On the automorphism groups of q-enveloping algebras of nilpotent Lie algebras
aa r X i v : . [ m a t h . R A ] D ec On the automorphism groups of q -enveloping algebrasof nilpotent Lie algebras. St´ephane Launois ∗ Abstract
We investigate the automorphism group of the quantised enveloping algebra U + q ( g ) of the positive nilpotent part of certain simple complex Lie algebras g inthe case where the deformation parameter q ∈ C ∗ is not a root of unity. Studyingits action on the set of minimal primitive ideals of U + q ( g ) we compute this group inthe cases where g = sl and g = so confirming a Conjecture of Andruskiewitschand Dumas regarding the automorphism group of U + q ( g ). In the case where g = sl ,we retrieve the description of the automorphism group of the quantum Heisenbergalgebra that was obtained independently by Alev and Dumas, and Caldero. In thecase where g = so , the automorphism group of U + q ( g ) was computed in [16] byusing previous results of Andruskiewitsch and Dumas. In this paper, we give a new(simpler) proof of the Conjecture of Andruskiewitsch and Dumas in the case where g = so based both on the original proof and on graded arguments developed in [17]and [18]. Introduction
In the classical situation, there are few results about the automorphism group of theenveloping algebra U ( L ) of a Lie algebra L over C ; except when dim L ≤
2, these groupsare known to possess “wild” automorphisms and are far from being understood. Forinstance, this is the case when L is the three-dimensional abelian Lie algebra [22], when L = sl [14] and when L is the three-dimensional Heisenberg Lie algebra [1].In this paper we study the quantum situation. More precisely, we study the automor-phism group of the quantised enveloping algebra U + q ( g ) of the positive nilpotent part of afinite dimensional simple complex Lie algebra g in the case where the deformation param-eter q ∈ C ∗ is not a root of unity. Although it is a common belief that quantum algebrasare ”rigid” and so should possess few symmetries, little is known about the automorphismgroup of U + q ( g ). Indeed, until recently, this group was known only in the case where g = sl ∗ This research was supported by a Marie Curie Intra-European Fellowship within the 6th EuropeanCommunity Framework Programme held at the University of Edinburgh. U q ( b + ), where b + is the positive Borel subalgebra of g , has been described in [9] in the general case.The automorphism group of U + q ( s l ) was computed independently by Alev-Dumas, [2],and Caldero, [8], who showed thatAut( U + q ( s l )) ≃ ( C ∗ ) ⋊ S . Recently, Andruskiewitsch and Dumas, [4] have obtained partial results on the automor-phism group of U + q ( so ). In view of their results and the description of Aut( U + q ( s l )), theyhave proposed the following conjecture. Conjecture (Andruskiewitsch-Dumas, [4, Problem 1]):
Aut( U + q ( g )) ≃ ( C ∗ ) rk( g ) ⋊ autdiagr( g ) , where autdiagr( g ) denotes the group of automorphisms of the Dynkin diagram of g .Recently we proved this conjecture in the case where g = so , [16], and, in collaborationwith Samuel Lopes, in the case where g = sl , [18]. The techniques in these two cases arevery different. Our aim in this paper is to show how one can prove the Andruskiewitsch-Dumas Conjecture in the cases where g = sl and g = so by first studying the action ofAut( U + q ( g )) on the set of minimal primitive ideals of U + q ( g ) - this was the main idea in[16] -, and then using graded arguments as developed in [17] and [18]. This strategy leadsus to a new (simpler) proof of the Andruskiewitsch-Dumas Conjecture in the case where g = so .Throughout this paper, N denotes the set of nonnegative integers, C ∗ := C \ { } and q is a nonzero complex number that is not a root of unity. In this section, we present the H -stratification theory of Goodearl and Letzter for thepositive part U + q ( g ) of the quantised enveloping algebra of a simple finite-dimensionalcomplex Lie algebra g . In particular, we present a criterion (due to Goodearl and Letzter)that characterises the primitive ideals of U + q ( g ) among its prime ideals. In the next section,we will use this criterion in order to describe the primitive spectrum of U + q ( g ) in the caseswhere g = sl and g = so . Let g be a simple Lie C -algebra of rank n . We denote by π = { α , . . . , α n } the set of simpleroots associated to a triangular decomposition g = n − ⊕ h ⊕ n + . Recall that π is a basis ofan euclidean vector space E over R , whose inner product is denoted by ( , ) ( E is usually2enoted by h ∗ R in Bourbaki). We denote by W the Weyl group of g , that is, the subgroupof the orthogonal group of E generated by the reflections s i := s α i , for i ∈ { , . . . , n } ,with reflecting hyperplanes H i := { β ∈ E | ( β, α i ) = 0 } , i ∈ { , . . . , n } . The length of w ∈ W is denoted by l ( w ). Further, we denote by w the longest element of W . We denoteby R + the set of positive roots and by R the set of roots. Set Q + := N α ⊕ · · · ⊕ N α n and Q := Z α ⊕ · · · ⊕ Z α n . Finally, we denote by A = ( a ij ) ∈ M n ( Z ) the Cartan matrixassociated to these data. As g is simple, a ij ∈ { , − , − , − } for all i = j .Recall that the scalar product of two roots ( α, β ) is always an integer. As in [5], weassume that the short roots have length √ i ∈ { , . . . , n } , set q i := q ( αi,αi )2 and (cid:20) mk (cid:21) i := ( q i − q − i ) . . . ( q m − i − q − mi )( q mi − q − mi )( q i − q − i ) . . . ( q ki − q − ki )( q i − q − i ) . . . ( q m − ki − q k − mi )for all integers 0 ≤ k ≤ m . By convention, (cid:20) m (cid:21) i := 1 . The quantised enveloping algebra U q ( g ) of g over C associated to the previous data is the C -algebra generated by the indeterminates E , . . . , E n , F , . . . , F n , K ± , . . . , K ± n subject tothe following relations: K i K j = K j K i K i E j K − i = q a ij i E j and K i F j K − i = q − a ij i F j E i F j − F j E i = δ ij K i − K − i q i − q − i and the quantum Serre relations: − a ij X k =0 ( − k (cid:20) − a ij k (cid:21) i E − a ij − ki E j E ki = 0 ( i = j ) (1)and − a ij X k =0 ( − k (cid:20) − a ij k (cid:21) i F − a ij − ki F j F ki = 0 ( i = j ) . We refer the reader to [5, 13, 15] for more details on this (Hopf) algebra. Further, asusual, we denote by U + q ( g ) (resp. U − q ( g )) the subalgebra of U q ( g ) generated by E , . . . , E n (resp. F , . . . , F n ) and by U the subalgebra of U q ( g ) generated by K ± , . . . , K ± n . More-over, for all α = a α + · · · + a n α n ∈ Q , we set K α := K a · · · K a n n .
3s in the classical case, there is a triangular decomposition as vector spaces: U − q ( g ) ⊗ U ⊗ U + q ( g ) ≃ U q ( g ) . In this paper we are concerned with the algebra U + q ( g ) that admits the following presenta-tion, see [13, Theorem 4.21]. The algebra U + q ( g ) is (isomorphic to) the C -algebra generatedby n indeterminates E , . . . , E n subject to the quantum Serre relations (1). U + q ( g ) . To each reduced decomposition of the longest element w of the Weyl group W of g , Lusztighas associated a PBW basis of U + q ( g ), see for instance [19, Chapter 37], [13, Chapter 8] or[5, I.6.7]. The construction relates to a braid group action by automorphisms on U + q ( g ).Let us first recall this action. For all s ∈ N and i ∈ { , . . . , n } , we set[ s ] i := q si − q − si q i − q − i and [ s ] i ! := [1] i . . . [ s − i [ s ] i . As in [5, I.6.7], we denote by T i , for 1 ≤ i ≤ n , the automorphism of U + q ( g ) defined by: T i ( E i ) = − F i K i ,T i ( E j ) = − a ij X s =0 ( − s − a ij q − si E ( − a ij − s ) i E j E ( s ) i , i = jT i ( F i ) = − K − i E i ,T i ( F j ) = − a ij X s =0 ( − s − a ij q si F ( s ) i F j F ( − a ij − s ) i , i = jT i ( K α ) = K s i ( α ) , α ∈ Q, where E ( s ) i := E si [ s ] i ! and F ( s ) i := F si [ s ] i ! for all s ∈ N . It was proved by Lusztig that theautomorphisms T i satisfy the braid relations, that is, if s i s j has order m in W , then T i T j T i · · · = T j T i T j . . . , where there are exactly m factors on each side of this equality.The automorphisms T i can be used in order to describe PBW bases of U + q ( g ) as follows.It is well-known that the length of w is equal to the number N of positive roots of g . Let s i · · · s i N be a reduced decomposition of w . For k ∈ { , . . . , N } , we set β k := s i · · · s i k − ( α i k ). Then { β , . . . , β N } is exactly the set of positive roots of g . Similarly, wedefine elements E β k of U q ( g ) by E β k := T i · · · T i k − ( E i k ) . Note that the elements E β k depend on the reduced decomposition of w . The followingwell-known results were proved by Lusztig and Levendorskii-Soibelman.4 heorem 1.1 (Lusztig and Levendorskii-Soibelman)
1. For all k ∈ { , . . . , N } , the element E β k belongs to U + q ( g ) .2. If β k = α i , then E β k = E i .3. The monomials E k β · · · E k N β N , with k , . . . , k N ∈ N , form a linear basis of U + q ( g ) .4. For all ≤ i < j ≤ N , we have E β j E β i − q − ( β i ,β j ) E β i E β j = X a k i +1 ,...,k j − E k i +1 β i +1 · · · E k j − β j − , where each a k i +1 ,...,k j − belongs to C . As a consequence of this result, U + q ( g ) can be presented as a skew-polynomial algebra: U + q ( g ) = C [ E β ][ E β ; σ , δ ] · · · [ E β N ; σ N , δ N ] , where each σ i is a linear automorphism and each δ i is a σ i -derivation of the appropriatesubalgebra. In particular, U + q ( g ) is a noetherian domain and its group of invertible elementsis reduced to nonzero complex numbers. U + q ( g ) . We denote by Spec( U + q ( g )) the set of prime ideals of U + q ( g ). First, as q is not a root ofunity, it was proved by Ringel [21] (see also [10, Theorem 2.3]) that, as in the classicalsituation, every prime ideal of U + q ( g ) is completely prime.In order to study the prime and primitive spectra of U + q ( g ), we will use the stratifica-tion theory developed by Goodearl and Letzter. This theory allows the construction of apartition of these two sets by using the action of a suitable torus on U + q ( g ). More precisely,the torus H := ( C ∗ ) n acts naturally by automorphisms on U + q ( g ) via:( h , . . . , h n ) .E i = h i E i for all i ∈ { , . . . , n } . (It is easy to check that the quantum Serre relations are preserved by the group H .)Recall (see [4, 3.4.1]) that this action is rational. (We refer the reader to [5, II.2.] for thedefintion of a rational action.) A non-zero element x of U + q ( g ) is an H -eigenvector of U + q ( g )if h.x ∈ C ∗ x for all h ∈ H . An ideal I of U + q ( g ) is H -invariant if h.I = I for all h ∈ H . Wedenote by H - Spec( U + q ( g )) the set of all H -invariant prime ideals of U + q ( g ). It turns outthat this is a finite set by a theorem of Goodearl and Letzter about iterated Ore extensions,see [11, Proposition 4.2]. In fact, one can be even more precise in our situation. Indeed,in [12], Gorelik has also constructed a stratification of the prime spectrum of U + q ( g ) usingtools coming from representation theory. It turns out that her stratification coincides withthe H -stratification, so that we deduce from [12, Corollary 7.1.2] that Proposition 1.2 (Gorelik) U + q ( g ) has exactly | W | H -invariant prime ideals. H on U + q ( g ) allows via the H -stratification theory of Goodearl and Letzter(see [5, II.2]) the construction of a partition of Spec( U + q ( g )) as follows. If J is an H -invariant prime ideal of U + q ( g ), we denote by Spec J ( U + q ( g )) the H -stratum of Spec( U + q ( g )) associated to J . Recall that Spec J ( U + q ( g )) := { P ∈ Spec( U + q ( g )) | T h ∈H h.P = J } . Thenthe H -strata Spec J ( U + q ( g )) ( J ∈ H -Spec( U + q ( g ))) form a partition of Spec( U + q ( g )) (see [5,II.2]): Spec( U + q ( g )) = G J ∈H - Spec( U + q ( g )) Spec J ( U + q ( g )) . Naturally, this partition induces a partition of the set Prim( U + q ( g )) of all (left) primi-tive ideals of U + q ( g ) as follows. For all J ∈ H -Spec( U + q ( g )), we set Prim J ( U + q ( g )) :=Spec J ( U + q ( g )) ∩ Prim( U + q ( g )). Then it is obvious that the H -strata Prim J ( U + q ( g )) ( J ∈ H -Spec( U + q ( g ))) form a partition of Prim( U + q ( g )):Prim( U + q ( g )) = G J ∈H - Spec( U + q ( g )) Prim J ( U + q ( g )) . More interestingly, because of the finiteness of the set of H -invariant prime ideals of U + q ( g ),the H -stratification theory provides a useful tool to recognise primitive ideals withouthaving to find all its irreductible representations! Indeed, following previous works ofHodges-Levasseur, Joseph, and Brown-Goodearl, Goodearl and Letzter have characterisedthe primitive ideals of U + q ( g ) as follows, see [11, Corollary 2.7] or [5, Theorem II.8.4]. Theorem 1.3 (Goodearl-Letzter)
Prim J ( U + q ( g )) ( J ∈ H - Spec( U + q ( g )) ) coincides withthose primes in Spec J ( U + q ( g )) that are maximal in Spec J ( U + q ( g )) . U + q ( g ) . In this section, we investigate the automorphism group of U + q ( g ) viewed as the algebragenerated by n indeterminates E , . . . , E n subject to the quantum Serre relations. Thisalgebra has some well-identified automorphisms. First, there are the so-called torus auto-morphisms; let H = ( C ∗ ) n , where n still denotes the rank of g . As U + q ( g ) is the C -algebragenerated by n indeterminates subject to the quantum Serre relations, it is easy to checkthat each ¯ λ = ( λ , . . . , , λ n ) ∈ H determines an algebra automorphism φ ¯ λ of U + q ( g ) with φ ¯ λ ( E i ) = λ i E i for i ∈ { , . . . , n } , with inverse φ − λ = φ ¯ λ − . Next, there are the so-called di-agram automorphisms coming from the symmetries of the Dynkin diagram of g . Namely,let w be an automorphism of the Dynkin diagram of g , that is, w is an element of thesymmetric group S n such that ( α i , α j ) = ( α w ( i ) , α w ( j ) ) for all i, j ∈ { , . . . , n } . Then onedefines an automorphism, also denoted w , of U + q ( g ) by: w ( E i ) = E w ( i ) . Observe that φ ¯ λ ◦ w = w ◦ φ ( λ w (1) ,...,,λ w ( n ) ) .
6e denote by G the subgroup of Aut( U + q ( g )) generated by the torus automorphismsand the diagram automorphisms. Observe that G ≃ H ⋊ autdiagr( g ) , where autdiagr( g ) denotes the set of diagram automorphisms of g .The group Aut( U + q ( s l )) was computed independently by Alev and Dumas, see [2,Proposition 2.3] , and Caldero, see [8, Proposition 4.4]; their results show that, in the casewhere g = sl , we have Aut( U + q ( s l )) = G. About ten years later, Andruskiewitsch and Dumas investigated the case where g = so ,see [4]. In this case, they obtained partial results that lead them to the following conjec-ture. Conjecture (Andruskiewitsch-Dumas, [4, Problem 1]):
Aut( U + q ( g )) = G. This conjecture was recently confirmed in two new cases: g = so , [16], and g = sl , [18].Our aim in this section is to show how one can use the action of the automorphism groupof U + q ( g ) on the primitive spectrum of this algebra in order to prove the Andruskiewitsch-Dumas Conjecture in the cases where g = sl and g = so . U + q ( g ) . Recall that an element a of U + q ( g ) is normal provided the left and right ideals generatedby a in U + q ( g ) coincide, that is, if aU + q ( g ) = U + q ( g ) a. In the sequel, we will use several times the following well-known result concerningnormal elements of U + q ( g ). Lemma 2.1
Let u and v be two nonzero normal elements of U + q ( g ) such that h u i = h v i .Then there exist λ, µ ∈ C ∗ such that u = λv and v = µu .Proof. It is obvious that units λ, µ exist with these properties. However, the set of unitsof U + q ( g ) is precisely C ∗ . (cid:3) .2 N -grading on U + q ( g ) and automorphisms. As the quantum Serre relations are homogeneous in the given generators, there is an N -grading on U + q ( g ) obtained by assigning to E i degree 1. Let U + q ( g ) = M i ∈ N U + q ( g ) i (2)be the corresponding decomposition, with U + q ( g ) i the subspace of homogeneous elementsof degree i . In particular, U + q ( g ) = C and U + q ( g ) is the n -dimensional space spannedby the generators E , . . . , E n . For t ∈ N set U + q ( g ) ≥ t = L i ≥ t U + q ( g ) i and define U + q ( g ) ≤ t similarly.We say that the nonzero element u ∈ U + q ( g ) has degree t , and write deg( u ) = t , if u ∈ U + q ( g ) ≤ t \ U + q ( g ) ≤ t − (using the convention that U + q ( g ) ≤− = { } ). As U + q ( g ) is adomain, deg( uv ) = deg( u ) + deg( v ) for u, v = 0. Definition 2.2
Let A = L i ∈ N A i be an N -graded C -algebra with A = C which is generatedas an algebra by A = C x ⊕ · · · ⊕ C x n . If for each i ∈ { , . . . , n } there exist = a ∈ A and a scalar q i,a = 1 such that x i a = q i,a ax i , then we say that A is an N -graded algebrawith enough q -commutation relations. The algebra U + q ( g ), endowed with the grading just defined, is a connected N -gradedalgebra with enough q -commutation relations. Indeed, if i ∈ { , . . . , n } , then there exists u ∈ U + q ( g ) such that E i u = q • uE i where • is a nonzero integer. This can be proved asfollows. As g is simple, there exists an index j ∈ { , . . . , n } such that j = i and a ij = 0,that is, a ij ∈ {− , − , − } . Then s i s j is a reduced expression in W , so that one can finda reduced expression of w starting with s i s j , that is, one can write w = s i s j s i . . . s i N . With respect to this reduced expression of w , we have with the notation of Section 1.2: β = α i and β = s i ( α j ) = α j − a ij α i Then it follows from Theorem 1.1 that E β = E i , E β = E α j − a ij α i and E i E β = q ( α i ,α j − a ij α i ) E β E i , that is, E i E β = q − ( α i ,α j ) E β E i . As a ij = 0, we have ( α i , α j ) = 0 and so q − ( α i ,α j ) = 1 since q is not a root of unity. So wehave just proved: Proposition 2.3 U + q ( g ) is a connected N -graded algebra with enough q -commutation re-lations. N -graded algebras with enough q -commutation relations isthat any automorphism of such an algebra must conserve the valuation associated to the N -graduation. More precisely, as U + q ( g ) is a connected N -graded algebra with enough q -commutation relations, we deduce from [18] (see also [17, Proposition 3.2]) the followingresult. Corollary 2.4
Let σ ∈ Aut( U + q ( g )) and x ∈ U + q ( g ) d \ { } . Then σ ( x ) = y d + y >d , forsome y d ∈ U + q ( g ) d \ { } and y >d ∈ U + q ( g ) ≥ d +1 . g = sl . In this section, we investigate the automorphism group of U + q ( g ) in the case where g = sl .In this case the Cartan matrix is A = (cid:18) − − (cid:19) , so that U + q ( s l ) is the C -algebragenerated by two indeterminates E and E subject to the following relations: E E − ( q + q − ) E E E + E E = 0 (3) E E − ( q + q − ) E E E + E E = 0 (4)We often refer to this algebra as the quantum Heisenberg algebra, and sometimes we denoteit by H , as in the classical situation the enveloping algebra of sl +3 is the so-called Heisenbergalgebra.We now make explicit a PBW basis of H . The Weyl group of sl is isomorphic to thesymmetric group S , where s is identified with the transposition (1 2) and s is identifiedwith (2 3). Its longest element is then w = (13); it has two reduced decompositions: w = s s s = s s s . Let us choose the reduced decomposition s s s of w in order toconstruct a PBW basis of U + q ( s l ). According to Section 1.2, this reduced decompositionleads to the following root vectors: E α = E , E α + α = T ( E ) = − E E + q − E E and E α = T T ( E ) = E . In order to simplify the notation, we set E := − E E + q − E E . Then, it follows fromTheorem 1.1 that • The monomials E k E k E k , with k , k , k nonnegative integers, form a PBW-basisof U + q ( s l ). • H is the iterated Ore extension over C generated by the indeterminates E , E , E subject to the following relations: E E = q − E E , E E = q − E E , E E = qE E + qE . In particular, H is a Noetherian domain, and its group of invertible elements isreduced to C ∗ . 9 It follows from the previous commutation relations between the root vectors that E is a normal element in H , that is, E H = H E .In order to describe the prime and primitive spectra of H , we need to introduce twoother elements. The first one is the root vector E ′ := T ( E ) = − E E + q − E E . Thisroot vector would have appeared if we have choosen the reduced decomposition s s s of w in order to construct a PBW basis of H . It follows from Theorem 1.1 that E ′ q -commuteswith E and E , so that E ′ is also a normal element of H . Moreover, one can describe thecentre of H using the two normal elements E and E ′ . Indeed, in [3, Corollaire 2.16], Alevand Dumas have described the centre of U + q ( s l n ); independently Caldero has described thecentre of U + q ( g ) for arbitrary g , see [7]. In our particular situation, their results show thatthe centre Z ( H ) of H is a polynomial ring in one variable Z ( H ) = C [Ω], where Ω = E E ′ .We are now in position to describe the prime and primitive spectra of H = U + q ( sl (3));this was first achieved by Malliavin who obtained the following picture for the poset ofprime ideals of H , see [20, Th´eor`eme 2.4]: hh E , E − β ii AAAAAAAAAAAAAAAA hh E , E ii (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) =============== hh E − α, E ii }}}}}}}}}}}}}}}}} h E i }}}}}}}}}}}}}}}} TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT h E i jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj AAAAAAAAAAAAAAAAA hh E ii OOOOOOOOOOOOOOOOOOOOOOOOOO hh Ω − γ ii hh E ′ ii oooooooooooooooooooooooooo h i where α, β, γ ∈ C ∗ .Recall from Section 1.3 that the torus H = ( C ∗ ) acts on U + q ( s l ) by automorphismsand that the H -stratification theory of Goodearl and Letzter constructs a partition of theprime spectrum of U + q ( s l ) into so-called H -strata, this partition being indexed by the H -invariant prime ideals of U + q ( s l ). Using this description of Spec( U + q ( s l )), it is easy toidentify the 6 = | W | H -invariant prime ideals of H and their corresponding H -strata. As E , E , E and E ′ are H -eigenvectors, the 6 H -invariant primes are: h i , h E i , h E ′ i , h E i , h E i and h E , E i . Moreover the corresponding H -strata are:Spec h i ( H ) = {h i} ∪ {h Ω − γ i | γ ∈ C ∗ } , 10pec h E i ( H ) = {h E i} ,Spec h E ′ i ( H ) = {h E ′ i} ,Spec h E i ( H ) = {h E i} ∪ {h E , E − β i | β ∈ C ∗ } ,Spec h E i ( H ) = {h E i} ∪ {h E − α, E i | α ∈ C ∗ } and Spec h E ,E i ( H ) = {h E , E i} .We deduce from this description of the H -strata and the the fact that primitive idealsare exactly those primes that are maximal within their H -strata, see Theorem 1.3, thatthe primitive ideals of U + q ( s l ) are exactly those primes that appear in double brackets inthe previous picture.We now investigate the group of automorphisms of H = U + q ( s l ). In that case, the torusacting naturally on U + q ( s l ) is H = ( C ∗ ) , there is only one non-trivial diagram automor-phism w that exchanges E and E , and so the subgroup G of Aut( U + q ( s l )) generated bythe torus and diagram automorphisms is isomorphic to the semi-direct product ( C ∗ ) ⋊ S .We want to prove that Aut( U + q ( s l )) = G .In order to do this, we study the action of Aut( U + q ( s l )) on the set of primitive idealsthat are not maximal. As there are only two of them, h E i and h E ′ i , an automorphism of H will either fix them or permute them.Let σ be an automorphism of U + q ( s l ). It follows from the previous observation thateither σ ( h E i ) = h E i and σ ( h E ′ i ) = h E ′ i , or σ ( h E i ) = h E ′ i and σ ( h E ′ i ) = h E i . As it is clear that the diagram automorphism w permutes the ideals h E i and h E ′ i , we getthat there exists an automorphism g ∈ G such that g ◦ σ ( h E i ) = h E i and g ◦ σ ( h E ′ i ) = h E ′ i . Then, as E and E ′ are normal, we deduce from Lemma 2.1 that there exist λ, λ ′ ∈ C ∗ such that g ◦ σ ( E ) = λE and g ◦ σ ( E ′ ) = λ ′ E ′ . In order to prove that g ◦ σ is an element of G , we now use the N -graduation of U + q ( s l )introduced in Section 2.2. With respect to this graduation, E and E are homogeneous ofdegree 1, and so E and E ′ are homogeneous of degree 2. Moreover, as ( q − − E E = E + q − E ′ , we deduce from the above discussion that g ◦ σ ( E E ) = 1 q − − (cid:0) λE + q − λ ′ E ′ (cid:1) has degree two. On the other hand, as U + q ( s l ) is a connected N -graded algebra withenough q -commutation relations by Proposition 2.3, it follows from Corollary 2.4 that σ ( E ) = a E + a E + u and σ ( E ) = b E + b E + v , where ( a , a ) , ( b , b ) ∈ C \{ (0 , } ,and u, v ∈ U + q ( s l ) are linear combinations of homogeneous elements of degree greater thanone. As g ◦ σ ( E ) .g ◦ σ ( E ) has degree two, it is clear that u = v = 0. To conclude that11 ◦ σ ∈ G , it just remains to prove that a = 0 = b . This can be easily shown byusing the fact that g ◦ σ ( − E E + q − E E ) = g ◦ σ ( E ) = λE ; replacing g ◦ σ ( E ) and g ◦ σ ( E ) by a E + a E and b E + b E respectively, and then identifying the coefficientsin the PBW basis, leads to a = 0 = b , as required. Hence we have just proved that g ◦ σ ∈ G , so that σ itself belongs to G the subgroup of Aut( U + q ( s l )) generated by thetorus and diagram automorphisms. Hence one can state the following result that confirmsthe Andruskiewitsch-Dumas Conjecture. Proposition 2.5
Aut( U + q ( s l )) ≃ ( C ∗ ) ⋊ autdiagr( sl )This result was first obtained independently by Alev and Dumas, [2, Proposition 2.3],and Caldero, [8, Proposition 4.4], but using somehow different methods; they studied thisautomorphism group by looking at its action on the set of normal elements of U + q ( s l ). g = so . In this section we investigate the automorphism group of U + q ( g ) in the case where g = so .In this case there are no diagram automorphisms, so that the Andruskiewitsch-DumasConjecture asks whether every automorphism of U + q ( so ) is a torus automorphism. In [16]we have proved their conjecture when g = so . The aim of this section is to present aslightly different proof based both on the original proof and on the recent proof by S.A.Lopes and the author of the Andruskiewitsch-Dumas Conjecture in the case where g is oftype A .In the case where g = so , the Cartan matrix is A = (cid:18) − − (cid:19) , so that U + q ( so ) isthe C -algebra generated by two indeterminates E and E subject to the following relations: E E − ( q + 1 + q − ) E E E + ( q + 1 + q − ) E E E + E E = 0 (5) E E − ( q + q − ) E E E + E E = 0 (6)We now make explicit a PBW basis of U + q ( so ). The Weyl group of so is isomorphicto the dihedral group D (4). Its longest element is w = − id ; it has two reduced decom-positions: w = s s s s = s s s s . Let us choose the reduced decomposition s s s s of w in order to construct a PBW basis of U + q ( so ). According to Section 1.2, this reduceddecomposition leads to the following root vectors: E α = E , E α + α = T ( E ) = 1( q + q − ) (cid:0) E E − q − ( q + q − ) E E E + q − E E (cid:1) , E α + α = T T ( E ) = − E E + q − E E and E α = T T T ( E ) = E . In order to simplify the notation, we set E := − E α + α and E := E α + α . Then, itfollows from Theorem 1.1 that • The monomials E k E k E k E k , with k , k , k , k nonnegative integers, form a PBW-basis of U + q ( so ). 12 U + q ( so ) is the iterated Ore extension over C generated by the indeterminates E , E , E , E subject to the following relations: E E = q − E E E E = E E − ( q + q − ) E , E E = q − E E ,E E = q E E − q E , E E = E E − q − q + q − E , E E = q − E E . In particular, U + q ( so ) is a Noetherian domain, and its group of invertible elementsis reduced to C ∗ .Before describing the automorphism group of U + q ( so ), we first describe the centre andthe primitive ideals of U + q ( so ). The centre of U + q ( g ) has been described in general byCaldero, [7]. In the case where g = so , his result shows that Z ( U + q ( so )) is a polynomialalgebra in two indeterminates Z ( U + q ( so )) = C [ z, z ′ ] , where z = (1 − q ) E E + q ( q + q − ) E and z ′ = − ( q − q − )( q + q − ) E E + q ( q − E . Recall from Section 1.3 that the torus H = ( C ∗ ) acts on U + q ( so ) by automorphismsand that the H -stratification theory of Goodearl and Letzter constructs a partition of theprime spectrum of U + q ( so ) into so-called H -strata, this partition being indexed by the8 = | W | H -invariant prime ideals of U + q ( so ). In [16], we have described these eight H -strata. More precisely, we have obtained the following picture for the poset Spec( U + q ( so )),13 h E , E − β ii CCCCCCCCCCCCCCCCC hh E , E ii zzzzzzzzzzzzzzzzz DDDDDDDDDDDDDDDDD hh E − α, E ii {{{{{{{{{{{{{{{{{ h E i RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR h E i lllllllllllllllllllllllllllllll hh z, z ′ − δ ii CCCCCCCCCCCCCCCCC hh E ii RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR hh z − γ, z ′ − δ ii hh E ′ ii lllllllllllllllllllllllllllllll hh z − γ, z ′ ii {{{{{{{{{{{{{{{{{ h z i DDDDDDDDDDDDDDDDDD
I h z ′ i zzzzzzzzzzzzzzzzz h i where α, β, γ, δ ∈ C ∗ , E ′ := E E − q E E and I = {h P ( z, z ′ ) i | P is a unitary irreductible polynomial of C [ z, z ′ ] , P = z, z ′ } . As the primitive ideals are those primes that are maximal in their H -strata, see Theorem1.3, we deduced from this description of the prime spectrum that the primitive ideals of U + q ( so ) are the following: • h z − α, z ′ − β i with ( α, β ) ∈ C \ { (0 , } . • h E i and h E ′ i . • h E − α, E − β i with ( α, β ) ∈ C such that αβ = 0.(They correspond to the “double brackets” prime ideals in the above picture.)Among them, two only are not maximal, h E i and h E ′ i . Unfortunately, as E and E ′ are not normal in U + q ( so ), one cannot easily obtain information using the fact thatany automorphism of U + q ( so ) will either preserve or exchange these two prime ideals.Rather than using this observation, we will use the action of Aut( U + q ( so )) on the setof maximal ideals of height two. Because of the previous description of the primitivespectrum of U + q ( so ), the height two maximal ideals in U + q ( so ) are those h z − α, z ′ − β i with ( α, β ) ∈ C \ { (0 , } . In [16, Proposition 3.6], we have proved that the group of unitsof the factor algebra U + q ( so ) / h z − α, z ′ − β i is reduced to C ∗ if and only if both α and β are nonzero. Consequently, if σ is an automorphism of U + q ( so ) and α ∈ C ∗ , we get that: σ ( h z − α, z ′ i ) = h z − α ′ , z ′ i or h z, z ′ − β ′ i , α ′ , β ′ ∈ C ∗ . Similarly, if σ is an automorphism of U + q ( so ) and β ∈ C ∗ , we get that: σ ( h z, z ′ − β i ) = h z − α ′ , z ′ i or h z, z ′ − β ′ i , (7)where α ′ , β ′ ∈ C ∗ .We now use this information to prove that the action of Aut( U + q ( so )) on the centre of U + q ( so ) is trivial. More precisely, we are now in position to prove the following result. Proposition 2.6
Let σ ∈ Aut( U + q ( so )) . There exist λ, λ ′ ∈ C ∗ such that σ ( z ) = λz and σ ( z ′ ) = λ ′ z ′ . Proof.
We only prove the result for z . First, using the fact that U + q ( so ) is noetherian, itis easy to show that, for any family { β i } i ∈ N of pairwise distinct nonzero complex numbers,we have: h z i = \ i ∈ N P ,β i and h z ′ i = \ i ∈ N P β i , , where P α,β := h z − α, z ′ − β i . Indeed, if the inclusion h z i ⊆ I := \ i ∈ N P ,β i is not an equality, then any P ,β i is a minimal prime over I for height reasons. As the P ,β i are pairwise distinct, I is a two-sided ideal of U + q ( so ) with infinitely many prime idealsminimal over it. This contradicts the noetherianity of U + q ( so ). Hence h z i = \ i ∈ N P ,β i and h z ′ i = \ i ∈ N P β i , , and so σ ( h z i ) = \ i ∈ N σ ( P ,β i ) . It follows from (7) that, for all i ∈ N , there exists ( γ i , δ i ) = (0 ,
0) with γ i = 0 or δ i = 0such that σ ( P ,β i ) = P γ i ,δ i . Naturally, we can choose the family { β i } i ∈ N such that either γ i = 0 for all i ∈ N , or δ i = 0 for all i ∈ N . Moreover, observe that, as the β i are pairwise distinct, so are the γ i or the δ i .Hence, either σ ( h z i ) = \ i ∈ N P γ i , , σ ( h z i ) = \ i ∈ N P ,δ i , that is, either h σ ( z ) i = σ ( h z i ) = h z ′ i or h σ ( z ) i = σ ( h z i ) = h z i . As z , σ ( z ) and z ′ are all central, it follows from Lemma 2.1 that there exists λ ∈ C ∗ such that either σ ( z ) = λz or σ ( z ) = λz ′ .To conclude, it just remains to show that the second case cannot happen. In order to dothis, we use a graded argument. Observe that, with respect to the N -graduation of U + q ( so )defined in Section 2.2, z and z ′ are homogeneous of degree 3 and 4 respectively. Thus, if σ ( z ) = λz ′ , then we would obtain a contradiction with the fact that every automorphismof U + q ( so ) preserves the valuation, see Corollary 2.4. Hence σ ( z ) = λz , as desired. Thecorresponding result for z ′ can be proved in a similar way, so we omit it. (cid:3) Andruskiewitsch and Dumas, [4, Proposition 3.3], have proved that the subgroup ofthose automorphisms of U + q ( so ) that stabilize h z i is isomorphic to ( C ∗ ) . Thus, as wehave just shown that every automorphism of U + q ( so ) fixes h z i , we get that Aut( U + q ( so ))itself is isomorphic to ( C ∗ ) . This is the route that we have followed in [16] in order toprove the Andruskiewitsch-Dumas Conjecture in the case where g = so . Recently, withSamuel Lopes, we proved this Conjecture in the case where g = sl using different methodsand in particular graded arguments. We are now using (similar) graded arguments toprove that every automorphism of U + q ( so ) is a torus automorphism (witout using resultsof Andruskiewitsch and Dumas).In the proof, we will need the following relation that is easily obtained by straightfor-ward computations. Lemma 2.7 ( q − E E ′ = ( q − zE + q z ′ . Proposition 2.8
Let σ be an automorphism of U + q ( so ) . Then there exist a , b ∈ C ∗ suchthat σ ( E ) = a E and σ ( E ) = b E . Proof.
For all i ∈ { , . . . , } , we set d i := deg( σ ( E i )). We also set d ′ := deg( σ ( E ′ )). Itfollows from Corollary 2.4 that d , d ≥ d , d ′ ≥ d ≥
3. First we prove that d = d = 1.Assume first that d + d >
3. As z = (1 − q ) E E + q ( q + q − ) E and σ ( z ) = λz with λ ∈ C ∗ by Proposition 2.6, we get: λz = (1 − q ) σ ( E ) σ ( E ) + q ( q + q − ) σ ( E ) . (8)Recall that deg( uv ) = deg( u ) + deg( v ) for u, v = 0, as U + q ( g ) is a domain. Thus, asdeg( z ) = 3 < deg( σ ( E ) σ ( E )) = d + d , we deduce from (8) that d + d = d . As z ′ = − ( q − q − )( q + q − ) E E + q ( q − E and deg( z ′ ) = 4 < d + d + d = d + d =16eg( σ ( E ) σ ( E )), we get in a similar manner that d + d = 2 d . Thus d + d = d . As d + d >
3, this forces d > d + d ′ >
4. Thus we deduce from Lemma 2.7 that d + d ′ = 3 + d . Hence d + d ′ = 3. As d ≥ d ′ ≥
2, this implies d = 1 and d ′ = 2.Thus we have just proved that d = deg( σ ( E )) = 1 and either d = 2 or d ′ = 2. Toprove that d = 1, we distinguish between these two cases.If d = 2, then as previously we deduce from the relation z ′ = − ( q − q − )( q + q − ) E E + q ( q − E that d + d = 4, so that d = 1, as desired.If d ′ = 2, then one can use the definition of E ′ and the previous expression of z ′ inorder to prove that z ′ = q − ( q − E ′ + E u , where u is a nonzero homogeneous elementof U + q ( so ) of degree 3. ( u is nonzero since h z ′ i is a completely prime ideal and E ′ / ∈ h z ′ i for degree reasons.) As d ′ = 2 and deg( σ ( z ′ )) = 4, we get as previously that d = 1.To summarise, we have just proved that deg( σ ( E )) = 1 = deg( σ ( E )), so that σ ( E ) = a E + a E and σ ( E ) = b E + b E , where ( a , a ) , ( b , b ) ∈ C \ { (0 , } . To concludethat a = b = 0, one can for instance use the fact that σ ( E ) and σ ( E ) must satisfy thequantum Serre relations. (cid:3) We have just confirmed the Andruskiewitsch-Dumas Conjecture in the case where g = so . Theorem 2.9
Every automorphism of U + q ( so ) is a torus automorphism, so that Aut( U + q ( so )) ≃ ( C ∗ ) . To finish this overview paper, let us mention that recently the Andruskiewitsch-DumasConjecture was confirmed by Samuel Lopes and the author, [18], in the case where g = sl .The crucial step of the proof is to prove that, up to an element of G , every normal elementof U + q ( sl ) is fixed by every automorphism. This step was dealt with by first computingthe Lie algebra of derivations of U + q ( sl ), and this already requires a lot of computations! Acknowledgments.
I thank Jacques Alev, Fran¸cois Dumas, Tom Lenagan and SamuelLopes for all the interesting conversations that we have shared on the topics of thispaper. I also like to thank the organisers of the Workshop ”From Lie Algebras toQuantum Groups” (and all the participants) for this wonderful meeting. Finally, I wouldlike to express my gratitude for the hospitality received during my subsequent visit to theUniversity of Porto, especially from Paula Carvalho Lomp, Christian Lomp and SamuelLopes.
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