In particular, J ( V ) = ker (id + c ) . (2.2)Another way to find relations in B ( V ) is through left skew derivations.Let f ∈ V ∗ and consider ∂ f ∈ End T ( V ) given by ∂ f (1) = 0 , ∂ f ( v ) = f ( v ) , v ∈ V,∂ f ( xy ) = ∂ f ( x ) y + X x i ∂ f i ( y ) , where c − ( f ⊗ x ) = X x i ⊗ f i . A pre-Nichols algebra B is any graded braided Hopf algebra intermediatebetween T ( V ) and B ( V ) ; in other words, any braided Hopf algebra of theform T ( V ) /I where I ⊆ J ( V ) is a homogeneous Hopf ideal.The skew derivations are well behaved with respect to pre-Nichols al-gebras, that is, for all f ∈ V ∗ we can define ∂ f ∈ End B satisfying theproperties above and compatible with the projection T ( V ) ։ B .Finally, we recall the following result which is very used to determineNichols algebras. Let x ∈ B n with n ≥ . If ∂ f ( x ) = 0 for all f ∈ V ∗ , then x ∈ J n ( V ) .See [A], [AS1] for more details.2.2. Upper triangular solutions of the Yang–Baxter equation inrank 3.
Let R ∈ End( V ⊗ V ) . We say that the map R satisfies the quantumYang-Baxter equation (QYBE) if R R R = R R R . (2.3)A solution of the QYBE is also called a R -matrix. The QYBE (2.3) isequivalent to the braid equation (1.1) via R ←→ c = τ R, where τ : V ⊗ V → V ⊗ V is the usual flip, τ ( x ⊗ y ) = y ⊗ x .In [Hi2], Hietarinta classified all the non-trivial upper triangular solutionsof the QYBE in rank 3. It turns out that there are 35 families of solu-tions, enumerated by ( R ,j ) j ∈ I , ( R ,j ) j ∈ I , ( R ,j ) j ∈ I , ( R ,j ) j ∈ I , ( R ,j ) j ∈ I , ( R ,j ) j ∈ I , ( R ,j ) j ∈ I and R , . All of them are invertible and rigid.For this study, we focus on the families ( R ,j ) j ∈ I . Notice that we ho-mogenize the original solutions by a parameter t .Given a solution of the Yang-Baxter equation R = (cid:0) R R R R R R R R R (cid:1) , where each column matrix R ij is the transpose of (cid:0) r ij r ij r ij r ij r ij r ij r ij r ij r ij (cid:1) , we associate the following braiding c ( x i ⊗ x j ) = X k, ℓ ∈ I r ijkℓ x k ⊗ x ℓ . (2.4) ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 5
Colexicographic order.
Let ( A, ≺ A ) and ( B, ≺ B ) be two partiallyordered sets. Define on the Cartesian product A × B the following relation: ( a , b ) ≺ colexA × B ( a , b ) iff b ≺ B b or ( b = b and a ≺ A a ) . It is a partial order and known as the colexicographical order on A × B . If A and B are totally ordered, then it is a total order also.More generally, one can define the colexicographic order on set of all (non-commutative) finite words A N with alphabet A . Write k A N for the corre-sponding algebra of finite words.Let x ∈ k A N , x = 0 . Then there is a unique writing x = P i ∈ I n λ i x i , λ i ∈ k , x i ∈ A N , such that λ n = 0 and x j ≺ x n for all j ∈ I n − . We call x n themaximal term of x and denote it by max x .Observe that the same constructions can be done for the context of com-mutative words/polynomials. Example 2.1.
Let A = { a ≺ b } be a totally ordered set. With respect tothe colexicographic order, we have that ≺ a ≺ a ≺ a ≺ ba ≺ ba ≺ aba ≺ b a ≺ b ≺ ab ≺ a b ≺ bab ≺ b ≺ ab ≺ b , where stands for the word of length .3. Nichols algebras of rank three with quadratic relations
For this section, we set the following notation: B = { x a x a x a : 0 ≤ a i < } ; (3.1) B = { x a x a x a : a i ≥ a , a < } ; (3.2) B = { x a x a x a : a i ≥ } . (3.3)We always assume that the homogenizing parameter t = 0 .3.1. Case R , . Let c be the braiding associated to the solution of QYBE R , = t · ta · · · − ta · − tab · t · · · t ( a − b ) · − ta tp · · t · · · · · − tb · · · t · ta · t ( b − a ) − tp · · · · t · · · ·· · · · · t · · ·· · · · · · t · tb · · · · · · · t ·· · · · · · · · t . Just for this initial case, we write the braiding explicitly. See (2.4) for details. c ( x ⊗ x ) = t x ⊗ x ; c ( x ⊗ x ) = t x ⊗ x ; c ( x ⊗ x ) = t x ⊗ x + ta x ⊗ x ; c ( x ⊗ x ) = t x ⊗ x ; c ( x ⊗ x ) = t x ⊗ x ; c ( x ⊗ x ) = t x ⊗ x + ta x ⊗ x + t ( a − b ) x ⊗ x ; GIRALDI AND SILVA c ( x ⊗ x ) = t x ⊗ x − ta x ⊗ x ; c ( x ⊗ x ) = t x ⊗ x + t ( b − a ) x ⊗ x − ta x ⊗ x ; c ( x ⊗ x ) = t x ⊗ x + tb x ⊗ x − tp x ⊗ x − tb x ⊗ x + tp x ⊗ x − tab x ⊗ x . Proposition 3.1. If t = 1 , then there are no quadratic relations. Other-wise, the Nichols algebras are as in Table 2, where h x x + x x , x x + x x , x x + x x − b x x , (3.4) x , x , x − b x x + p x x i ; h x x − x x , x x − x x − a x , x x − x x + ( b − a ) x x i . (3.5) Table 2.
Nichols algebras of type R , . t J ( V ) Basis
GK-dim − (3.4) B (3.1) , dim = 81 (3.5) B (3.3) Proof.
The quadratic relations are obtained by (2.2). If t = − , then the pre-Nichols algebra T ( V ) /I where I is the ideal (3.4) is a 8-dimensional algebrawhose basis is B . In this case, it is clear that no more relations exist andthen B ( V ) = T ( V ) /I .If t = 1 , then we write B = T ( V ) /I where I is the ideal (3.5). Observethat B generates linearly B since the relations x x a = x a x + a a x a +11 and x x a = x a x + a (2 a − b ) x x a (3.6)hold in B , for all a , a ≥ . An easy calculation shows that c ( x a x a ⊗ x ) = ( x − a ( a + a ) x ) ⊗ x a x a + a ( b − a ) x ⊗ x a +11 x a − , for a , a ≥ , what implies that ∂ ( x a x a x a ) = x a x a ∂ ( x a ) , a , a , a ≥ . (3.7)As ∂ ( x a ) ∈ B a − , there are µ a ,c ,c ,c ∈ k , with c + c + c = a − and c i ≥ , such that ∂ ( x a ) = X c + c + c = a − , c i ≥ µ a ,c ,c ,c x c x c x c , a ≥ . (3.8)We claim that µ a , , ,a − = a . Indeed, the case a = 1 is obvious. Assumethat µ a , , ,a − = a , then ∂ ( x a +13 ) = x a + ( x + b x ) ∂ ( x a )= x a + ( x + b x ) X c + c + c = a − , c i ≥ µ a ,c ,c ,c x c x c x c . By (3.6), the claim holds.
ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 7
Suppose that the natural projection π : B → B ( V ) is not an isomorphism.Pick a linear homogeneous relation of minimal degree n ≥ r = X a + a + a = n, a i ≥ λ a ,a ,a x a x a x a . Thus, by (3.7) and (3.8), ∂ ( r ) = X a + a + a = na i ≥ λ a ,a ,a X c + c + c = a − c i ≥ µ a ,c ,c ,c x a + c x a + c x c . (3.9)Observe that the term x n − appears only one time in (3.9). By the min-imality of n and the claim above, we get λ , ,n = 0 . Inductively on k , weprove that λ j,k − j,n − k = 0 , j ∈ I ,k , for all k = 0 , , · · · , n − since, at eachstep k , the term x j x k − j x n − k shows up just one time in (3.9).Hence, the relation r can be rewritten as r = P λ a ,a , x a x a . Since ∂ ( x a x a ) = a x a − x a and ∂ ( x a x a ) = a x a x a − , we conclude that r = 0 , a contradiction. Therefore, there are no more relations and π is anisomorphism. (cid:3) Case R , . Let c be the braiding associated to the solution of QYBE R , = t · tb · · · tp · ta · t · · · t ( p − q ) · tq tk · · t · · · · · tp · · · t · tb · · − tk · · · · t · · · ·· · · · · t · · ·· · · · · · t · tb · · · · · · · t ·· · · · · · · · t . For the explicit presentation of c , use (2.4).By (2.2), it is clear that the Nichols algebra associated to c has a quadraticrelation if and only if t = 1 . To simplify the study of these Nichols algebras,we consider three subcases:(i) t = 1 and b = p − q ; (ii) t = − and b = − p ; (iii) t = 1 and b = p − q , or t = − and b = − p .Each subcase is treated separately in the next three subsections.3.2.1. Case (i) . Assume that t = 1 and b = p − q . Here, we classify theNichols algebras under these conditions. To do so, first we consider a suitableenvironment to deal with the problem. Then, we show two lemmas neededto prove the main result which is Proposition 3.4.By (2.2), we get that x x − x x , x x − x x + ( p − b )2 x (3.10) GIRALDI AND SILVA are the quadratic relations of B ( V ) . For n ≥ , define recursively z n = ( x − nb x ) z n − − z n − x + ( n − q − p ) (cid:18) − b − p (cid:19) n − x n x , where z := x . By induction on n , we prove that the terms z n satisfy thefollowing properties in B ( V ) : z n x i = x i z n , z n z m = z n z m ,c ( z n ⊗ x i ) = x i ⊗ z n , c ( z n ⊗ x ) = ( x + ( np + q ) x ) ⊗ z n ,∂ ( z n ) = 0 , ∂ ( z n ) = γ n x n , ∂ ( z n ) = β n z n − , (3.11)for all n ≥ , m ∈ I n − and i ∈ I , where γ n = n !( q − p ) (cid:18) − p − b (cid:19) n − and β n = − n (cid:18) n + 12 b + n − p + q (cid:19) . (3.12)We leave to reader the proof of these properties. We just observe that z n +1 z n − z n z n +1 , n ≥ , (3.13)are new relations and proved through derivations. We also mention someadditional identities that are necessary to prove the properties (3.11): c ( x n ⊗ x ) = ( x + np x ) ⊗ x n ,x x n = x n x − n (cid:18) p − b (cid:19) x n +11 , n ≥ . (3.14)Set B = T ( V ) /I where I ⊆ J ( V ) is the ideal generated by (3.10) and(3.13). Observe that properties (3.11) remain true in B . In particular, x z mn = z mn x + m z m − n z n +1 + m ( n + 1) b x z mn − m n !( q − p ) (cid:18) − b − p (cid:19) n x n +11 x z m − n (3.15)holds in B for all n, m ≥ . Hence, it follows that B ∞ = { x a x a z b z b · · · z b n n x a : n ≥ a i , b j ≥ } generates linearly the pre-Nichols algebra B since by (3.14) and (3.15), k B ∞ is a left ideal of B which contains its unit.Using the properties (3.11), we prove that ∂ i ( z b n n ) = b n ∂ i ( z n ) z b n − n , i ∈ I , n ≥ . Thus, we obtain that ∂ i ( x a x a z b · · · z b n n ) = ∂ i ( x a ) x a z b · · · z b n n + x a ∂ i ( x a ) z b · · · z b n n + x a x a ∂ i ( z b ) · · · z b n n + · · · + x a x a z b · · · ∂ i ( z b n n ) , for all i ∈ I . In particular, ∂ ( x a x a z b · · · z b n n ) = 0 . ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 9
Lemma 3.2.
For a i , b j ≥ and n > , the following identities hold in B : ∂ i ( Y x a ) = ∂ i ( Y ) x a + Y ( ∂ i ( x a ) + δ i, α ∂ ( x a )) , i ∈ I , (3.16) ∂ ( x n ) = X i ∈ I ,n − (cid:18) ni + 1 (cid:19) x i x n − − i Y j ∈ I i (cid:18) b + ( j − b − p (cid:19) , (3.17) where Y = x a x a z b · · · z b n n and α = a p + a q + P j ∈ I n b j ( jp + q ) .Proof. For all n, m ≥ , we have c ( z mn ⊗ x ) = ( x + m ( np + q ) x ) ⊗ z mn ,c ( x m ⊗ x ) = ( x + mp x ) ⊗ x m . Then, c ( Y ⊗ x ) = ( x + αx ) ⊗ Y . In particular, (3.16) follows since c ( Y ⊗ x i ) = x i ⊗ Y , for each i ∈ I . To obtain (3.17), first observe that ∂ ( x n +13 ) = ( bx + x ) ∂ ( x n ) . Then, ∂ ( x n ) = X i ∈ I ,n − ( bx + x ) i x n − − i . But, for n ≥ , ( bx + x ) n = P i ∈ I ,n (cid:0) ni (cid:1) x i x n − i Q j ∈ I i (cid:16) b + ( j − b − p (cid:17) . Therefore, ∂ ( x n ) = X ℓ ∈ I ,n − (cid:18) X i ∈ I ,ℓ (cid:18) ℓi (cid:19) x i x ℓ − i Y j ∈ I i (cid:18) b + ( j − b − p (cid:19) (cid:19) x n − − ℓ = X i ∈ I ,n − (cid:18) ni + 1 (cid:19) x i x n − − i Y j ∈ I i (cid:18) b + ( j − b − p (cid:19) . (cid:3) Lemma 3.3.
For n ≥ , define the matrix M n of order n + 1 given by β n P · · · β n β n − P β n − P · · · Q j ∈ I n − ,n β j P β n − β n − P β n − P · · · ... ... ... ... . . . ... ... Q j ∈ I n β j P n Q j ∈ I n − β j P n Q j ∈ I n − β j P n Q j ∈ I n − β j · · · P n − n β P nn γ n γ n − γ n − γ n − · · · γ , (3.18) where β j and γ j are as in (3.12) , and P ji = i ! / ( i − j )! . Then, det M n = (cid:18) n + 12 (cid:19) ( p − b − q ) Y j ∈ I n − j ! β j . (3.19) Proof.
After a cumbersome computation, we obtain that det M n = (cid:18) Y i ∈ I n i ! (cid:19) X j ∈ I ,n ( − j γ j ( n − j )! Y i ∈ I j +1 ,n β i , n ≥ , where γ = 1 . Then (3.19) follows since X j ∈ I n − ℓ,n ( − j γ j ( n − j )! Y i ∈ I j +1 ,n β i = n ( − n − ℓ ℓ ! ( n − ℓ ) γ n − ℓ Y i ∈ I n − ℓ,n − β i , ℓ ∈ I ,n − . (cid:3) Proposition 3.4.
Assume that t = 1 and b = p − q . (a) If β n = 0 for all n ∈ N , then J ( V ) = h (3.10) , (3.13) i , B ∞ is a PBW-basis of B ( V ) and GK-dim B ( V ) = ∞ .(b) Otherwise, there is a unique N ∈ N such that β N = 0 ; then J ( V ) = h (3.10) , (3.13) , z N +1 +( N +1)( b + p ) x z N i , B N +3 = { x a x a z b z b · · · z b N N x a : a i , b j ≥ } is a PBW-basis of B ( V ) and GK-dim B ( V ) = N + 3 . Proof.
Write B = T ( V ) /I where I = h (3.10) , (3.13) i . We already provedthat I ⊂ J ( V ) and B ∞ generates linearly B . Let π : B → B ( V ) be thenatural projection.(a) Assume that β n = 0 for all n ∈ N . Suppose that π is not an isomorphismand pick a linear homogeneous relation of minimal degree d ≥ = r = X λ b , ··· ,b n a ,a ,a x a x a z b z b · · · z b n n x a ∈ k B ∞ . We also denote λ b , ··· ,b n a ,a ,a by λ y , if y = x a x a z b · · · z b n n x a . Order the monomials B ∞ via the colexicographic order ≺ considering that x ≺ x ≺ z ≺ z ≺ · · · ≺ z n ≺ · · · ≺ x . See §2.3 for details and notation. Let X = x ˜ a x ˜ a z ˜ b z ˜ b · · · z ˜ b n n x ˜ a be themaximal term of the relation r . In particular, λ X = 0 .If ˜ a = 0 , then the maximal term of ∂ ( X ) is max ∂ ( X ) = x ˜ a − x ˜ a z ˜ b z ˜ b · · · z ˜ b n n x ˜ a . Define B d ≺ X = { y ∈ B ∞ : deg( y ) = d and y ≺ X } . Observe that the terms of ∂ ( y ) are lower than max ∂ ( X ) for all y ∈ B d ≺ X . Thus, the term max ∂ ( X ) appears just one time in ∂ ( r ) = 0 what leads to λ X = 0 what contradictsthe maximality of X . Hence ˜ a = 0 and X = x ˜ a z ˜ b z ˜ b · · · z ˜ b n n x ˜ a .Similarly, ˜ a = 0 ; otherwise, max ∂ ( X ) = x ˜ a − z ˜ b z ˜ b · · · z ˜ b n n x ˜ a and weapply the previous argument again. Consequently, X = z ˜ b z ˜ b · · · z ˜ b n n x ˜ a .Let B On the other hand, max ∂ ( ω , ) = x z ˜ b − z ˜ b · · · z ˜ b n n x ˜ a and, for all y ∈ B d ≺ X − { ω , } , the terms of ∂ ( y ) are lower than max ∂ ( ω , ) . In particular, ∂ ( ω , )) ∗ ( ∂ ( r ))= (max ∂ ( ω , )) ∗ ( λ X ∂ ( X ) + λ ω , ∂ ( ω , )) = ˜ b γ λ X + λ ω , . (3.21)Equations (3.20) and (3.21) give rise to a homogeneous system whoseassociated matrix is (cid:18) ˜ b β b γ (cid:19) . By Lemma 3.3, the determinant of suchmatrix is ˜ b ( p − b − q ) = 0 what implies that λ X = 0 , a contradiction.Hence ˜ b = 0 and X = z ˜ b · · · z ˜ b n n x ˜ a .Inductively, assume that ˜ b i = 0 for i ∈ I j − and suppose ˜ b j = 0 . Write ω j,i = x i z j − i z ˜ b j − j z ˜ b j +1 j +1 · · · z ˜ b n n x ˜ a , for i ∈ I ,j . Note that ω j, = X and ω j,i ∈ B d ≺ X , i ∈ I j . It holds that max ∂ ℓ ( X ) = z j − ℓ z ˜ b j − j z ˜ b j +1 j +1 · · · z ˜ b n n x ˜ a , ℓ ∈ I j .Moreover, if i > ℓ , then all the terms of ∂ ℓ ( ω j,i ) are lower than max ∂ ℓ ( X ) ;if i ≤ ℓ , then max ∂ ℓ ( ω j,i ) = max ∂ ℓ ( X ) and the list { ω j,i } i ∈ I j is exhaustive,that is, when we apply ∂ ℓ , ℓ ∈ I j , just the elements { ω j,i } i ∈ I ℓ of B d ≺ X “con-tribute” in the direction of max ∂ ℓ ( X ) . In particular, for all ℓ ∈ I j , ∂ ℓ ( X )) ∗ ( ∂ ℓ ( r )) = (max ∂ ℓ ( X )) ∗ (cid:18) X i ∈ I ,ℓ λ ω j,i ∂ ℓ ( ω j,i ) (cid:19) = λ X ˜ b j Y θ ∈ I j − ℓ +1 ,j β θ + X i ∈ I ℓ − λ ω j,i P iℓ Y θ ∈ I j − ℓ +1 ,j − i β θ . (3.22)Further, max ∂ ( ω j,j ) = x j z ˜ b j − j z ˜ b j +1 j +1 · · · z ˜ b n n x ˜ a and the terms of ∂ ( y ) arelower than max ∂ ( ω j,j ) for all y ∈ B d ≺ X − { ω j,i } i ∈ I j . Then, ∂ ( ω j,j )) ∗ ( ∂ ( r )) = (max ∂ ( ω j,j )) ∗ (cid:18) X i ∈ I ,j λ ω j,i ∂ ( ω j,i ) (cid:19) = ˜ b j γ j λ X + X i ∈ I j − γ j − i λ ω j,i + λ ω j,j . (3.23)The matrix associated to the homogeneous system obtained from equa-tions (3.22) and (3.23) is equals to the matrix M j (3.18) with exception of thefirst column, which is the original one multiplied by the scalar ˜ b j . By Lemma3.3, the determinant of this matrix is ˜ b j (cid:0) j +12 (cid:1) ( p − b − q ) Q i ∈ I j − i ! β i = 0 ,what implies that λ X = 0 , a contradiction. Hence ˜ b j = 0 .After the whole inductive process, we get X = x ˜ a . As ˜ a = d ≥ , then max ∂ ( X ) = x ˜ a − by Lemma 3.2. Arguing as above, we have another con-tradiction. Therefore, there are no more relations in B and π is, in fact, anisomorphism. In particular, B ∞ is a basis of B ( V ) and GK-dim B ( V ) = ∞ . (b) Assume that β N = 0 for some N ∈ N . An easy calculation shows thatsuch N is unique. Clearly ∂ i ( z N +1 + ( N + 1)( b + p ) x z N ) = 0 for all i ∈ I .Then consider the pre-Nichols algebra e B = T ( V ) / e I where e I = h (3.10) , (3.13) , z N +1 + ( N + 1)( b + p ) x z N i . By induction on i , we see that z N + i = ( N + i )! N ! (cid:18) − b − p (cid:19) i x i z N holds in e B , i ≥ . In particular, B N +3 = { x a x a z b z b · · · z b N N x a : a i , b j ≥ } generates lin-early e B .To get that B ( V ) = e B , we proceed as in the proof of case (a): we supposethe existence of another relation what leads to a contradiction. The fact that β N = 0 does not affect the previous arguments because to verify that ˜ b j = 0 for j ∈ I N , we just need that β i = 0 for i ∈ I N − . (cid:3) Case (ii) . Assume that t = − and b = − p . We adapt the strategyused in §3.2.1 to classify the Nichols algebras under these conditions.By (2.2), the quadratic relations of B ( V ) are x x + x x , x i , i ∈ I . (3.24)Set x = x x + x x and note that x x = x x . Further, the relations x x − x x and x x − x x + ( p − b ) x x (3.25)hold in B ( V ) .For n ≥ , define recursively z n = ( x − nb x ) z n − ++ ( + z n − x + (cid:0) n − (cid:1) !( p − q )( − p − b ) n − x x x n − if n is odd , − z n − x − (cid:0) n − (cid:1) !( p − q )( − p − b ) n − x x n if n is even , assuming that z := x . Then, the following properties z n x i = ( − n +1 x i z n , z n x = x z n ,z n z m = ( − ( n +1)( m +1) z n z m , z n = 0 c ( z n ⊗ x i ) = ( − n +1 x i ⊗ z n ,c ( z n ⊗ x ) = ( − n +1 ( x + ( np + q ) x ) ⊗ z n ,∂ ( z n ) = 0 , ∂ ( z n ) = e γ n x x n − , ∂ ( z n ) = e β n z n − , (3.26)hold in B ( V ) for all n ≥ , m ∈ I n − and i ∈ I , where e γ n = χ o ( n ) (cid:18) n − (cid:19) !( q − p ) ( − p − b ) n − and e β n = ( − (cid:0) n +12 b + n − p + q (cid:1) if n is odd , − n ( b + p ) if n is even . (3.27)We observe that the relations z n z n − − z n − z n , z n , n ≥ , (3.28) ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 13 are new and proved by derivations. We present the following identities thatare necessary to show the properties (3.26): c ( x ⊗ x i ) = x i ⊗ x ,∂ j ( x n ) = − δ j, n ( b + p ) x x n − ,c ( x n ⊗ x ) = ( x + 2 np x ) ⊗ x n , n ≥ , i ∈ I , j ∈ I . Consider the ideal I ⊆ J ( V ) generated by (3.24), (3.25) and (3.28), andset B = T ( V ) /I . Thus, for m, i, j ≥ , i odd and j even, x x = x − x x ,x x m = x m x − m ( p − b ) x x m ,x z mi = z mi x + m z m − i z i +1 + m ( i + 1) b x z mi + m (cid:18) i − (cid:19) !( p − q ) ( − p − b ) i − x x i +12 z m − i ,x z j = z j +1 − z j x + ( j + 1) b x z j − (cid:18) j (cid:19) !( p − q ) ( − p − b ) j x x x j , hold in B . In particular, it follows that e B ∞ = { x a x a x a z b · · · z b n n x a : n ≥ a i , b j ≥ a , a , b j < } (3.29)is a system of linear generators of B . From now on, we use the notation a i , b j admitting always that they are suitable, that is, as in (3.29).Observe that ∂ i ( z n z m ) = ∂ i ( z n ) z m + ( − n +1 z n ∂ i ( z m ) , i ∈ I , n, m ≥ . Then, ∂ i ( z b n n ) = b n ∂ i ( z n ) z b n − n , for all i ∈ I , n ≥ . Hence, we get that ∂ i ( x a x a x a z b · · · z b n n ) = ∂ i ( x a ) x a x a z b · · · z b n n + ( − a x a ∂ i ( x a ) x a z b · · · z b n n + ( − a + a x a x a ∂ i ( x a ) z b · · · z b n n + · · · + ( − a + a +2 a + P i ∈ I n − ( i +1) b i x a x a x a z b · · · ∂ i ( z b n n ) , for all i ∈ I . Lemma 3.5. For a i , b j as in (3.29) and n > , the derivations ∂ i ( Y x a ) = ∂ i ( Y ) x a + ( − η Y ( ∂ i ( x a ) + δ i, α ∂ ( x a )) ,∂ ( x n ) = − P i ∈ I , n − (cid:0) n i +1 (cid:1) x x i x n − i − Q j ∈ I i +1 jb − ( j − p if n is even , P i ∈ I , n − (cid:0) n − i (cid:1) x i x n − i − Q j ∈ I i jb − ( j − p − P i ∈ I , n − (cid:0) n − i +1 (cid:1) x x i x n − i − Q j ∈ I i +1 jb − ( j − p if n is odd , hold in B , where Y = x a x a x a z b · · · z b n n , η = a + a + 2 a + P i ∈ I n ( i + 1) b i and α = a p + a q + 2 a p + P j ∈ I n b j ( jp + q ) .Proof. Similar to Lemma 3.2. We just mention that, for n > , ( − bx − x ) n = x n + P i ∈ I n (cid:0) n i (cid:1) x i x n − i Q j ∈ I i jb − ( j − p if n is even , − x n − P i ∈ I n − (cid:0) n − i (cid:1) x i x n − i Q j ∈ I i jb − ( j − p − P i ∈ I , n − (cid:0) n − i (cid:1) x x i x n − i − Q j ∈ I i +1 jb − ( j − p if n is odd . (cid:3) Lemma 3.6. For n ≥ , let f M n = ( a ij ) i, j ∈ I n +1 be the matrix given by a ij = if j ≥ i + 2 , Q s ∈ I ℓ e β s if j = i + 1 , if i, j ∈ I n , j ≤ i and i, j are even, (cid:0) ℓθ (cid:1) Q s ∈ I θ e β s Q s ∈ I i − j +1 e β n +2 − j − s if i, j ∈ I n , j ≤ i and i or j is odd, χ o ( j ) e γ n +1 − j if i = n + 1 and j ∈ I n , ( − n if i = j = n + 1 . where ℓ = (cid:4) i (cid:5) , θ = j j − k , e β j and e γ j are as in (3.27) . Then, for m ≥ , det f M m = ( − m Y j ∈ I m e β j − e β m − j )+12 j , det f M m − = ( − m m e β Y j ∈ I m − e β j − e β m − j )2 j . Proof. Analogous to Lemma 3.3. (cid:3) Proposition 3.7. Assume that t = − and b = − p . (a) If e β n = 0 for all n ∈ N , then J ( V ) = h (3.24) , (3.25) , (3.28) i , e B ∞ is aPBW-basis of B ( V ) and GK-dim B ( V ) = ∞ .(b) Otherwise, there is a unique odd number N ∈ N such that e β N = 0 ;then J ( V ) = h (3.24) , (3.25) , (3.28) , z N +1 + ( N + 1)( b + p ) x z N i , e B N +52 = { x a x a x a z b z b · · · z b N N x a : a i , b j ≥ a , a , b j < } is a PBW-basis of B ( V ) and GK-dim B ( V ) = N +52 . Proof. We follow the same strategy and notation adopted in Proposition 3.4.Set x ≺ x ≺ x ≺ z ≺ z ≺ · · · ≺ x and X = x ˆ a x ˆ a x ˆ a z ˆ b z ˆ b · · · z ˆ b n n x ˆ a .By the same reason that ˜ a vanishes, we have that ˆ a i = 0 , i ∈ I . We alsoget ˆ b j = 0 applying Lemma 3.6; here, ω j,i = x i − ⌊ i ⌋ x ⌊ i ⌋ z j − i z ˆ b j − j z ˆ b j +1 j +1 · · · z ˆ b n n x ˆ a , i ∈ I ,j . ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 15 Hence, X = x ˆ a . If b = 0 and ˆ a is even, then ∂ ( x ˆ a ) = 0 by Lemma 3.5.To avoid this problem, we see that ( x ˆ a − ) ∗ ( ∂ ( x ˆ a )) = ( − ˆ a +1 p j ˆ a k since ∂ ( x n +13 ) = − ( x + b x ) ∂ ( x n ) + ( q − p ) x ∂ ( x n ) − ( p x + k x + a x ) ∂ ( x n ) , n > . For part (b), use that the following identity holds in e B : z N + i = (cid:4) N + i (cid:5) ! (cid:0) N − (cid:1) ! ( − b − p ) ⌊ i +12 ⌋ x i − ⌊ i ⌋ x ⌊ i ⌋ z N , i ≥ . (cid:3) Case (iii) . Here we classify the Nichols algebras associated to R , such that t = 1 and are not covered by cases (i) and (ii). Proposition 3.8. Assume that t = 1 and b = p − q , or t = − and b = − p .Then, the Nichols algebras are as in Table 3, where h x x − x x , x x − x x + q x , x x − x x + (2 q − p ) x x i ; (3.30) h x x + x x , x x + x x , x x + x x + p x x , x , x i ; (3.31) h x x + x x , x x + x x , x x + x x + p x x , (3.32) x , x , x + k x x + p x x i . Table 3. Nichols algebras of type R , (iii) Case t b a J ( V ) Basis GK-dim (a) p − q (3.30) B (3.3) (b) − − p = − p (3.31) B (3.2) (c) − − p − p (3.32) B (3.1) , dim = 8 Proof. By (2.2), the quadratic relations hold. Now we proceed the analysisof each case enumerated above separately.Case (a): This case follows analogously to the case t = 1 in Proposition 3.1.For completeness, we present here just some of the necessary identities: x x a = x a x − a q x a +11 , x x a = x a x + a ( p − q ) x x a ,c ( x a x a ⊗ x ) = ( x + ( a p + a q ) x ) ⊗ x a x a . Case (b): Write B = T ( V ) /I for the pre-Nichols algebra where I is the ideal(3.31). It is clear that B = { x a x a x a : a i ≥ a , a < } generateslinearly B . Observe that the following derivations hold in B : ∂ ( x a x a x a ) = δ a , x a x a − δ a , ( − a + a ⌊ ( a + a ) / ⌋ p x a x a − + δ a , ( − a ( ⌊ ( a + a ) / ⌋ p + χ o ( a ) q ) ( − x ) a x x a − − ⌊ a / ⌋ k (( − x ) a x a +12 x a − + χ o ( a ) q x a +11 x a +12 x a − ) + ⌊ a / ⌋ ( ⌊ ( a − / ⌋ p + a pq − a ) x a +11 x a x a − ,∂ ( x a x a x a ) = δ a , ( − x ) a x a + ⌊ a / ⌋ k x a +11 x a x a − ,∂ ( x a x a x a ) = ( − a + a χ o ( a ) x a x a x a − + ⌊ a / ⌋ p x a +11 x a x a − , for all a , a ∈ I , and a ≥ .Suppose that the natural projection π : B → B ( V ) is not an isomorphism.Let = r = λ x n + λ x x n − + λ x x n − + λ x x x n − ∈ ker π a relationof minimal degree n ≥ .If n is even, then we obtain λ = 0 and λ = nk λ since ∂ ( r ) = 0 . From ∂ ( r ) = 0 , we get λ = np λ . However ∂ ( r ) = − n ( p + a ) λ x x n − what implies λ = 0 because a = − p .If n is odd, then r = 0 using the following strategy: we seek a term thatshows up just one time in some of the equations ∂ i ( r ) = 0 , i ∈ I , and we usethe linear independence given by the minimality of n to vanish the respective λ j . This argument is applied until we get all λ j = 0 .Hence r = 0 in both circumstances and, in particular, π is an isomorphism.Case (c): Analogous to the case t = − of Proposition 3.1. (cid:3) Case R , . Let c be the braiding associated to the solution of QYBE R , = t · ta · · · − ta · − tab · t · · · · · − tb − tp · · t · · · · · − tb · · · t · tb · · tp · · · · t · · · tq · · · · · t · · ·· · · · · · t · tb · · · · · · · t ·· · · · · · · · t . Apply (2.4) for the explicit presentation of c . Proposition 3.9. If t = 1 , then there are no quadratic relations. Other-wise, the Nichols algebras are as in Table 4, where h x x − x x , x x − x x − a x , x x − x x − b x x i ; (3.33) h x x + x x , x x + x x , x x + x x − b x x , x , x i ; (3.34) h x x + x x , x x + x x , x x + x x − b x x , (3.35) x , x , x − b x x − p x x i . Proof. We get the quadratic relations through (2.2). Next we study the casesindividually.Case (a): It follows similarly to case t = 1 in Proposition 3.1. Some of thenecessary identities to prove it are: x x a = x a x + a a x a +11 , x x a = x a x + a b x x a , ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 17 Table 4. Nichols algebras of type R , Case t q J ( V ) Basis GK-dim (a) (3.33) B (3.3) (b) − = 0 (3.34) B (3.2) (c) − (3.35) B (3.1) , dim = 8 c ( x a x a ⊗ x ) = ( x − ( a a + a b ) x ) ⊗ x a x a . Case (b): Imitate case (b) of Proposition 3.8. For a , a ∈ I , and a ≥ ,the corresponding derivations are: ∂ ( x a x a x a ) = δ a , x a x a + p ⌊ a / ⌋ ( − x ) a x a +12 x a − + ( ⌊ a / ⌋ ( − a b − χ o ( a )( a a + a b )) ( − x ) a ( − x ) a x a − + b ⌊ a / ⌋ ( aχ e ( a ) + b ( a + ⌊ ( a − / ⌋ )) x a +11 x a x a − − pa ⌊ a / ⌋ χ o ( a ) x a +11 x a +12 x a − ,∂ ( x a x a x a ) = δ a , ( − x ) a x a − ⌊ a / ⌋ q ( − x ) a x a +12 x a − − ⌊ a / ⌋ x a +11 x a ( p x − χ o ( a ) qb x ) x a − ,∂ ( x a x a x a ) = χ o ( a ) ( − x ) a ( − x ) a x a − − ⌊ a / ⌋ b x a +11 x a x a − . Case (c): Analogous to case t = − of Proposition 3.1. (cid:3) Case R , . Let c be the braiding associated to the solution of QYBE R , = t · ta · · · − ta · − tab · t · · · · · t ( a − b ) tp · · t · · · · · − tb · · · t · t (2 b − a ) · · ·· · · · t · · · ·· · · · · t · · ·· · · · · · t · tb · · · · · · · t ·· · · · · · · · t . See (2.4) for details on the presentation of c . Proposition 3.10. If t = 1 , then there are no quadratic relations. Other-wise, the Nichols algebras are as in Table 5, where h x x − x x , x x − x x − a x , x x − x x + ( a − b ) x x i ; (3.36) h x x + x x , x x + x x , x x + x x + ( a − b ) x x , x , x i ; (3.37) h x x + x x , x x + x x , x x + x x + ( a − b ) x x , (3.38) x , x , x − b x x i . Table 5. Nichols algebras of type R , Case t p J ( V ) Basis GK-dim (a) (3.36) B (3.3) (b) − = 0 (3.37) B (3.2) (c) − (3.38) B (3.1) , dim = 8 Proof. Similar to Proposition 3.9. For completeness, we present next justsome of the necessary identities. For case (a): x x a = x a x + a a x a +11 , x x a = x a x − a ( a − b ) x x a ,c ( x a x a ⊗ x ) = ( x + ( a ( a − b ) − a a ) x ) ⊗ x a x a , and for case (b): ∂ ( x a x a x a ) = δ a , x a x a − p ⌊ a / ⌋ ( − x ) a x a +12 x a − + ( ⌊ a / ⌋ ( − a b − χ o ( a )( a a + a (2 b − a ))) ( − x ) a ( − x ) a x a − + b ⌊ a / ⌋ ( a ( χ e ( a ) − a ) + b (2 a + ⌊ ( a − / ⌋ )) x a +11 x a x a − + pb ⌊ a / ⌋ χ o ( a ) x a +11 x a +12 x a − ,∂ ( x a x a x a ) = δ a , ( − x ) a x a ,∂ ( x a x a x a ) = χ o ( a ) ( − x ) a ( − x ) a x a − − ⌊ a / ⌋ b x a +11 x a x a − . (cid:3) Case R , . Let c be the braiding associated to the solution of QYBE R , = t tℓ · − tℓ − tℓ t ( ℓq − k ) tp tk ta · t · · − tℓ · · tq tb · · t · · − tℓ · · tp · · · t tℓ · · t ( p − q ) − tb · · · · t · · · ·· · · · · t · · ·· · · · · · t tℓ ·· · · · · · · t ·· · · · · · · · t . Use (2.4) for the explicit presentation of c .The Nichols algebra associated to c has a quadratic relation if and only if t = 1 . In the next two results, we compute the Nichols algebras when t = 1 and p = 2 q , and t = − and p = 0 , respectively. Proposition 3.11. Assume that t = 1 and p = 2 q . Then J ( V ) = h x x − x x − ℓ x , x x − x x + q x ,x x − x x + ℓ x x + q x x + ( k − qℓ ) x i ,B (3.3) is a PBW-basis of B ( V ) and GK-dim B ( V ) = 3 . ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 19 Proof. The relations above follow by (2.2). Let B = T ( V ) /I be the pre-Nichols algebra where I is the ideal generated by these quadratic relations.Observe that, for a , a ≥ , the relations x x a = x a x + a ℓ x a +11 , x x a = x a x − a q x a +11 ,x x a = x a x − a ℓx x a − x − a qx x a + ( qℓ − k ) X i ∈ I a a ! ℓ i − ( a − i )! i x i +11 x a − i , hold in B . They guarantee that B is a system of linear generators of B .As c ( x a x a ⊗ x ) ∈ B ⊗ B a + a , we have it is equal to P i ∈ I x i ⊗ h i for some h i ∈ B a + a . We prove that h = P i ∈ I ,a a ! ℓ i ( a − i )! x a + i x a − i byinduction. Then, ∂ ( x a x a x a ) = X i ∈ I ,a a ! a ℓ i ( a − i )! x a + i x a − i x a − , a , a , a ≥ , (3.39)since ∂ ( x a ) = a x a − .Suppose that the natural projection π : B → B ( V ) is not injective. Picka linear homogeneous relation of minimal degree n ≥ r = X a + a + a = n, a i ≥ λ a ,a ,a x a x a x a . By (3.39), we have that ∂ ( r ) = X a + a + a = n, a i ≥ λ a ,a ,a a X i ∈ I ,a a ! ℓ i ( a − i )! x a + i x a − i x a − = X a ∈ I ,n − X i ∈ I ,a X a + a = n − a a ≥ , a ≥ λ a ,a ,a a ! a ℓ i ( a − i )! x a + i x a − i x a − . (3.40)Note that the term x n − appears just one time in (3.40). By the min-imality of n , we get λ ,n − , = 0 . Now, for each j ∈ I , , note that theterm x j x n − x − j shows up only one time in (3.40) and then we obtain λ j,n − , − j = 0 . Inductively on k , we have that λ j,n − k,k − j = 0 , j ∈ I ,k − , forall k ∈ I n . In particular, we can rewrite the relation r as P λ a ,a , x a x a .Applying that ∂ ( x a x a ) = X i ∈ I a a ! ℓ i − ( a − i )! i x a + i − x a − i , a ≥ , a ≥ , and repeating the previous procedure, we obtain that r = 0 , a contradiction.Hence, there are no more relations and π is an isomorphism. (cid:3) Proposition 3.12. Assume that t = − and p = 0 . If a = bℓ , then J ( V ) = h x x + x x , x x + x x , x x + x x − ℓ x x + q x x , x , x − ℓ x x i , B (3.2) is a PBW-basis of B ( V ) and GK-dim B ( V ) = 1 . If a = bℓ , then J ( V ) = h x x + x x , x x + x x , x x + x x − ℓ x x + q x x ,x , x − ℓ x x , x + b x x i ,B (3.1) is a PBW-basis of B ( V ) and dim B ( V ) = 8 .Proof. By (2.2), we get the relations above. If a = bℓ , then it is clear thatno more relations exist.If a = bℓ , this case follows the same lines of case (b) of Proposition 3.8.We present here only the derivations: ∂ ( x a ) = − ⌊ a / ⌋ ( a x x + b x x + χ o ( a ) qb x x ) x a − ,∂ ( x x a ) = x a + ⌊ a / ⌋ b x x x a − ,∂ ( x x a ) = ⌊ a / ⌋ ( bℓ − a ) x x x a − − χ o ( a ) ( k x + q x ) x a − ,∂ ( x x x a ) = x x a + ℓ x x a + χ o ( a ) q x x x a − ,∂ ( x a x a x a ) = δ a , x a (( − a x + χ o ( a ) qx ) x a − + ⌊ a / ⌋ bx a +11 x a x a − ,∂ ( x a x a x a ) = χ o ( a ) ( − x ) a ( − ℓ x − x ) a x a − . (cid:3) Case R , . Let c be the braiding associated to the solution of QYBE R , = t ta tp − ta − tab − t (2 pa + k ) − tp tk − tpq · t · · − tb · · − tp ·· · t · · − ta · · − tq · · · t tb tp · · ·· · · · t · · · ·· · · · · t · · ·· · · · · · t ta tq · · · · · · · t ·· · · · · · · · t . For the presentation of c apply (2.4). Proposition 3.13. If t = 1 , then there are no quadratic relations. Other-wise, the Nichols algebras are as in Table 6, where h x x + x x , x x + x x , x x + x x − a x x − p x x , (3.41) x , x − b x x , x − q x x i ; h x x − x x − a x , x x − x x − p x , (3.42) x x − x x + a x x − p x x + ( k + pa ) x i . Proof. The relations above hold by (2.2). Clearly, there are no more relationsif t = − . If t = 1 , then this case is similar to Proposition 3.11. Next, wepresent just some of necessary identities: x x a = x a x + a a x a +11 , x x a = x a x + a p x a +11 , ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 21 Table 6. Nichols algebras of type R , t J ( V ) Basis GK-dim − (3.41) B (3.1) , dim = 81 (3.42) B (3.3) x x a = x a x − a ax x a − x + a px x a − ( pa + k ) X i ∈ I a a ! a i − ( a − i )! i x i +11 x a − i ,∂ ( x a x a x a ) = X i ∈ I a (cid:18) a i (cid:19) Y j ∈ I ,i − ( ja + b ) x a − i x a − i x a ,∂ ( x a x a ) = X i ∈ I a (cid:18) a i (cid:19) Y j ∈ I ,i − ( jp + q ) x a − i x a − i . (cid:3) Case R , . Let c be the braiding associated to the solution of QYBE R , = t tk tp − tk − tkℓ td ta t ( k ( a − q ) − d ) tb · t · · − tℓ t ( p − q ) · ta ·· · t · · − tk · · ta · · · t tℓ tq · · ·· · · · t · · · ·· · · · · t · · ·· · · · · · t tk tp · · · · · · · t ·· · · · · · · · t . For details on the explicit presentation of c see (2.4).The Nichols algebra associated to c has a quadratic relation if and only if t = 1 . We calculate it for t = 1 and a = p − q , and t = − and a = − p . Proposition 3.14. Assume that t = 1 and a = p − q , or t = − and a = − p . Then, the Nichols algebras are as in Table 7, where h x x − x x − k x , x x − x x − q x , (3.43) x x − x x + k x x − q x x − ( d + qk ) x i ; h x x + x x , x x + x x , x x + x x − k x x − q x x , (3.44) x , x − ℓ x x i ; h x x + x x , x x + x x , x x + x x − k x x − q x x , (3.45) x , x − ℓ x x , x − p x x i . Proof. We get the quadratic relations through (2.2). Next we study the casesindividually. Table 7. Nichols algebras of type R , Case t a b J ( V ) Basis GK-dim (a) p − q (3.43) B (3.3) (b) − − p = − p (3.44) B (3.2) (c) − − p − p (3.45) B (3.1) , dim = 8 Case (a): It follows similarly to Proposition 3.11. Some of the necessaryidentities to prove it are: x x a = x a x + a k x a +11 , x x a = x a x + a q x a +11 ,x x a = x a x − a kx x a − x + a qx x a + ( d + qk ) X i ∈ I a a ! k i − ( a − i )! i x i +11 x a − i ,∂ ( x a x a x a ) = X i ∈ I a (cid:18) a i (cid:19) Y j ∈ I ,i − ( jk + ℓ ) x a − i x a − i x a ,∂ ( x a x a ) = X i ∈ I a (cid:18) a i (cid:19) Y j ∈ I ,i − ( p + jq ) x a − i x a − i . Case (b): Analogous to case (b) of Proposition 3.8. We just present thederivations: ∂ ( x a ) = ⌊ a / ⌋ (cid:0)(cid:0) ⌊ ( a − / ⌋ p − b (cid:1) x + ( − a p x (cid:1) x a − ,∂ ( x x a ) = x a + ( − a +1 ⌊ ( a + 1) / ⌋ p x x a − ,∂ ( x x a ) = ⌊ a / ⌋ (cid:0) ⌊ a / ⌋ p − b (cid:1) x x x a − + χ o ( a ) ( qk + d ) x x a − + ( − a +1 ⌊ ( a + 1) / ⌋ p ( x + k x ) x a − ,∂ ( x x x a ) = ( x + k x ) x a + ( − a ( ⌊ ( a + 1) / ⌋ + χ o ( a )) p x x x a − ∂ ( x a x a x a ) = δ a , ( − x ) a x a ,∂ ( x a x a x a ) = χ o ( a )( − x ) a ( − k x − x ) a x a − −⌊ a / ⌋ p x a +11 x a x a − . Case (c): Similar to the case t = − of Proposition 3.13. (cid:3) Case R , . Let c be the braiding associated to the solution of QYBE R , = t ta tq − ta − ta − tb − tq tb tap · t · · − ta tq · · tp · · t · · − ta · · ·· · · t ta · · − tq − tp · · · · t · · · ·· · · · · t · · ·· · · · · · t ta ·· · · · · · · t ·· · · · · · · · t . ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 23 For the presentation of c , utilize (2.4). Proposition 3.15. If t = 1 , then there are no quadratic relations. Other-wise, the Nichols algebras are as in Table 8, where h x x + x x , x x + x x , x x + x x − a x x + q x x , (3.46) x , x − a x x , x + p x x i ; h x x − x x − a x , x x − x x − q x , (3.47) x x − x x + a x x − q x x + b x i . Table 8. Nichols algebras of type R , t J ( V ) Basis GK-dim − (3.46) B (3.1) , dim = 81 (3.47) B (3.3) Proof. Analogous to Proposition 3.13. Some of the necessary equations toprove case t = 1 are: x x a = x a x + a a x a +11 , x x a = x a x + a q x a +11 ,x x a = x a x − a a x x a − x + a q x x a − b X i ∈ I a a ! a i − ( a − i )! i x i +11 x a − i ,∂ ( x a x a x a ) = a X i ∈ I ,a a ! a i ( a − i )! x a + i x a − i x a − ,∂ ( x a x a ) = X i ∈ I a a ! a i − ( a − i )! i x a − i x a − i . (cid:3) Case R , . Let c be the braiding associated to the solution of QYBE R , = t t · − t − t ta tb t ( b − a ) tp · t t · − t − t · tb t ( b − q ) · · t · · − t · · tb · · · t t · − t − t tq · · · · t t · − t − t · · · · · t · · − t · · · · · · t t ·· · · · · · · t t · · · · · · · · t . Apply (2.4) for the presentation of c .As previously, the Nichols algebra associated to c has a quadratic relationiff t = 1 . We compute it just when b = 1 also. Proposition 3.16. Assume that t = 1 and b = 1 . Then the Nichols algebrasare as in Table 9, where h x x − x x − x , x x − x x − x x , (3.48) x x − x x − x + x x + x x − a x i ; h x x + x x , x x + x x + x x , x x + x x − x x , (3.49) x , x − x x i ; h x x + x x , x x + x x + x x , x x + x x − x x , (3.50) x , x − x x , x − x x + x x − q x x i . Table 9. Nichols algebras of type R , Case t b p J ( V ) Basis GK-dim (a) (3.48) B (3.3) (b) − = − a − q (3.49) B (3.2) (c) − − a − q (3.50) B (3.1) , dim = 8 Proof. The quadratic relations above hold by (2.2). Next we proceed theanalysis of each case enumerated above separately.Case (a): We proceed as in case t = 1 of Proposition 3.1. First, we prove x x a = x a ( x + a x ) + (cid:18) a (cid:19) x a +11 , x a x a = X i ∈ I ,a T ia ,a x a + i x a − i ,x x a = x a x + a x a +12 − a x x a − x − (cid:18) a + 12 (cid:19) x x a + a X i ∈ I a a !( a − i )! i x i +11 x a − i , for all a , a ≥ , where T ia ,a = ( a + i − a − (cid:0) a i (cid:1) if a ≥ , and T i ,a = δ i, . Wealso have ∂ ( x a x a x a ) = X i ∈ I ,a a !( a − i )! x a + i x a − i ∂ ( x a ) , what implies that X a + a + a = na i ≥ λ a ,a ,a X c + c + c = a − c i ≥ µ a ,c ,c ,c X i ∈ I ,a X j ∈ I ,a − i a ! T jc ,a − i ( a − i )! x a + c + i + j x a + c − i − j x c . Finally we vanish all λ a i as in Proposition 3.11. To do so, we also use that ∂ ( x a x a ) = X i ∈ I a a !( a − i )! i x a − i x a − i . ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 25 Case (b): Similar to case (b) of Proposition 3.8. We only present the deriva-tions. ∂ ( x a ) = ( − a +1 j a k (cid:22) a + 12 (cid:23) x a − − ( p + χ o ( a ) a ) x x a − + j a k q x x a − + j a k(cid:18)j a k + χ o ( a ) a ( a − (cid:19) x ( x + x ) x a − − j a k(cid:18) j a k + 12 j a k − (cid:19) ( x x + x x + x x ) x a − − (cid:18) χ e ( a ) ( a − q + a )2 + χ o ( a )( p + a ) (cid:19) x x x a − ,∂ ( x x a ) = ( − a (cid:22) a + 12 (cid:23) (cid:22) a + 22 (cid:23) ( x + x ) x a − + j a k(cid:18)j a k + 2 j a k + j a k + (cid:22) a + 32 (cid:23) − p − q − a (cid:19) x x x a − + χ o ( a ) j a k (cid:18) j a k + 3 j a k − a (cid:19) x x x a − + χ o ( a ) a x x a − ,∂ ( x x a x a ) = x a x a + j a k (cid:18) j a k + 32 j a k + 56 − q (cid:19) x x a x a − + (cid:18) ( − a (cid:22) a + 12 (cid:23)(cid:22) a + 2 + 2 a (cid:23) − a χ o ( a ) (cid:19) x ( − x ) a x a − + x a x a ,∂ ( x a x a x a ) = δ a , ( − x ) a x a − ⌊ a / ⌋ q x a +11 x a ( χ o ( a ) x + x ) x a − − j a k(cid:18) j a k + (cid:18) χ o ( a ) + 4 a − (cid:19)j a k + 16 (cid:19) x a +11 x a ( x + x ) x a − +( − a ( ⌊ ( a + a + a ) / ⌋ + a a ( − a − )( − x ) a ( − x − x ) a x a − + ⌊ a / ⌋ ⌊ ( a + 2 a + 3 a ) / ⌋ ( − x ) a ( − x ) a x x a − ,∂ ( x a x a x a ) = ⌊ a / ⌋ ⌊ ( a − / ⌋ x a +11 x a ( x + x ) x a − + χ o ( a )( − x ) a ( − x − x ) a x a − + ( a + 1) ⌊ a / ⌋ ( − x ) a ( − x ) a +1 x a − . Case (c): Analogous to case t = − of Proposition 3.13. (cid:3) Case R , . Let c be the braiding associated to the solution of QYBE R , = t · − − a a · − a ( a + 7) · · − a − a − a · a ( a + 1) 6 a ( a + 2)( a − · · · · − a · · − a (1 − a ) · · · a − a (1 − a ) − − a ( a + 1) − a (1 + 3 a )( a − · · · · a · − a a (1 − a ) · · · · · · · − a · · · · · · a − − a )(1 − a ) · · · · · · · a − · · · · · · · · . Utilize (2.4) for the explicit presentation of c . Proposition 3.17. If t = 1 , then there are no quadratic relations. Other-wise, the Nichols algebras are as in Table 10, where h x x − x x − x , x x − x x − x x , x x − x x (3.51) − a x + (4 a − x x + (8 a − x x − a ( a + 1) x i ; h x x + x x , x x + x x + 2 x x , x x + x x (3.52) + 2(1 − a ) x x + 4(1 − a ) x x , x , x − a x x i ; h x x + x x , x x + x x + 2 x x , x x + x x + 2(1 − a ) x x (3.53) + 4(1 − a ) x x , x , x − a x x , x − a − x x − a ( a − a − x x − a ( a − a + 4) x x i . Table 10. Nichols algebras of type R , Case t a J ( V ) Basis GK-dim (a) (3.51) B (3.3) (b) − / ∈ {− , , } (3.52) B (3.2) (c) − ∈ {− , , } (3.53) B (3.1) , dim = 8 Proof. The relations above are obtained via (2.2). Next we study the casesindividually.Case (a): This case follows similarly to the case ( a ) of Proposition 3.16. Wehave that x x a = x a (cid:18) x + 2 a x + 4 (cid:18) a (cid:19) x (cid:19) , x a x a = X i ∈ I ,a i T ia ,a x a + i x a − i ,x x a = x a x + 2 a a x a +12 + X i ∈ I a A a i x i +11 x a − i + X i ∈ I a − B a i x i +11 x a − − i x − a (4 a − x x a − x − a (2 a − a ( a − 1) + 4) x x a , for some A a i , B a i ∈ k and T ia ,a as in Proposition 3.16. It also holds that ∂ ( x a x a x a ) = X i ∈ I ,a ⌊ i ⌋ (cid:18) a i (cid:19) C i D ⌊ i − ⌋ x a + i x a − i ∂ ( x a ) ,∂ ( x a x a ) = X i ∈ I a i − (cid:18) a i (cid:19) D i − x a − i x a − i , where C i = Q j ∈ I , ⌊ i − ⌋ (2 a + 2 j − and D i = Q j ∈ I ,i ( a + j ) . ICHOLS ALGEBRAS ASSOCIATED TO SOLUTIONS OF THE QYBE IN RANK 3 27 Case (b): We proceed similarly to the case (b) of Proposition 3.8. We justpresent the derivations of the term x a . ∂ ( x a ) = ( − a +1 a − ⌊ a / ⌋ ((2 a − ⌊ ( a − / ⌋ + a ) x a − − / ⌊ a / ⌋ (cid:16) E , a ⌊ a / ⌋ x x − E , a ⌊ a / ⌋ x x + E , a ⌊ a / ⌋ x x (cid:17) x a − ,∂ ( x a ) = 2(2 a − ⌊ a / ⌋ (( − a x + (( a − a − 1) + χ e ( a )) x ) x a − − / ⌊ a / ⌋ (cid:16) F , a ⌊ a / ⌋ x x a − + F , a ⌊ a / ⌋ x x x a − (cid:17) ,∂ ( x a ) = 4 a (2 a − (cid:4) ( a − / (cid:5) x x x a − − a − ⌊ a / ⌋ x x a − + 4(2 a − ⌊ a / ⌋ ((2 a − ⌊ ( a − / ⌋ + a ) x x a − + χ o ( a ) x a − , where E ,nr = a (64 r + 24(1 − χ e ( n )) r + (5 + 12 χ e ( n ))) − a (32 r + 8(1 − χ e ( n )) r + 7(1 + 2 χ e ( n )))+ 6 a (8 r + (1 − χ e ( n )) r + (1 + 7 χ e ( n ))) − r − χ e ( n ) r + (1 + 3 χ e ( n ))) ,E ,nr = 96 a (4 r − χ e ( n ) r − (1 − χ e ( n )) r − χ e ( n )) − a (48 r − χ e ( n )) r − − χ e ( n )) r + (1 − χ e ( n )))+ 6 a (96 r − χ e ( n )) r − − χ e ( n )) r + (1 − χ e ( n ))) − a (32 r − χ e ( n )) r − − χ e ( n )) r − χ e ( n )))+ 8(3 r − χ e ( n )) r + 12 χ e ( n ) r + (1 − χ e ( n ))) ,E ,nr = 2 a (64 r − χ e ( n )) r + (5 + 39 χ e ( n ))) − a (192 r + 16(7 − χ e ( n )) r − − χ e ( n )) r + (29 + 24 χ e ( n )))+ 6 a (256 r +16(1 − χ e ( n )) r − − χ e ( n )) r +(29 − χ e ( n ))) − a (144 r − χ e ( n )) r − − χ e ( n )) r − (8 + 33 χ e ( n )))+ 4 a (96 r − χ e ( n )) r − − χ e ( n )) r − χ e ( n ))) − r − (1 + 2 χ e ( n )) r − (1 − χ e ( n )) r + (1 − χ e ( n ))) ,F ,nr = a (128 r − χ e ( n )) r − (5 − χ e ( n ))) − a (64 r − χ e ( n )) r − (7 − χ e ( n )))+ 6 a (16 r − χ e ( n )) r − (1 − χ e ( n ))) − r − χ e ( n )) r − (1 − χ e ( n ))) ,F ,nr = 2 a (64 r − χ e ( n )) r + (5 + 51 χ e ( n ))) − a (32 r − χ e ( n )) r + (31 + 57 χ e ( n ))) − a (8 r + 3(3 − χ e ( n )) r − (8 − χ e ( n ))) + 8 a (10 r + 3(1 − χ e ( n )5) r − − χ e ( n ))) − r − χ e ( n ) r − (2 − χ e ( n ))) . 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