"Classical" flag varieties for quantum groups: the standard quantum SL(n,C)

28 Pages·0.37 MB·English

by  Christian Ohn

“CLASSICAL” FLAG VARIETIES FOR QUANTUM GROUPS:
THE STANDARD QUANTUM SL(n,C)
1
0
0
CHRISTIAN OHN
2

n
a Abstract. We suggest a possible programme to associate geometric “flag-
J like”datatoanarbitrarysimplequantumgroup,inthespiritofthenoncom-
mutative algebraic geometry developed by Artin, Tate, and Van den Bergh.
5 WethencarryoutthisprogrammeforthestandardquantumSL(n)ofDrinfel′d
1
andJimbo,wherethevarietiesinvolvedarecertainT-stablesubvarietiesofthe

(ordinary)flagvariety.
]
A
Q
.
h
t Introduction
a
m The study of quantum analogues of flag varieties, first suggested by Manin [31],
[ hasbeenundertakenduringthepastdecadebyseveralauthors,fromvariouspoints

of view; see e.g. [1, 8, 13, 16, 17, 23, 26, 29, 35, 38, 39, 40]. Around the same time,
2
an approach to noncommutative projective algebraic geometry was initiated by
v
5 Artin, Tate, and Van den Bergh [4, 5, 6], and considerably developed since (see
0 e.g. [3, 7, 9, 24, 28, 37, 41, 42, 44]). One attractive feature of their approachis the
0 association of actual geometric data to certain classes of graded noncommutative
7
algebras.
0
The present work is an attempt to study quantum flag varieties from this point
0
0 of view. As a consequence, our “quantum flag varieties” will be actual varieties
/ (with some bells and whistles).
h
Recall the originalidea of [4,5, 6]. If A is the homogeneouscoordinatering of a
t
a projectiveschemeE,thenthepointsofEareinone-to-onecorrespondencewiththe
m
isomorphismclassesofso-calledpoint modules ofA,i.e.N-gradedcyclicA-modules
: P suchthatdimP =1foralln. NowifAisanN-gradednoncommutativealgebra,
v n
i one may still try to parametrizeits point modules by the points of some projective
X
scheme E. Of course, one cannot hope to reconstructA from E alone,but there is
r now an additional ingredient: the shift operation σ : P 7→ P[1], where P[1] is the
a
N-gradedA-moduledefinedbyP[1] :=P . (WhenAiscommutative,this shift
n n+1
is trivial: P[1] ≃ P for every point module P.) Assume that σ may be viewed as
an automorphism of E: one may then hope, at least in “good” cases, to recover A
fromthetriple(E,σ,L),whereListhelinebundleoverE definedbyitsembedding
into a projective space. The first step of this recovery is the construction of the
twisted homogeneous coordinate ring B(E,σ,L) of a triple (E,σ,L), defined in [6]
as follows:
B(E,σ,L)= B , B :=H0(E,L⊗Lσ⊗...⊗Lσn−1)
n n
n∈N
M
Date:June30,2000; thisversionJanuary12,2001.
2000 Mathematics Subject Classification. Primary20G42; Secondary14M15,16S38,17B37.
1
2 CHRISTIAN OHN
(where Lσ denotes the pullback of L along σ), the multiplication being given by
αβ :=α⊗βσm for allα∈B , β ∈B . (When σ is the identity, this algebrais the
m n
(commutative)homogeneouscoordinate ring of E w.r.t.the polarizationL.) If the
triple (E,σ,L) comes from an algebra A as above, the second step then consists
in analysing the canonical morphism A → B(E,σ,L). The initial success of this
method has been a complete study of all regular algebras of dimension three [4]
(where the kernel of A→B(E,σ,L) turns out to be generated by a single element
of degree three).
The present paper is organized as follows.
In Part 1, we give a general outline of a possible theory of flag varieties for
quantumgroups,usingamultigradedversionoftheideasof[4,5,6]recalledabove,
some of which have already been introduced by Chan [10]. This Part is largely
conjecturalandcontainsno(significant)newresults;its purposeisrathertosetup
a framework that will be used in Parts 2 and 3.
More specifically, we proceed as follows. Let G be a simple complex group;
our interest in flag varieties allows us to assume without harm that G is simply
connected. Let P+ be the monoid of dominant integral weights of G (w.r.t. some
BorelsubgroupB ⊂G): theshape algebra M ofGisaP+-gradedG-algebrawhose
term of degree λ is the irreducible representation of G of highest weight λ. Now
considerthedefinitionofapointmodule(seeabove),butwithN-gradingsreplaced
by P+-gradings: we obtain the notion of a flag module of M (Definition 2.1); this
terminology is justified by the fact that the isomorphism classes of such modules
are indeed parametrized by the points of the flag variety G/B (Proposition 2.2).
If a quantum group has the “same” representation theory as G (in the sense of
Definition1.1),then thewe maystilldefine a(P+-graded)shapealgebra. We then
discussthe possibilitytoparametrizethelatter’sflagmodules(uptoisomorphism)
by the points of some scheme E, and to realize shifts F 7→ F[λ] (λ ∈ P+) as
automorphisms σ of E. It will of course be sufficient to know the automorphisms
λ
σ ,...,σ associatedto the fundamental weights ̟ ,...,̟ , which freely generate
1 ℓ 1 ℓ
P+. Moreover, since we are in a multigraded situation, it will be more natural to
view E as a subscheme of a product of ℓ projective spaces, corresponding to ℓ line
bundles L ,...,L over E.
1 ℓ
We thenconsiderthe converseproblemofreconstructingtheshapealgebrafrom
E,theσ ,andtheL ,usingChan’sconstruction[10]ofatwistedmultihomogeneous
i i
coordinate ring: this is the P+-graded algebra
B(E,σ ,...,σ ,L ,...,L ):= H0(E,L ),
1 ℓ 1 ℓ λ
λM∈P+
where the line bundles L are constructed inductively from the rules L = L ,
λ ̟i i
L = L ⊗ Lσi. (Again, if E = G/B, if each σ is the identity, and if the
̟i+λ i λ i
L are the line bundles associated to the fundamental G-modules V1,...,Vℓ, then
i
this algebra is the (commutative) multihomogeneous coordinate ring of G/B ⊂
P(V1)×···×P(Vℓ), which in turn is equal to the shape algebra O(G/U), U the
unipotent radical of B.)
WestressthattheideasdevelopedinthisPartarenot restrictedtothestandard
quantumgroupsofDrinfel′d[14]andJimbo[22],butcould,inprinciple,beapplied
to other quantum groups as well, as long as they have the “same” representation
theory as a given simple complex group. (Potential other examples include the
A “CLASSICAL” FLAG VARIETY FOR STANDARD QUANTUM SL(n) 3
multiparameter quantum groups of Artin, Schelter, and Tate [2, 19], the quantum
SL(n) of Cremmer and Gervais [11, 18], or the quantum SL(3)’s classified in [32].)
InParts2and3,wedo restrictourselvestoastandardDrinfel′d-Jimboquantum
groupODJ(G),withq notarootofunity. Thankstothe resultsofLusztig[30]and
q
Rosso [34], ODJ(G) has the “same” representation theory as the group G, so one
q
can define a shape algebra MDJ.
In Part2, we construct geometric data EDJ, σ , and L , and we conjecture that
i i
thesedataindeedcorrespondtoMDJ asdescribedabove(Conjectures9.1and9.2).
The scheme EDJ will actually be a union of certain T-stable subvarieties of the
(ordinary) flag variety G/B. Since the latter may be of independent interest to
algebraicgeometers, we have decided to describe them in a separate note [33] (but
we recall their construction here, without proofs).
InPart3,weproveConjecture9.1forG=SL(n),thusobtaininga“flagvariety”
for the standard Drinfel′d-Jimbo quantum SL(n). The proof uses special features
ofthe groupSL(n)(the Weylgroupis thesymmetricgroup,allfundamentalrepre-
sentations are minuscule, etc.) and is essentially combinatorial; it is therefore not
likely to be extendable to an arbitrary G.
Acknowledgement. The author would like to thank the Universit´e de Reims
for granting a sabbatical leave during the year 1999–2000, when part of this work
has been done.
Conventions. All vector spaces, dimensions, algebras, tensor products, vari-
eties, schemes, etc. will be over the field C of complex numbers. If G is a linear
algebraicgroup,wedenotebyO(G)theHopfalgebraofpolynomialfunctionsonG.
If A is a (co)algebra,then the dual of a left A-(co)module is a rightA-(co)module,
and vice-versa; morphisms of A-(co)modules will simply be called A-morphisms.
When V is a vector space and v ∈V is nonzero, we will sometimes still denote by
v the corresponding point in the projective space P(V).
Contents
Introduction 1
Part 1. An approach to quantum flag varieties: general outline 4
1. Simple quantum groups and their shape algebras 4
2. The scheme of flag modules 6
3. Braided tuples and reconstruction of shape algebras 8
Part 2. A conjectural flag tuple for the standard Drinfel′d-Jimbo
quantum groups 10
4. Recollections on UDJ(g) and ODJ(G) 10
q q
5. Recollections from [33] 11
6. Monogressive orthocells and the variety EDJ 11
7. The automorphisms σ ,...,σ 12
1 ℓ
8. The line bundles L ,...,L 12
1 ℓ
9. Main conjectures and result 13
Part 3. The standard quantum SL(n) 13
10. The varieties E(C) 13
4 CHRISTIAN OHN
11. Monogressivity 14
12. The automorphisms σ 15
i
13. Proof of Conjecture 9.1 for G=SL(n) 16
Appendix A. Proof of Proposition 1.3 20
Appendix B. Proof of Proposition 7.1 21
Appendix C. Proof of Lemma 13.4 22
References 26
Part 1. An approach to quantum flag varieties: general outline
This Part contains no (significant) new results. It rather discusses a possible
theory of flag varieties for simple quantum groups, asking several questions along
the way (as well as two ambitious problems at the end).
Mostofwhatwewillsayhereis amultigradedversionofsomeofthe mainideas
of [4, 5, 6], applied in a Lie-theoretic setting.
1. Simple quantum groups and their shape algebras
Let G be a simply connected simple complex group, B ⊂ G a Borel subgroup,
and P+ the set of dominant integral weights of G w.r.t. B. For each λ,µ,ν ∈P+,
denote by
• dλ the dimension of the simple G-module of highest weight λ, and by
• cλµ the multiplicity of the simple G-module of highest weight ν inside the
ν
tensor product of those of highest weights λ and µ.
BearinginmindthatthealgebraO(G)ofpolynomialfunctionsonGisacommuta-
tiveHopf algebra,andthat(finite-dimensional) leftG-modules correspondto right
O(G)-comodules, recall the following definition from [32].
Definition 1.1. We call a quantum G any (not necessarily commutative) Hopf
algebra A (over C) such that
(a) there is a family {Vλ |λ∈P+}ofsimple andpairwisenonisomorphic(right)
A-comodules, with dimVλ =dλ,
(b) every A-comodule is isomorphic to a direct sum of these,
(c) for every λ,µ∈P+, Vλ⊗Vµ is isomorphic to cλµVν.
ν ν
For convenience, we will write L
V :=(Vλ)∗.
λ
For every λ,µ ∈ P+, Definition 1.1(c) yields an injective A-morphism Vλ+µ →
Vλ⊗Vµ that is unique up to scalars. Denote by
m :V ⊗V →V
λµ λ µ λ+µ
the corresponding projection. Gluing these together on
M := V ,
A λ
λ
M
we get a (not necessarily associative) multiplication m:M ⊗M →M .
A A A
Definition 1.2. The algebra M is called the shape algebra of A.
A
A “CLASSICAL” FLAG VARIETY FOR STANDARD QUANTUM SL(n) 5
Question A. Is it possible to renormalize the m in such a way that the multi-
λµ
plication m becomes associative?
Recall that this Question has a positive answer in the commutative case A =
O(G): if U is a maximal unipotent subgroup, then by the Borel-Weil theorem, we
may set
M =O(G/U):={f ∈O(G)|f(gu)=f(g) ∀g ∈G, ∀u∈U}.
O(G)
The next Propositionprovides a criterionfor a positive answer to Question A. We
first introduce some more notation: let ℓ be the rank of G, denote by ̟ ,...,̟
1 ℓ
the fundamental weights, and let us use the shorthand notation
V :=V , Vi :=V̟i, 1≤i≤ℓ.
i ̟i
For every 1 ≤ i,j,k ≤ ℓ, Definition 1.1(c) implies that V ⊗V ⊗V contains a
i j k
unique subcomodule isomorphic to V ; denote this subcomodule by W .
̟i+̟j+̟k ijk
Proposition 1.3. Question A has a positive answer (for a given A) if and only if
there exist A-isomorphisms R :V ⊗V →V ⊗V for all i>j, such that the braid
ij i j j i
relation
(1.1) (R ⊗id)(id⊗R )(R ⊗id)| =(id⊗R )(R ⊗id)(id⊗R )|
jk ik ij Wijk ij ik jk Wijk
holds for all i>j >k.
We defer the proof to Appendix A.
Corollary 1.4. Question A has a positive answer in each of the following situa-
tions:
• when G is of rank 2,
• when A is dual quasitriangular,
• when G=SL(n) (by the main result of [25]).
Since ̟ ,...,̟ generatethe monoidP+, andsincethe m aresurjective,the
1 ℓ λµ
algebra M is generated by
A
M :=V ⊕···⊕V .
1 1 ℓ
Inthis way,M maybe viewedasanN-gradedalgebra. Moreexplicitly, ifλ∈P+
A
decomposes as a ̟ (with each a ∈ N), and if we write h(λ) := a for the
i i i i i
height of λ, then the N-grading on M is given by M := V .
P A k h(λ)=k λP
Question B. Is the shape algebra M quadratic (as an NL-graded algebra)?
A
In the commutative case A = O(G), the shape algebra O(G/U) is indeed qua-
draticbyawellknowntheoremofKostant(see[27,Theorem1.1]foraproof). This
remains true for the standard Drinfel′d-Jimbo quantum SL(n): a presentation of
the corresponding shape algebra by generators and (quadratic) relations has been
given by Taft and Towber [40].
Question C. Is the shape algebra M a Koszul algebra?
A
To finish this Section, let us take a closer look at the quadratic relations in
M . For every 1 ≤ i,j ≤ ℓ, let K be the kernel of the multiplication V ⊗V →
A ij i j
V . ByDefinition 1.1(c),theA-comodulesV ⊗V andV ⊗V areisomorphic,
̟i+̟j i j j i
and, rescaling the A-isomorphism R : V ⊗ V → V ⊗ V of Proposition 1.3
ij i j j i
if necessary, we may assume that the diagram (A.1) (in Appendix A) commutes.
6 CHRISTIAN OHN
UsingDefinition1.1(c),weseethatthequadraticrelationsinM ofdegree̟ +̟
A i j
are of two kinds:
(I) ξ =0, for ξ ∈K ;
ij ij
(II) ξ =R (ξ), for ξ ∈V ⊗V .
ij ij i j
Remark 1.5. By Definition 1.1(c), relations (I) and (II) for arbitrary i,j are
ij ij
consequences of relations (I) for i≥j only and relations (II) for i>j only.
ij ij
2. The scheme of flag modules
Assume that Question A has a positive answer. The following definition is a
multigraded analogue of the point modules introduced in [5].
Definition 2.1. A flag module is a P+-graded right M -module F such that
A
(a) dimF =1 for each λ∈P+,
λ
(b) F is cyclic.
The terminology is justified by the commutative case. Indeed, let B ⊂ G be a
Borel subgroup and U the unipotent radical of B. Then we have the following
Proposition 2.2. The isomorphism classes of flag modules of M = O(G/U)
O(G)
are parametrized by the points of the flag variety G/B.
Proof. First,recallfromtheBorel-WeiltheoremthatthedecompositionO(G/U)=
V is given by
λ∈P+ λ
V ={f ∈O(G)|f(gb)=λ(b)f(g) ∀g ∈G, ∀b∈B}.
L λ
Now fix g ∈ G and endow a vector space F = Ce with the flag module
λ∈P+ λ
structure defined by
L
e .f =f(g)e for all f ∈V .
λ λ+µ µ
If we replace g by gb for some b ∈ B, the expression for e .f is just multiplied
λ
by µ(b), so up to isomorphism (of graded modules), the flag module thus obtained
only depends on the class gB ∈G/B.
Conversely, assume that F is a flag module of O(G/U), and choose a graded
basis {e |λ∈P+} of F. For each λ,µ∈P+, let vµ ∈Vµ be defined by
λ λ
e .f =hf,vµie for all f ∈V .
λ λ λ+µ µ
Since the algebra O(G/U) is commutative, we have (e .f).f′ =(e .f′).f for every
0 0
f ∈V , f′ ∈V , hence
λ µ
(2.1) vλ⊗vµ =vλ⊗vµ.
0 λ µ 0
It follows that vµ is a multiple of vµ =: vµ, say vµ = a vµ. Inserting back into
λ 0 λ λ
(2.1) yields a =a , for all λ,µ∈P+. Since a =1, we get a =1 for all λ∈P+.
λ µ 0 λ
Therefore,
e .f =hf,vµie for all f ∈V .
0 λ λ
The collection {vλ |λ∈P+} defines a linear form v on O(G/U). Furthermore, we
have e .(ff′) = (e .f).f′ for all f ∈ V , f′ ∈ V , so hff′,vλ+µi = hf,vλihf′,vµi,
0 0 λ µ
which shows that the linear form v is a character on O(G/U), corresponding to a
point x of the affine variety G/U. Moreover, since F is cyclic, each vλ must be
nonzero,so x actually lies in G/U, sayx=gU. This yields anelement gB ∈G/B.
It is clear that these two constructions are inverse to each other.
A “CLASSICAL” FLAG VARIETY FOR STANDARD QUANTUM SL(n) 7
Wewillnowdiscussapossiblepictureofthiskindinthenoncommutativesituation:
if A is a quantum G, we would like to parametrize the isomorphism classes of flag
modules over the shape algebra M by the (closed) points of some scheme E.
A
Moreover, given a flag module F and a weight λ ∈ P+, consider the shifted
flag module F[λ], defined as the P+-graded module such that F[λ] := F . We
µ λ+µ
would then like that, for each λ, the shift operation F 7→ F[λ] corresponds to an
automorphism of schemes σ :E →E.
λ
To achieve this, let us encode the structure ofa flagmodule more geometrically,
as follows. If F is a flag module with basis {e | λ ∈ P+}, then for each λ ∈ P+
λ
and each 1≤i≤ℓ, let vi ∈Vi be defined by
λ
(2.2) e .f =hf,viie for all f ∈V .
λ λ λ+̟i i
Replacing F by an isomorphic flag module (i.e. rescaling the e ) only multiplies
λ
each vi by a scalar, so let pi be the corresponding point in P(Vi). To simplify
λ λ
notation, let us write
P1...ℓ :=P(V1)×···×P(Vℓ)
and denote by pri :P1...ℓ →P(Vi) the natural projection. For any point p∈P1...ℓ,
we use the shorthand notation pi := pri(p). Thus, to an isomorphism class of flag
modules, we associate a collection of points {p |λ∈P+} in P1...ℓ.
λ
From now on, we assume that Question B has a positive answer. The quadratic
relations (I) and (II) in M (see the end of Section 1) impose some conditions on
A
this collection of points, which we now analyse.
For relations of type (I), identify P(Vi)×P(Vj) with its image in P(Vi ⊗Vj)
under the Segre embedding. Relations (I) may be viewed as equations defining a
ij
subscheme Γij of P(Vi)×P(Vj). We then have
(2.3) (pi,pj )∈Γij
λ λ+̟i
for all λ∈P+ and all 1≤i,j ≤ℓ.
Similarly, for relations of type (II), we consider the map P(Rji):P(Vj ⊗Vi)→
P(Vi⊗Vj), where Rji denotes the transpose of R . Then we must have
ij
(2.4) (pi,pj )=P(Rji)(pj,pi )
λ λ+̟i λ λ+̟j
(again identifying P(Vi)×P(Vj) with its image under the Segre embedding).
Gluing together conditions (2.3) and (2.4) for all i,j, we are led to consider the
subscheme Γ⊂(P1...ℓ)ℓ+1 of all (ℓ+1)-tuples (p ,p ,...,p ) satisfying
0 1 ℓ
(pi,pj)∈Γij,
0 i
(pi,pj)=P(Rji)(pj,pi)
0 i 0 j
for all 1≤i,j ≤ℓ. We may now rephraseconditions (2.3) and (2.4)by sayingthat
the collection {p |λ∈P+} satisfies
λ
(2.5) (p ,p ,...,p )∈Γ for all λ∈P+.
λ λ+̟1 λ+̟ℓ
Proposition 2.3. Assume that M is quadratic (as an N-graded algebra). Then
A
there is a one-to-one correspondence between isomorphism classes of flag modules
over M and families {p |λ∈P+} of points of P1...ℓ satisfying (2.5).
A λ
8 CHRISTIAN OHN
Proof. It remains to show that the above construction can be reversed, so assume
that {p | λ ∈ P+} is a collection of points in P1...ℓ satisfying (2.5). Choose a
λ
(nonzero)representativevi ∈Viforeachpi,andendowavectorspace Ce
λ λ λ∈P+ λ
with the flag module structure defined by the rule (2.2). By (2.3), this rule is
L
compatible with relations of type (I) in M . By (2.4), it is also compatible with
A
relations of type (II), provided that, for each λ ∈ P+ and each 1 ≤ i,j ≤ ℓ, we
suitably rescale one of vi, vj , vj, vi . Proceeding by induction over the
λ λ+̟i λ λ+̟j
height h(λ), we may perform this rescaling in a consistent way.
It is clear that the two constructions are inverse to each other.
Remark 2.4. Rescalingthem onlymultipliestheR byscalars. Therefore,the
λµ ij
scheme Γ does not depend on the normalizations of the multiplication in M , but
A
only on A itself.
The following Question is inspired by the description given in the Introduction
of [4].
Question D. Do there exist a subscheme E of P1...ℓ and ℓ pairwise commuting
automorphisms σ ,...,σ :E →E such that the scheme Γ is given by
1 ℓ
(2.6) Γ={(p,σ (p),...,σ (p))|p∈E}?
1 ℓ
A positive answer to this Question would fulfill the aim of parametrizing flag
modules, as expressed at the beginning of this Section. Indeed, assume that E
and σ ,...,σ as in Question D do exist. For each weight λ = a ̟ , define
1 ℓ i i
σ := σa1...σaℓ; since the σ commute, we have σ = σ σ . Then for every
faλmily {1p |λ∈ℓ P+} of pointsi in P1...ℓ satisfying (2.5λ)+,µthe reλaliµzatiPon (2.6) shows
λ
that p = σ (p ) for all λ ∈ P+, with p ∈ E. Conversely, for every p ∈ E, the
λ λ 0 0
family {σ (p) | λ ∈ P+} satisfies (2.5) and thus defines an isomorphism class of
λ
flag modules by Proposition 2.3. Therefore, if Question D had a positive answer,
flag modules (up to isomorphism) would be parametrized by the points of E, with
σ corresponding to the shift operation F 7→F[λ].
λ
Finally,forfuturereference,wedefine,foreach1≤i≤ℓ,thelinebundleL over
i
E as the pullback of OP(Vi)(1) along pri (restricted to E), and we call the tuple
T(M ):=(E,σ ,...,σ ,L ,...,L )
A 1 ℓ 1 ℓ
the flag tuple associated to A.
3. Braided tuples and reconstruction of shape algebras
Chan [10] has given a construction in the opposite direction, starting from a
scheme E, automorphisms σ ,...,σ of E, and line bundles L ,...,L over E
1 ℓ 1 ℓ
(satisfying some compatibility conditions; see Definition 3.1), and building a P+-
graded algebra from these data. Let us briefly recall his construction. To improve
legibility, we will write Lσ for the pullback of a line bundle L along a map σ.
Definition 3.1. We call a tuple T =(E,σ ,...,σ ,L ,...,L ) as above a braided
1 ℓ 1 ℓ
tuple if
(a) the σ pairwise commute,
i
(b) for every i>j, there exists an equivalence R :L ⊗Lσi →∼ L ⊗Lσj of line
ij i j j i
bundles such that the braid relation
(3.1) (R ⊗id)(id⊗Rσj)(R ⊗id)=(id⊗Rσk)(R ⊗id)(id⊗Rσi)
jk ik ij ij ik jk
A “CLASSICAL” FLAG VARIETY FOR STANDARD QUANTUM SL(n) 9
holds for every i> j > k (both sides being equivalences L ⊗Lσi ⊗Lσiσj →∼
i j k
L ⊗Lσk ⊗Lσkσj).
k j i
Note that if we set R := id for all i and R := R−1 for all i > j, then (3.1)
ii ji ij
becomes true for all i,j,k.
If λ ∈ P+ decomposes as λ = a ̟ , then define σ := σa1...σaℓ, as before
i i λ 1 ℓ
(so σ =σ σ ). Define a line bundle L over E by the following inductive rules
λ+µ λ µ λ
P
(with L the trivial bundle):
0
L =L , 1≤i≤ℓ,
(3.2) ̟i i
L =L ⊗Lσλ.
λ+µ λ µ
As is shown in [10], this procedure is, thanks to (3.1), well defined up to unique
equivalencesoflinebundlesbuiltfromtheR (cf.alsotheproofofProposition1.3).
ij
Now define the product of two sections α∈H0(E,L ) and β ∈H0(E,L ) by
λ µ
(3.3) αβ :=α⊗βσλ ∈H0(E,L ).
λ+µ
Theorem 3.2 (Chan [10]). The product rule (3.3) turns the direct sum
B(T):= H0(E,L )
λ
λM∈P+
into an associative P+-graded algebra.
The algebraB(T) is notquadratic in general,so we consider its quadraticcover
M(T):=B(T)(2).
(IfBisanyN-gradedalgebra,wedefineitsquadraticcover B(2)asfollows: consider
the canonical homomorphism T(B ) → B and its kernel J = J , then set
1 k≥2 k
B(2) := T(B )/(J ). Here we view B(T) as an N-graded algebra via the height
1 2 L
function h(λ).)
The quadratic algebra M(T) may also be described more directly in terms of
the braided tuple T, as follows. For each 1≤i≤ℓ, set V :=H0(E,L ), denote by
i i
Vi the dual of V , and consider the map Pli :E →P(Vi) corresponding to the line
i
bundle Li. For every 1 ≤ i,j ≤ ℓ, the map Pli⊠(Plj)σi corresponding to the line
bundle L ⊗Lσi is then given by the composite
i j
(3.4) E −d−i−a→g. E×E −id−×−σ→i E×E −P−li−×−−P→lj P(Vi)×P(Vj)−S−e−gr→e P(Vi⊗Vj).
Denote by Γij the image of this map and by K ⊂ V ⊗V the subspace of linear
ij i j
forms on Vi⊗Vj vanishing on Γij.
For every 1 ≤ i,j ≤ ℓ, Definition 3.1 implies that there exists a linear isomor-
phism Rji :Vj ⊗Vi →Vi⊗Vj such that the following diagram commutes:
(3.5) E×E id×σi // E×EPli×Plj// P(Vi)×P(Vj) Segre // P(Vi⊗Vj)
mdmiamgm.mmm66 OO
E QdQiaQgQ.QQQ(( P(Rji)
E×E //E×E // P(Vj)×P(Vi) //P(Vj ⊗Vi).
id×σj Plj×Pli Segre
Let R : V ⊗V → V ⊗V be the transpose of Rji. It is clear that modulo K
ij i j j i ij
and K , the map R is unique up to a scalar.
ji ij
10 CHRISTIAN OHN
ThealgebraM(T)isthengeneratedbyV ⊕···⊕V ,withrelationsgivenby(I)
1 ℓ ij
and (II) for all 1≤i,j ≤ℓ (see the end of Section 1; Remark 1.5 still applies).
ij
Question E. What is the kernel of the canonical morphism M(T)→B(T)?
HavingconstructedthealgebraM(T)fromabraidedtupleT,wemayformulate
a converse to Question D:
Question F. Assume that A is a quantum G such that the shape algebra M is
A
quadratic. Does there exist a braided tuple T such that M =M(T)?
A
ThisQuestionisapriori weakerthanQuestionD,forthefollowingreason. IfM
A
isquadraticanddoesadmitaflagtupleT asinQuestionD,thenthereconstructed
algebra M(T) is canonically isomorphic to M . However, we might also have
A
M(T′) = M for some subtuple T′ of T (i.e. a subscheme E′ of E stabilized by
A
each σ , with σ′ and L′ the obvious restrictions).
i i i
Problem G. Given a simple complex group G, characterize the flag tuples of all
quantum G’s intrinsically (i.e. as braided tuples).
For G = SL(2), this is elementary: E must be the projective line P1, σ can be
an arbitrary automorphism of infinite order, and L = OP1(1). The three possible
formsofσ correspondto three differentquantumSL(2)’s, namely,O(SL(2))(when
σ = id), the standard Drinfel′d-Jimbo quantum SL(2) for q not a root of unity
(when σ has two fixed points), and the Jordanian quantum SL(2) [12] (when σ
has one fixed point). These are known [43] to be the only quantum SL(2)’s (in
the sense of Definition 1.1). The associated shape algebras are Chx,yi/(xy−yx),
Chx,yi/(xy−qyx), and Chx,yi/(xy−yx−y2), respectively.
Problem H. Reconstruct not only a shape algebra, but a quantum G itself from
a braided tuple satisfying the conditions found in Problem G.
Part 2. A conjectural flag tuple for the standard Drinfel′d-Jimbo
quantum groups
InthisPart,wedescribeingredientsforapotentialbraidedtuple,andweconjec-
ture that these geometric data provide positive answers to Questions F and D for
the standard quantum groups of Drinfel′d and Jimbo. (The conjecture concerning
Question F will be proved for SL(n) in Part 3.)
Again, G will denote a simply-connected simple complex group.
4. Recollections on UDJ(g) and ODJ(G)
q q
Let g be the Lie algebra of G. Drinfel′d [14] and Jimbo [22] have defined (in-
dependently) a Hopf algebra UDJ(g) that depends on a parameter q ∈ C∗ and
q
that is a “quantum analogue” of the universal enveloping algebra U(g) (in the
sense that its comultiplication is no longer cocommutative). When q is not a root
of unity, finite-dimensional UDJ(g)-modules have been studied (independently) by
q
Lusztig [30] and by Rosso [34]: in particular, discarding unwanted nontrivial one-
dimensional modules, there still exists a family {V | λ ∈ P+} of U (g)-modules
λ q
satisfying conditions (a) and (c) of Definition 1.1 (condition (c) follows e.g. from
Theorem 4.12(b) of [30]).
Therefore, if ODJ(G) denotes the subspace of UDJ(g)∗ spanned by the matrix
q q
coefficientsofthemodulesV ,thenODJ(G)(forq notarootofunity)isaquantum
λ q