A finite-dimensional Lie algebra arising from a Nichols algebra of diagonal type (rank 2)
Nicolás Andruskiewitsch, Iván Angiono, Fiorela Rossi Bertone
aa r X i v : . [ m a t h . QA ] M a r A FINITE-DIMENSIONAL LIE ALGEBRA ARISING FROMA NICHOLS ALGEBRA OF DIAGONAL TYPE (RANK 2)
NICOL ´AS ANDRUSKIEWITSCH, IV ´AN ANGIONO, FIORELA ROSSI BERTONE
Abstract.
Let B q be a finite-dimensional Nichols algebra of diagonaltype corresponding to a matrix q ∈ k θ × θ . Let L q be the Lusztig algebraassociated to B q [AAR]. We present L q as an extension (as braided Hopfalgebras) of B q by Z q where Z q is isomorphic to the universal envelopingalgebra of a Lie algebra n q . We compute the Lie algebra n q when θ = 2. Introduction k be a field, algebraically closed and of characteristic zero. Let θ ∈ N , I = I θ := { , , ..., θ } . Let q = ( q ij ) i,j ∈ I be a matrix with entries in k × , V a vector space with a basis ( x i ) i ∈ I and c q ∈ GL ( V ⊗ V ) be given by c q ( x i ⊗ x j ) = q ij x j ⊗ x i , i, j ∈ I . Then ( c q ⊗ id)(id ⊗ c q )( c q ⊗ id) = (id ⊗ c q )( c q ⊗ id)(id ⊗ c q ), i.e. ( V, c q ) is abraided vector space and the corresponding Nichols algebra B q := B ( V ) iscalled of diagonal type. Recall that B q is the image of the unique map ofbraided Hopf algebras Ω : T ( V ) → T c ( V ) from the free associative algebraof V to the free associative coalgebra of V , such that Ω | V = id V . Forunexplained terminology and notation, we refer to [AS].Remarkably, the explicit classification of all q such that dim B q < ∞ isknown [H2] (we recall the list when θ = 2 in Table 1). Also, for every q inthe list of [H2], the defining relations are described in [A2, A3].1.2. Assume that dim B q < ∞ . Two infinite dimensional graded braidedHopf algebras e B q and L q (the Lusztig algebra of V ) were introduced andstudied in [A3, A5], respectively [AAR]. Indeed, e B q is a pre-Nichols, and L q a post-Nichols, algebra of V , meaning that e B q is intermediate between T ( V )and B q , while L q is intermediate between B q and T c ( V ). This is summarized Mathematics Subject Classification. in the following commutative diagram: T ( V ) Ω * * / / & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ B q & & ▲▲▲▲▲▲▲▲▲▲▲▲▲ / / T c ( V ) e B q π ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) L q < < ③③③③③③③③③ The algebras e B q and L q are generalizations of the positive parts of the DeConcini-Kac-Procesi quantum group, respectively the Lusztig quantum di-vided powers algebra. The distinguished pre-Nichols algebra e B q is defineddiscarding some of the relations in [A3], while L q is the graded dual of e B q .1.3. The following notions are discussed in Section 2. Let ∆ q + be the gener-alized positive root system of B q and let O q ⊂ ∆ q + be the set of Cartan rootsof q . Let x β be the root vector associated to β ∈ ∆ q + , let N β = ord q ββ andlet Z q be the subalgebra of e B q generated by x N β β , β ∈ O q . By [A5, Theorems4.10, 4.13], Z q is a braided normal Hopf subalgebra of e B q and Z q = co π e B q .Actually, Z q is a true commutative Hopf algebra provided that q N β αβ = 1 , ∀ α, β ∈ O q . (1)Let Z q be the graded dual of Z q ; under the assumption (1) Z q is a co-commutative Hopf algebra, hence it is isomorphic to the enveloping algebra U ( n q ) of the Lie algebra n q := P ( Z q ). We show in Section 3 that L q is anextension (as braided Hopf algebras) of B q by Z q : B q π ∗ ֒ → L q ι ∗ ։ Z q . (2)The main result of this paper is the determination of the Lie algebra n q when θ = 2 and the generalized Dynkin diagram of q is connected. Theorem 1.1.
Assume that dim B q < ∞ and θ = 2 . Then n q is either 0or isomorphic to g + , where g is a finite-dimensional semisimple Lie algebralisted in the last column of Table 1. Assume that there exists a Cartan matrix a = ( a ij ) of finite type, thatbecomes symmetric after multiplying with a diagonal ( d i ), and a root ofunit q of odd order (and relatively prime to 3 if a is of type G ) suchthat q ij = q d i a ij for all i, j ∈ I . Then (2) encodes the quantum Frobeniushomomorphism defined by Lusztig and Theorem 1.1 is a result from [L].The penultimate column of Table 1 indicates the type of q as establishedin [AA]. Thus, we associate Lie algebras in characteristic zero to some con-tragredient Lie (super)algebras in positive characteristic. In a forthcomingpaper we shall compute the Lie algebra n q for θ > IE ALGEBRAS ARISING FROM NICHOLS ALGEBRAS 3
Row Generalized Dynkin diagrams parameters Type of B q n q ≃ g + ❡ ❡ q q − q q = 1 Cartan A A ❡ ❡ q q − − ❡ ❡ − q − q = ± A A ❡ ❡ q q − q q = ± B B ❡ ❡ q q − − ❡ ❡ − q − q − q / ∈ G Super
B A ⊕ A ❡ ❡ ζ q − q ❡ ❡ ζ ζ − qζq − ζ ∈ G q br (2 , a ) A ⊕ A ❡ ❡ ζ − ζ − ❡ ❡ ζ − − ζ − − ζ ∈ G ′ Standard B ❡ ❡ − ζ − − ζ − ζ ❡ ❡ − ζ − ζ − − ❡ ❡ − ζ − ζ − ζ ∈ G ′ ufo (7) 0 ❡ ❡ − ζ ζ − ❡ ❡ − ζ − ζ − − ❡ ❡ − ζ ζ − ζ ❡ ❡ − ζ ζ − ❡ ❡ − ζ − − ζ − ζ ∈ G ′ ufo (8) A ❡ ❡ − ζ ζ − ζ ❡ ❡ ζ ζ − − ❡ ❡ − ζ ζ − ζ ∈ G ′ brj (2; 3) A ⊕ A ❡ ❡ q q − q q / ∈ G ∪ G Cartan G G ❡ ❡ ζ ζ ζ − ❡ ❡ ζ − ζ − − ❡ ❡ ζ − ζ − ζ ∈ G ′ Standard G A ⊕ A ❡ ❡ ζ − ζ − − ζ − ❡ ❡ ζ ζ ζ − ζ ∈ G ′ ufo (9) A ⊕ A ❡ ❡ − ζ − ζ − ❡ ❡ ζ ζ − − ❡ ❡ ζ ζ − ❡ ❡ − ζ − ζ − − ζ ∈ G ′ brj (2; 5) B ❡ ❡ ζ ζ − − ❡ ❡ − ζ − ζ − − ζ ∈ G ′ ufo (10) A ⊕ A ❡ ❡ − ζ − ζ − ❡ ❡ − ζ − − ζ − ❡ ❡ − ζ − ζ − ζ ❡ ❡ ζ − ζ − ζ − ζ ∈ G ′ ufo (11) A ⊕ A ❡ ❡ ζ − ζ − − ❡ ❡ ζ − ζ − ❡ ❡ − ζ − ζ − − ❡ ❡ − ζ − − ζ − ζ ∈ G ′ ufo (12) G Table 1.
Lie algebras arising from Dynkin diagrams of rank 2.
ANDRUSKIEWITSCH; ANGIONO; ROSSI BERTONE n q is the matter of Section 4. We denote by G N the group of N -th roots of 1, and by G ′ N its subset of primitive roots.2. Preliminaries
The Nichols algebra, the distinguished-pre-Nichols algebra andthe Lusztig algebra.
Let q be as in the Introduction and let ( V, c q ) be thecorresponding braided vector space of diagonal type. We assume from nowon that B q is finite-dimensional. Let ( α j ) j ∈ I be the canonical basis of Z θ .Let q : Z θ × Z θ → k × be the Z -bilinear form associated to the matrix q , i.e. q ( α j , α k ) = q jk for all j, k ∈ I . If α, β ∈ Z θ , we set q αβ = q ( α, β ). Considerthe matrix ( c q ij ) i,j ∈ I , c ij ∈ Z defined by c q ii = 2, c q ij := − min { n ∈ N : ( n + 1) q ii (1 − q nii q ij q ji ) = 0 } , i = j. (3)This is well-defined by [R]. Let i ∈ I . We recall the following definitions: ⋄ The reflection s q i ∈ GL ( Z θ ), given by s q i ( α j ) = α j − c q ij α i , j ∈ I . ⋄ The matrix ρ i ( q ), given by ρ i ( q ) jk = q ( s q i ( α j ) , s q i ( α k )), j, k ∈ I . ⋄ The braided vector space ρ i ( V ) of diagonal type with matrix ρ i ( q ).A basic result is that B q ≃ B ρ i ( q ) , at least as graded vector spaces.The algebras T ( V ) and B q are Z θ -graded by deg x i = α i , i ∈ I . Let ∆ q + be the set of Z θ -degrees of the generators of a PBW-basis of B q , countedwith multiplicities [H1]. The elements of ∆ q + are called (positive) roots. Let∆ q = ∆ q + ∪ − ∆ q + . Let X := { ρ j . . . ρ j N ( q ) : j , . . . , j N ∈ I , N ∈ N } . Then the generalized root system of q is the fibration ∆ → X , where the fiberof ρ j . . . ρ j N ( q ) is ∆ ρ j ...ρ jN ( q ) . The Weyl groupoid of B q is a groupoid, de-noted W q , that acts on this fibration, generalizing the classical Weyl group,see [H1]. We know from loc. cit. that W q is finite (and this characterizesfinite-dimensional Nichols algebras of diagonal type).Here is a useful description of ∆ q + . Let w ∈ W q be an element of maximallength. We fix a reduced expression w = σ q i σ i · · · σ i M . For 1 ≤ k ≤ M set β k = s q i · · · s i k − ( α i k ) , (4)Then ∆ q + = { β k | ≤ k ≤ M } [CH, Prop. 2.12]; in particular | ∆ q + | = M .The notion of Cartan root is instrumental for the definitions of e B q and L q . First, following [A5] we say that i ∈ I is a Cartan vertex of q if q ij q ji = q c q ij ii , for all j = i, (5)Then the set of Cartan roots of q is O q = { s q i s i . . . s i k ( α i ) ∈ ∆ q + : i ∈ I is a Cartan vertex of ρ i k . . . ρ i ρ i ( q ) } . IE ALGEBRAS ARISING FROM NICHOLS ALGEBRAS 5
Given a positive root β ∈ ∆ q + , there is an associated root vector x β ∈ B q defined via the so-called Lusztig isomorphisms [H3]. Set N β = ord q ββ ∈ N , β ∈ ∆ q + . Also, for h = ( h , . . . , h M ) ∈ N M we write x h = x h M β M x h M − β M − · · · x h β . Let e N k = ( N β k if β k / ∈ O q , ∞ if β k ∈ O q . . For simplicity, we introduce H = { h ∈ N M : 0 ≤ h k < e N k , for all k ∈ I M } . (6)By [A5, Theorem 3.6] the set { x h | h ∈ H } is a basis of e B q .As said in the Introduction, the Lusztig algebra associated to B q is thebraided Hopf algebra L q which is the graded dual of e B q . Thus, it comesequipped with a bilinear form h , i : e B q × L q → k , which satisfies for all x, x ′ ∈ e B q , y, y ′ ∈ L q h y, xx ′ i = h y (2) , x ih y (1) , x ′ i and h yy ′ , x i = h y, x (2) ih y ′ , x (1) i . If h ∈ H , then define y h ∈ L q by h y h , x j i = δ h , j , j ∈ H . Let ( h k ) k ∈ I M denotethe canonical basis of Z M . If k ∈ I M and β = β k ∈ ∆ q + , then we denote theelement y n h k by y ( n ) β . Then the algebra L q is generated by { y α : α ∈ Π q } ∪ { y ( N α ) α : α ∈ O q , x N α α ∈ P ( e B q ) } , by [AAR]. Moreover, by [AAR, 4.6], the following set is a basis of L q : { y ( h ) β · · · y ( h M ) β M | ( h , . . . , h M ) ∈ H } . Lyndon words, convex order and PBW-basis.
For the compu-tations in Section 4 we need some preliminaries on Kharchenko’s PBW-basis. Let ( V, q ) be as above and let X be the set of words with letters in X = { x , . . . , x θ } (our fixed basis of V ); the empty word is 1 and for u ∈ X we write ℓ ( u ) the length of u . We can identify k X with T ( V ). Definition 2.1.
Consider the lexicographic order in X . We say that u ∈ X − { } is a Lyndon word if for every decomposition u = vw , v, w ∈ X − { } ,then u < w . We denote by L the set of all Lyndon words.A well-known theorem, due to Lyndon, established that any word u ∈ X admits a unique decomposition, named Lyndon decomposition , as a non-increasing product of Lyndon words:(7) u = l l . . . l r , l i ∈ L, l r ≤ · · · ≤ l . Also, each l i ∈ L in (7) is called a Lyndon letter of u .Now each u ∈ L − X admits at least one decomposition u = v v with v , v ∈ L . Then the Shirshov decomposition of u is the decomposition u = u u , u , u ∈ L , such that u is the smallest end of u between allpossible decompositions of this form. ANDRUSKIEWITSCH; ANGIONO; ROSSI BERTONE
For any braided vector space V , the braided bracket of x, y ∈ T ( V ) is(8) [ x, y ] c := multiplication ◦ (id − c ) ( x ⊗ y ) . Using the identification T ( V ) = k X and the decompositions describedabove, we can define a k -linear endomorphism [ − ] c of T ( V ) as follows:[ u ] c := u, if u = 1 or u ∈ X ;[[ v ] c , [ w ] c ] c , if u ∈ L − X, u = vw its Shirshov decomposition ;[ u ] c . . . [ u t ] c , if u ∈ X − L, u = u . . . u t its Lyndon decomposition . We will describe PBW-bases using this endomorphism.
Definition 2.2.
For l ∈ L , the element [ l ] c is the corresponding hyperletter .A word written in hyperletters is an hyperword ; a monotone hyperword isan hyperword W = [ u ] k c . . . [ u m ] k m c such that u > · · · > u m .Consider now a different order on X , called deg-lex order [K]: For eachpair u, v ∈ X , we have that u ≻ v if ℓ ( u ) < ℓ ( v ), or ℓ ( u ) = ℓ ( v ) and u > v for the lexicographical order. This order is total, the empty word 1 is themaximal element and it is invariant by left and right multiplication.Let I be a Hopf ideal of T ( V ) and R = T ( V ) /I . Let π : T ( V ) → R bethe canonical projection. We set: G I := { u ∈ X : u / ∈ k X ≻ u + I } . Thus, if u ∈ G I and u = vw , then v, w ∈ G I . So, each u ∈ G I is anon-increasing product of Lyndon words of G I .Let S I := G I ∩ L and let h I : S I → { , , . . . } ∪ {∞} be defined by:(9) h I ( u ) := min (cid:8) t ∈ N : u t ∈ k X ≻ u t + I (cid:9) . Theorem 2.3. [K]
The following set is a PBW-basis of R = T ( V ) /I : { [ u ] k c . . . [ u m ] k m c : m ∈ N , u > . . . > u m , u i ∈ S I , < k i < h I ( u i ) } . (cid:3) We refer to this base as
Kharchenko’s PBW-basis of T ( V ) /I (it dependson the order of X ). Definition 2.4. [A2, 2.6] Let ∆ + q be as above and let < be a total order on∆ + q . We say that the order is convex if for each α, β ∈ ∆ + q such that α < β and α + β ∈ ∆ + q , then α < α + β < β . The order is called strongly convex if for each ordered subset α ≤ α ≤ · · · ≤ α k of elements of ∆ + q such that α = P i α i ∈ ∆ + q , then α < α < α k . Theorem 2.5. [A2, 2.11]
The following statements are equivalent: • The order is convex. • The order is strongly convex. • The order arises from a reduced expression of a longest element w ∈ W q ,cf. (4) . (cid:3) IE ALGEBRAS ARISING FROM NICHOLS ALGEBRAS 7
Now, we have two PBW-basis of B q (and correspondingly of e B q ), namelyKharchenko’s PBW-basis and the PBW-basis defined from a reduced ex-pression of a longest element of the Weyl groupoid. But both basis arereconciled by [AY, Theorem 4.12], thanks to [A2, 2.14]. Indeed, each gen-erator of Kharchenko’s PBW-basis is a multiple scalar of a generator of thesecondly mentioned PBW-basis. So, for ease of calculations, in the rest ofthis work we shall use the Kharchenko generators.The following proposition is used to compute the hyperword [ l β ] c associ-ated to a root β ∈ ∆ + q : Proposition 2.6. [A2, 2.17]
For β ∈ ∆ + q , l β = ( x α i , if β = α i , i ∈ I ;max { l δ l δ : δ , δ ∈ ∆ + q , δ + δ = β, l δ < l δ } , if β = α i , i ∈ I . (cid:3) We give a list of the hyperwords appearing in the next section:Root Hyperword Notation α i x i x i nα + α (ad c x ) n x x ... α + 2 α [ x α + α , x ] c [ x , x ] c α + 2 α [ x α + α , x α + α ] c [ x , x ] c α + 3 α [ x α +2 α , x α + α ] c [[ x , x ] c , x ] c α + 3 α [ x α + α , x α +2 α ] c [ x , [ x , x ] c ] c We use an analogous notation for the elements of L q : for example we write y , when we refer to the element of L q which corresponds to [ x , x ] c .3. Extensions of braided Hopf algebras
We recall the definition of braided Hopf algebra extensions given in [AN];we refer to [BD, GG] for more general definitions. Below we denote by ∆ thecoproduct of a braided Hopf algebra A and by A + the kernel of the counit.First, if π : C → B is a morphism of Hopf algebras in HH YD , then we set C co π = { c ∈ C | (id ⊗ π )∆( c ) = c ⊗ } , co π C = { c ∈ C | ( π ⊗ id)∆( c ) = 1 ⊗ c } . Definition 3.1. [AN, § H be a Hopf algebra. A sequence of mor-phisms of Hopf algebras in HH YD k → A ι → C π → B → k (10)is an extension of braided Hopf algebras if(i) ι is injective,(ii) π is surjective,(iii) ker π = Cι ( A + ) and(iv) A = C co π , or equivalently A = co π C .For simplicity, we shall write A ι ֒ → C π ։ B instead of (10). ANDRUSKIEWITSCH; ANGIONO; ROSSI BERTONE
This Definition applies in our context: recall that B q ≃ e B q / h x N β β , β ∈ O q i .Let Z q be the subalgebra of e B q generated by x N β β , β ∈ O q . Then ◦ The inclusion ι : Z q → e B q is injective and the projection π : e B q → B q issurjective. ◦ [A5, Theorem 4.10] Z q is a normal Hopf subalgebra of e B q ; since ker π isthe two-sided ideal generated by ι ( Z + q ), ker π = e B q ι ( Z + q ). ◦ [A5, Theorem 4.13] Z q = co π e B q .Hence we have an extension of braided Hopf algebras Z q ι ֒ → e B q π ։ B q . (11)The morphisms ι and π are graded. Thus, taking graded duals, we obtaina new sequence of morphisms of braided Hopf algebras B q π ∗ ֒ → L q ι ∗ ։ Z q . (2) Proposition 3.2.
The sequence (2) is an extension of braided Hopf algebras.Proof.
The argument of [A, 3.3.1] can be adapted to the present situation,or more generally to extensions of braided Hopf algebras that are gradedwith finite-dimensional homogeneous components. The map π ∗ : B q → L q is injective because B q ≃ B ∗ q ; ι ∗ : L q ι ∗ → Z q is surjective being the transposeof a graded monomorphism between two locally finite graded vector spaces.Now, since Z q = co π e B q = e B co π q , we haveker ι ∗ = L q B + q = B + q L q . (12)Similarly L co ι ∗ q = B ∗ q because ker π ⊥ = B q . (cid:3) From now on, we assume the condition (1) on the matrix q mentioned inthe Introduction, that is q N β αβ = 1 , ∀ α, β ∈ O q . The following result is our basic tool to compute the Lie algebra n q . Theorem 3.3.
The braided Hopf algebra Z q is an usual Hopf algebra, iso-morphic to the universal enveloping algebra of the Lie algebra n q = P ( Z q ) .The elements ξ β := ι ∗ ( y ( N β ) β ) , β ∈ O q , form a basis of n q .Proof. Let A q be the subspace of L q generated by the ordered monomials y ( r N βi ) β i . . . y ( r k N βik ) β ik where β i < · · · < β i k are all the Cartan roots of B q and r , . . . , r k ∈ N . We claim that the restriction of the multiplication µ : B q ⊗ A q → L q is an isomorphism of vector spaces. Indeed, µ is surjective IE ALGEBRAS ARISING FROM NICHOLS ALGEBRAS 9 by the commuting relations in L q . Also, the Hilbert series of L q , B q and A q are respectively: H L q = Y β k ∈ O q − T deg β . Y β k / ∈ O q − T N β deg β − T deg β ; H B q = Y β k ∈ ∆ + q − T N β deg β − T deg β ; H A q = Y β k ∈ O q − T N β deg β . Since the multiplication is graded and H L q = H B q H A q , µ is injective. Theclaim follows and we have L q = A q ⊕ B + q A q . (13)We next claim that ι ∗ : A q → Z q is an isomorphism of vector spaces.Indeed, by (12), ker ι ∗ = B + q L q = B + q ( B q A q ) = B + q A q . By (13), the claimfollows.By (1), Z q is a commutative Hopf algebra, see [A5]; hence Z q is a co-commutative Hopf algebra. Now the elements ξ β := ι ∗ ( y ( N β ) β ), , β ∈ O q ,are primitive, i.e. belong to n q = P ( Z q ). The monomials ξ r β i . . . ξ r k β ik , β i < · · · < β i k ∈ O q , r , . . . , r k ∈ N form a basis of Z q , hence Z q = k h ξ β : β ∈ O q i ⊆ U ( n q ) ⊆ Z q . We conclude that ( ξ β ) β ∈ O q is a basis of n q and that Z q = U ( n q ). (cid:3) Proof of Theorem 1.1
In this section we consider all indecomposable matrices q of rank 2 whoseassociated Nichols algebra B q is finite-dimensional; these are classified in[H2] and we recall their diagrams in Table 1. For each q we obtain anisomorphism between Z q and U ( g + ), the universal enveloping algebra of thepositive part of g . Here g is the semisimple Lie algebra of the last columnof Table 1, with Cartan matrix A = ( a ij ) ≤ i,j ≤ . By simplicity we denote g by its type, e.g. g = A .We recall that we assume (1) and that ξ β = ι ∗ ( y ( N β ) β ) ∈ Z q . Thus,[ ξ α , ξ β ] c = ξ α ξ β − ξ β ξ α = [ ξ α , ξ β ] , for all α, β ∈ O q . The strategy to prove the isomorphism F : U ( g + ) → Z q is the following:(1) If O q = ∅ , then g + = 0. If | O q | = 1, then g = sl , i.e. of type A .(2) If | O q | = 2, then g is of type A ⊕ A . Indeed, let O q = { α, β } . As Z q is N θ -graded, [ ξ α , ξ β ] ∈ n q has degree N α α + N β β . Thus [ ξ α , ξ β ] = 0.(3) Now assume that | O q | >
2. We recall that Z q is generated by { ξ β | x N β β is a primitive element of e B q } . We compute the coproduct of all x N β β in e B q , β ∈ O q , using that ∆ is agraded map and Z q is a Hopf subalgebra of e B q . In all cases we get twoprimitive elements x N β β and x N β β , thus Z q is generated by ξ β and ξ β .(4) Using the coproduct again, we check that(ad ξ β i ) − a ij ξ β j = 0 , ≤ i = j ≤ . (14)To prove (14), it is enough to observe that n q has a trivial componentof degree N β i (1 − a ij ) β i + N β j β j . Now (14) implies that there exists asurjective map of Hopf algebras F : U ( g + ) ։ Z q such that e i ξ β i .(5) To prove that F is an isomorphism, it suffices to see that the restriction g + ∗ → n q is an isomorphism; but in each case we see that ∗ is surjective,and dim g + = dim n q = | O q | .We refer to [A1, AAY, A4] for the presentation, root system and Cartanroots of braidings of standard, super and unidentified type respectively. Row 1.
Let q ∈ G ′ N , N ≥
2. The diagram ❡ ❡ q q − q corresponds to abraiding of Cartan type A whose set of positive roots is ∆ + q = { α , α + α , α } . In this case O q = ∆ + q and N β = N for all β ∈ O q . By hypothesis, q N = q N = 1. The elements x , x ∈ e B q are primitive and∆( x ) = x ⊗ ⊗ x + (1 − q − ) x ⊗ x . Then the coproducts of the elements x N , x N , x N ∈ e B q are:∆( x N ) = x N ⊗ ⊗ x N ; ∆( x N ) = x N ⊗ ⊗ x N ;∆( x N ) = x N ⊗ ⊗ x N + (1 − q − ) N q N ( N − x N ⊗ x N . As [ ξ , ξ ], [ ξ , ξ ] ∈ n q have degree N α +2 N α , respectively 2 N α + N α ,and the components of these degrees of n q are trivial, we have[ ξ , ξ ] = [ ξ , ξ ] = 0 . Again by degree considerations, there exists c ∈ k such that [ ξ , ξ ] = cξ .By the duality between Z q and Z q we have that[ ξ , ξ ] = (1 − q − ) N q N ( N − ξ . Then there exists a morphism of algebras F : U ( A +2 ) → Z q given by e ξ , e ξ . This morphism takes a basis of A +2 to a basis of n q , so Z q ≃ U ( A +2 ). Row 2.
Let q ∈ G ′ N , N ≥
3. These diagrams correspond to braidings ofsuper type A with positive roots ∆ + q = { α , α + α , α } . IE ALGEBRAS ARISING FROM NICHOLS ALGEBRAS 11
The first diagram is ❡ ❡ q q − − . In this case the unique Cartan root is α with N α = N . The element x N ∈ e B q is primitive and Z q is generated by ξ . Hence Z q ≃ U ( A +1 ).The second diagram gives a similar situation, since O q = { α + α } . Row 3.
Let q ∈ G ′ N , N ≥
3. The diagram ❡ ❡ q q − q corresponds to abraiding of Cartan type B with ∆ + q = { α , α + α , α + α , α } . In thiscase O q = ∆ + q . The coproducts of the generators of e B q are:∆( x ) = x ⊗ ⊗ x ; ∆( x ) = x ⊗ ⊗ x ;∆( x ) = x ⊗ ⊗ x + (1 − q − ) x ⊗ x ;∆( x ) = x ⊗ ⊗ x + (1 − q − )(1 − q − ) x ⊗ x + q (1 − q − ) x ⊗ x . We have two different cases depending on the parity of N .(1) If N is odd, then N β = N for all β ∈ ∆ + q . In this case,∆( x N ) = x N ⊗ ⊗ x N ; ∆( x N ) = x N ⊗ ⊗ x N ;∆( x N ) = x N ⊗ ⊗ x N + (1 − q − ) N x N ⊗ x N ;∆( x N ) = x N ⊗ ⊗ x N + (1 − q − ) N (1 − q − ) N x N ⊗ x N + C x N ⊗ x N , for some C ∈ k . Hence, in Z q we have the relations[ ξ , ξ ] = (1 − q − ) N ξ ;[ ξ , ξ ] = C ξ ;[ ξ , ξ ] c = (1 − q − ) N (1 − q − ) N ξ + (1 − q − ) N ξ ξ ;[ ξ , ξ ] = [ ξ , ξ ] = 0 . Thus there exists an algebra map F : U ( B +2 ) → Z q given by e ξ , e ξ .Moreover, F is an isomorphism, and so Z q ≃ U ( B +2 ). Using the relations of U ( B +2 ) we check that C = 2(1 − q − ) N (1 − q − ) N .(2) If N = 2 M >
2, then N α = N α + α = N and N α + α = N α = M .In this case we have∆( x N ) = x N ⊗ ⊗ x N ; ∆( x M ) = x M ⊗ ⊗ x M ;∆( x N ) = x N ⊗ ⊗ x N + (1 − q − ) N q M ( N − x N ⊗ x M + (1 − q − ) M q M x M ⊗ x M ;∆( x M ) = x M ⊗ ⊗ x M + (1 − q − ) M (1 − q − ) M q M ( M − x N ⊗ x M . Hence, the following relations hold in Z q :[ ξ , ξ ] = (1 − q − ) M (1 − q − ) M q M ( M − ξ ;[ ξ , ξ ] = (1 − q − ) M q M ξ ;[ ξ , ξ ] = [ ξ , ξ ] = 0 . Thus F : U ( C +2 ) → Z q , e ξ , e ξ , is an isomorphism of algebras.(Of course C ≃ B but in higher rank we will get different root systemsdepending on the parity of N ). Row 4.
Let q ∈ G ′ N , N = 2 ,
4. These diagrams correspond to braidings ofsuper type B with ∆ + q = { α , α + α , α + α , α } .If the diagram is ❡ ❡ q q − − , then the Cartan roots are α and α + α ,with N α = N , N α + α = M ; here, M = N if N is odd and M = N if N is even. The elements x N , x M ∈ e B q are primitive in e B q . Thus, in Z q ,[ ξ , ξ ] = 0 and Z q ≃ U (( A ⊕ A ) + ).If we consider the diagram ❡ ❡ − q − q − , then O q = { α , α + α } , N α = M and N α + α = N . The elements x M , x N ∈ e B q are primitive, so [ ξ , ξ ] = 0and Z q ≃ U (( A ⊕ A ) + ). Row 5.
Let q ∈ G ′ N , N = 3, ζ ∈ G ′ . The diagram ❡ ❡ ζ q − q correspondsto a braiding of standard type B , so ∆ + q = { α , α + α , α + α , α } . Theother diagram ❢ ❢ ζ qζ − ζq − is obtained by changing the parameter q ↔ ζq − .The Cartan roots are 2 α + α and α , with N α + α = M := ord( ζq − )and N α = N . The elements x M , x N ∈ e B q are primitive. Thus, in Z q , wehave [ ξ , ξ ] = 0. Hence, Z q ≃ U (( A ⊕ A ) + ). Row 6.
Let ζ ∈ G ′ . The diagrams ❡ ❡ ζ − ζ − and ❡ ❡ ζ − − ζ − − correspondto braidings of standard type B , thus ∆ + q = { α , α + α , α + α , α } . Inboth cases O q is empty so the corresponding Lie algebras are trivial. Row 7.
Let ζ ∈ G ′ . The diagrams of this row correspond to braidings oftype ufo (7). In all cases O q = ∅ and the associated Lie algebras are trivial. Row 8.
Let ζ ∈ G ′ . The diagrams of this row correspond to braidings oftype ufo (8). For ❡ ❡ − ζ ζ − ζ , ∆ + q = { α , α + α , α + α , α + 2 α , α } .In this case O q = { α + α } , N α + α = 12. Hence Z q ≃ U ( A +1 ). The sameresult holds for the other braidings in this row. Row 9.
Let ζ ∈ G ′ . The diagrams of this row correspond to braidings oftype brj (2; 3). If q has diagram ❡ ❡ − ζ ζ ζ , then∆ + q = { α , α + α , α + 2 α , α + α , α + 2 α , α } . IE ALGEBRAS ARISING FROM NICHOLS ALGEBRAS 13
In this case O q = { α , α + α } and N α = N α + α = 18. Thus [ ξ , ξ ] = 0,so Z q ≃ U (( A ⊕ A ) + ).If q has diagram ❡ ❡ ζ ζ − , ❡ ❡ − ζ ζ − the set of positive roots are,respectively, { α , α + α , α + 2 α , α + 3 α , α + α , α } , { α , α + α , α + α , α + α , α + α , α } ;the Cartan roots are, respectively, α + α , α + α and α , α + α . Hence,in both cases, Z q ≃ U (( A ⊕ A ) + ). Row 10.
Let q ∈ G ′ N , N ≥
4. The diagram ❡ ❡ q q − q corresponds to abraiding of Cartan type G , so O q = ∆ + q = { α , α + α , α + α , α + α , α + 2 α , α } . The coproducts of the PBW-generators are:∆( x ) = x ⊗ ⊗ x ; ∆( x ) = x ⊗ ⊗ x ;∆( x ) = x ⊗ ⊗ x + (1 − q − ) x ⊗ x ;∆( x ) = x ⊗ ⊗ x + (1 + q )(1 − q − ) x ⊗ x + (1 − q − )(1 − q − ) x ⊗ x ;∆( x ) = x ⊗ ⊗ x + q (1 − q − ) x ⊗ x + ( q − − q − ) x ⊗ x + (1 − q − )(1 − q − )(1 − q − ) x ⊗ x ;∆([ x , x ] c ) = [ x , x ] c ⊗ ⊗ [ x , x ] c + ( q − q − ) x ⊗ x + (1 − q − )(1 + q )(1 − q − + q ) x x ⊗ x − qq (1 − q − )(1 + q − q ) x ⊗ x + q q (1 − q − ) x ⊗ [ x , x ] c + (1 − q − ) ( q − x ⊗ x x + q (1 − q − ) (1 − q − )(1 − q − ) x ⊗ x . We have two cases.(1) If 3 does not divide N , then N β = N for all β ∈ ∆ + q . Thus, in e B q ,∆( x N ) = x N ⊗ ⊗ x N ; ∆( x N ) = x N ⊗ ⊗ x N ;∆( x N ) = x N ⊗ ⊗ x N + a x N ⊗ x N ;∆( x N ) = x N ⊗ ⊗ x N + a x N ⊗ x N + a x N ⊗ x N ;∆( x N ) = x N ⊗ ⊗ x N + a x N ⊗ x N + a x N ⊗ x N + a x N ⊗ x N ;∆([ x , x ] Nc ) = [ x , x ] Nc ⊗ ⊗ [ x , x ] Nc + a x N ⊗ x N + a x N ⊗ x N + a x N ⊗ x N + a x N ⊗ x N x N + a x N x N ⊗ x N + a x N ⊗ x N ; for some a i ∈ k . Since a = (1 − q − ) N q N ( N − = 0 ,a = (1 − q − ) N (1 − q − ) N = 0 ,a = (1 − q − ) N (1 − q − ) N (1 − q − ) N q N ( N − = 0 ,a = (1 − q − ) N (1 − q − ) N (1 − q − ) N = 0 , the elements x N , x N , x N and [ x , x ] Nc are not primitive. Hence Z q isgenerated by ξ and ξ ; also[ ξ , ξ ] = a ξ ; [ ξ , ξ ] = a ξ ;[ ξ , ξ ] = a ξ ; [ ξ , ξ ] = [ ξ , ξ ] = 0 . Thus, we have Z q ≃ U ( G +2 ).(2) If N = 3 M , then N α = N α + α = N α + α = N and N α + α = N α +2 α = N α = M . In this case we have∆( x N ) = x N ⊗ ⊗ x N ; ∆( x M ) = x M ⊗ ⊗ x M ;∆( x N ) = x N ⊗ ⊗ x N + (1 − q − ) M q N ( M − [ x , x ] Mc ⊗ x M + (1 − q − ) M x M ⊗ x M + (1 − q − ) N q N ( N − x N ⊗ x M ;∆( x N ) = x M ⊗ ⊗ x M + b x N ⊗ x N + b x M ⊗ [ x , x ] Mc + b x N ⊗ x M + b x M ⊗ x M + b x M x N ⊗ x M + b x N ⊗ x M ;∆( x M ) = x M ⊗ ⊗ x M + b x N ⊗ x M [ x , x ] Mc ;∆([ x , x ] Mc ) = x M ⊗ ⊗ x M + b x N ⊗ x M + b x M ⊗ x M ;for some b i ∈ k . We compute some of them explicitly: b = (1 + q ) M (1 − q − ) M q M q N ( M − ,b = (1 − q − ) M (1 − q − ) M (1 − q − ) M q N ( M − ,b = (1 − q − ) M (1 − q − ) M (1 − q − ) M q M . As these scalars are not zero, the elements x N , x N , x M and [ x , x ] Mc are not primitive. Thus Z q ≃ U ( G +2 ). Row 11.
Let ζ ∈ G ′ . The diagrams of this row correspond to braidings ofstandard type G , so ∆ + q = { α , α + α , α + α , α + 2 α , α + α , α } .If q has diagram ❡ ❡ ζ ζ ζ − , then the Cartan roots are 2 α + α and α with N α + α = N α = 8. The elements x , x ∈ e B q are primitive and[ ξ , ξ ] = 0 in Z q . Hence Z q ≃ U (( A ⊕ A ) + ). An analogous result holdsfor the other diagrams of the row. IE ALGEBRAS ARISING FROM NICHOLS ALGEBRAS 15
Row 12.
Let ζ ∈ G ′ . This row corresponds to type ufo (9). If q hasdiagram ❡ ❡ ζ ζ ζ , then∆ + q = { α , α + α , α + α , α + 2 α , α + 3 α , α + α , α + 2 α , α } and O q = { α + α , α + α } . Here, N α + α = N α + α = 24, and x , x ∈ e B q are primitive. In Z q we have the relation [ ξ , ξ ] = 0;thus Z q ≃ U (( A ⊕ A ) + ).For the other diagrams, ❡ ❡ ζ ζ ζ − , ❡ ❡ ζ ζ − and ❡ ❡ ζ ζ − , thesets of positive roots are, respectively, { α , α + α , α + α , α + α , α + 2 α , α + 2 α , α + 3 α , α } , { α , α + α , α + α , α + 2 α , α + 3 α , α + 3 α , α + 4 α , α } , { α , α + α , α + α , α + α , α + α , α + α , α + 2 α , α } . The Cartan roots are, respectively, 2 α + α , α ; α + α , α +3 α ; α , α +2 α . Hence, in all cases, Z q ≃ U (( A ⊕ A ) + ). Row 13.
Let ζ ∈ G ′ . The braidings in this row are associated to theLie superalgebra brj (2; 5) [A5, § q has diagram ❡ ❡ ζ ζ − , then∆ + q = { α , α + α , α + α , α + 3 α , α + 2 α , α + 3 α , α + α , α } .In this case the Cartan roots are α , α + α , 2 α + α and 3 α + α , with N α = N α +2 α = 5 and N α + α = N α + α = 10. In e B q ,∆( x ) = x ⊗ ⊗ x ;∆( x ) = x ⊗ ⊗ x + (1 − ζ ) x ⊗ x ;∆( x ) = x ⊗ ⊗ x + (1 + ζ )(1 − ζ ) x ⊗ x + (1 − ζ )(1 − ζ ) x ⊗ x ;∆([ x , x ] c ) = [ x , x ] c ⊗ ⊗ [ x , x ] c − ζ (1 − ζ )(1 + ζ ) x ⊗ x − ζq x x ⊗ x + (1 + q + ζ q ) x x ⊗ x + ζ (1 − ζ ) x x x ⊗ x + (1 − ζ )(1 − ζ ) x ⊗ x x . Hence the coproducts of x , x , x , [ x , x ] c , ∈ e B q are:∆( x ) = x ⊗ ⊗ x ; ∆( x ) = x ⊗ ⊗ x ;∆( x ) = x ⊗ ⊗ x + a x ⊗ x + a x ⊗ [ x , x ] c ;∆([ x , x ] c ) = [ x , x ] c ⊗ ⊗ [ x , x ] c + a x ⊗ x . for some a i ∈ k . Thus, the following relations hold in Z q [ ξ , ξ ] = a ξ , ; [ ξ , , ξ ] = a ξ ; [ ξ , ξ , ] = [ ξ , ξ ] = 0 . Since a = − (1 − ζ ) (1 + ζ ) (1 + 62 ζ − ζ − ζ + 70 ζ ) = 0; a = − (1 − ζ ) (1 + ζ ) (4 − ζ − ζ − ζ − ζ ) = 0 , the elements x , [ x , x ] c are not primitive, so ξ , ξ generate Z q . Hence, Z q ≃ U ( B +2 ).If q has diagram ❡ ❡ − ζ ζ − , then∆ + q = { α , α + α , α + α , α + 2 α , α + α , α + 2 α , α + α , α } , O q = { α , α + α , α + α , α + α } , with N α = N α + α = 10, N α + α = N α + α = 5. The generators of Z q are ξ and ξ and they satisfy the following relations[ ξ , ξ ] = b ξ , [ ξ , ξ ] = b ξ , [ ξ , ξ ] = [ ξ , ξ ] = 0 , for some b , b ∈ k × . Hence Z q ≃ U ( C +2 ). Row 14.
Let ζ ∈ G ′ . This row corresponds to type ufo (10). If q hasdiagram ❡ ❡ ζ ζ − , then ∆ + q = { α , α + α , α + α , α + 3 α , α +2 α , α + 3 α , α + α , α } . The Cartan roots are α and 3 α + 2 α with N α = N α +2 α = 20. The elements x , [ x , x ] c ∈ e B q are primitive;thus [ ξ , ξ , ] = 0 in Z q and Z q ≃ U (( A ⊕ A ) + ). The same holds whenthe diagram of q is another one in this row: Z q ≃ U (( A ⊕ A ) + ). Row 15.
Let ζ ∈ G ′ . This row corresponds to type ufo (11). If q hasdiagram ❡ ❡ − ζ − ζ ζ , then ∆ + q = { α , α + α , α + 2 α , α + α , α +2 α , α + α , α + 2 α , α } . The Cartan roots are α and 3 α + 2 α with N α = N α +2 α = 30. In Z q we have [ ξ , ξ , ] = 0, thus Z q ≃ U (( A ⊕ A ) + ). The same result holds if we consider the other diagrams of this row. Row 16.
Let ζ ∈ G ′ . This row corresponds to type ufo (12). If q hasdiagram ❡ ❡ − ζ − ζ − , then∆ + q = { α , α + α , α + α , α + 2 α , α + α , α + 3 α , α + 2 α , α + 3 α , α + α , α + 2 α , α + α , α } . Also, O q = { α , α + α , α + α , α +2 α , α + α , α + α } with N β = 14for all β ∈ O q . In e B q we have∆( x ) = x ⊗ ⊗ x ;∆( x ) = x ⊗ ⊗ x + (1 + ζ ) x ⊗ x ;∆( x ) = x ⊗ ⊗ x + (1 − ζ )(1 − ζ ) x ⊗ x + (1 − ζ )(1 + ζ ) x ⊗ x ; IE ALGEBRAS ARISING FROM NICHOLS ALGEBRAS 17 ∆( x ) = x ⊗ ⊗ x + (1 + ζ − ζ )(1 + ζ ) x ⊗ x + ζ ( ζ − x ⊗ x + ζ (1 − ζ )(1 + ζ ) x ⊗ x ;∆( x ) = x ⊗ ⊗ x − ζ (1 − ζ )(1 − ζ ) x ⊗ x + (1 − ζ ) x ⊗ x − (1 − ζ )(1 − ζ ) x ⊗ x + ζ (1 − ζ )(1 − ζ ) x ⊗ x ;∆([ x , x ] c ) = [ x , x ] c ⊗ ⊗ [ x , x ] c − (1 − ζ )(1 + ζ ) (1 − ζ + 2 ζ ) x ⊗ x − q (1 − ζ )(1 − ζ ) x ⊗ [ x , x ] c − (1 − ζ ) (4 + 4 ζ + ζ − ζ − ζ ) x ⊗ x x + q (1 − ζ ) ζ (1 − ζ − ζ − ζ + ζ ) x ⊗ x + (1 − ζ ) (1 + ζ ) (1 + ζ ) x ⊗ x x − ζ (1 − ζ )(1 − ζ ) x ⊗ x − q ζ (1 − ζ ) (1 − ζ )(1 + 2 ζ ) x ⊗ x x + q ζ (1 − ζ ) (1 − ζ )(1 + ζ ) x ⊗ x − q (1 + ζ )(1 − ζ )(1 − ζ + ζ ) x ⊗ x + ζq (1 + ζ )(1 − ζ )(1 − ζ )(1 + ζ − ζ ) x x ⊗ x − ζ (1 − ζ ) (1 + ζ )(1 − ζ − ζ − ζ ) x x ⊗ x + (1 − ζ )(1 + ζ + ζ − ζ − ζ ) x x ⊗ x + ζq (1 − ζ ) (2 + ζ − ζ ) x ⊗ x . Hence∆( x ) = x ⊗ ⊗ x ; ∆( x ) = x ⊗ ⊗ x ;∆( x ) = x ⊗ ⊗ x + a x ⊗ x ;∆( x ) = x ⊗ ⊗ x + a x ⊗ x + a x ⊗ x ;∆( x ) = x ⊗ ⊗ x + a x ⊗ x + a x ⊗ x + a x ⊗ x ;∆([ x , x ] c ) = [ x , x ] c ⊗ ⊗ [ x , x ] c + a x ⊗ x + a x ⊗ x + a x ⊗ x + a x ⊗ x + a x ⊗ x x + a x x ⊗ x ;with a i ∈ k . For instance, a = q (1 − ζ ) (1 − ζ ) (cid:0) − ζ + 35105 ζ + 31472 ζ − ζ +19299 ζ + 40124 ζ (cid:1) = 0 , because ζ ∈ G ′ . Also, a = 26686268 + 39070423 ζ − ζ − ζ + 52678504 ζ − ζ − ζ = 0 . Since a , a , a , a = 0 then x , x , x and [ x , x ] c are notprimitive elements in e B q . Thus, ξ and ξ generates Z q .Also, in Z q we have[ ξ , ξ ] = a ξ ; [ ξ , ξ ] = a ξ ;[ ξ , ξ ] = a ξ ; [ ξ , ξ ] = [ ξ , ξ ] = 0 . So, Z q ≃ U ( G +2 ).In the case of the diagram ❡ ❡ − ζ − ζ − Z q is generated by ξ , ξ and[ ξ , ξ ] = b ξ ; [ ξ , ξ ] = b ξ , ;[ ξ , ξ , ] = b ξ (112 , , ; [ ξ , ξ ] = [ ξ , ξ (112 , , ] = 0 , where b , b , b ∈ k × . Hence, we also have Z q ≃ U ( G +2 ). Remark . The results of this paper are part of the thesis of one of theauthors [RB], where missing details of the computations can be found.
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FaMAF-CIEM (CONICET), Universidad Nacional de C´ordoba, Medina A-llende s/n, Ciudad Universitaria, 5000 C´ordoba, Rep´ublica Argentina.
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