Quantization of Poisson Groups

50 Pages·0.48 MB·English

by  Fabio Gavarini

QUANTIZATION OF POISSON GROUPS
F. Gavarini
6
Dipartimento di Matematica, Istituto G. Castelnuovo,
9
9 Universit`a degli studi di Roma ”La Sapienza”
1

n
a
J Abstract. Thequantizationof well-knownpairsof Poissongroupsof awideclassisstudied

9 by means of Drinfeld’s double construction and dualization via formal Hopf algebras; new
2 quantizedenvelopingalgebrasUM,∞(h)andquantumformalgroupsFM,∞[G]areintroduced
q,ϕ q,ϕ
(both in one-parameter and multi-parametercase), and their specializations at roots of 1 (in
6
v particular, their classical limits) are studied: a new insight into the classical Poisson duality
2 is thus obtained.
2
0
1
1
5
9
/
g Introduction
l
a
- ”Dualitas dualitatum
q et omnia dualitas”
:
v N. Barbecue, ”Scholia”
i
X
Let G be a semisimple, connected, and simply connected affine algebraic group over an
r
a algebraicallyclosed field of characteristic zero; we consider a family of structures of Poisson
group on G, indexed by a multiparameter τ, which generalize the well-known Sklyanin-
Drinfeld one (cf. e. g. [DP], §11). Then every such Poisson group Gτ has a well defined
corresponding dual Poisson group Hτ, and gτ := Lie(Gτ) and hτ := Lie(Hτ) are Lie
bialgebras dual of each other.
In1985Drinfeld([Dr])andJimbo([Ji])providedaquantizationUQ(g)of U(g) = U(g0),
q
namely a Hopf algebra UQ(g) over k(q), presented by generators and relations with a
q
k[q,q−1]-formUQ(g)whichfor q → 1 specializestoU(g)asaPoissonHopfcoalgebra. This
q
has been extended to general parameter τ introducing multiparameter quantum groups
UQ (g) (cf. [Rb], and [CV-1], [CV-2]). Dually, by means of a Peter-Weyl type axiomatic
q,ϕ
trick one constructs a Hopf algebra FP[G] of matrix coefficients of UQ(g) with a k[q,q−1]-
q q
form FP[G] which specializes to F[G], as a Poisson Hopf algebra, for q → 1; in particular
q
FP[G] is nothing but the Hopf subalgebra of ”functions” in FP[G] which take values in
q q
b
b Revised version of January 29th 1996
Typeset by AMS-TEX
1
2 F. GAVARINI
k[q,q−1] when ”evaluated” on UQ(g) (in a word, the k[q,q−1]-integer valued functions on
q
UQ(g)). This again extends to general τ (cf. [CV-2]).
q
So far the quantization procebdure only dealt with the Poisson group Gτ; the dual group
Hb is involved defining a different k[q,q−1]-form UP(g) (of a quantum group UP(g)) which
q q
specializes to F[H] (as a Poisson Hopf algebra) for q → 1 (cf. [DP]), with generalization
to the general case possible again. Here a new pheenomenon of ”crossing dualities” (duality
among enveloping and function algebra and duality among Poisson groups) occurs which
was described (in a formal setting) by Drinfel’d (cf. [Dr], §7). This leads to consider
the following: let FQ[G] be the quantum function algebra dual of UP(g), and look at the
q q
”dual” ofUP(g)withinFQ[G], callit FQ[G], which shouldbe theHopf algebra ofk[q,q−1]-
q q q
integer valued functions on UP(g); then this should specialize to U(h) (as a Poisson Hopf
q
coalgebra)efor q → 1; the same conjeceture can be formulated in the general case too.
The starting aim of the peresent work was to achieve this goal, i. e. to construct FQ[G]
q
and itsk[q,q−1]-form FQ[G], and to prove thatFQ[G] is a deformation of the Poisson Hopf
q q
coalgebra U(h). This goal is succesfully attained by performing a suitable dualization of
Drinfeld’s quantum doeuble; but by the way, thiseleads to discover a new ”quantum group”
(both one-parameter and multiparameter), which we call UM,∞(h), which is to U(h) as
q
UM(g) is to U(g); in particular it has an integer form UQ(h) which is a quantization
q q
of U(h), and an integer form UP,∞(h) which is a quantization of F∞[G] (the function
q
algebra of the formal Poisson group associated to G); thisbis also produced in the shape
of a quantum formal group FP,∞e[G]. Moreover, we exhibit a quantum analog of the Hopf
q
pairings F[H]⊗U(h) → k, F[G]⊗U(g) → k, F∞[G]⊗U(g) → k, and also a quantum
analog of the Lie bialgebra pbairing h⊗g → k, which are strictly related with each other
and with the already known quantum pairing FP[G]⊗UQ(g) → k(q): all this is stressed
q q
by introducing the definition of quantum Poisson pairing. Once again all this extend to
the multiparameter case. Thus in particular we provide a quantization for a wide class
of Poisson groups (the Hτ’s); now, in the summer of 1995 (when the present work was
already accomplished) a quantization of any Poisson group was presented in [EK-1] and
[EK-2], but greatest generality implies lack of concreteness: in contrast, our construction is
extremely concrete, by hands; moreover, it allows specialization at roots of 1, construction
of quantum Frobenius morphisms, and so on (like for UQ(g) and UP(g)), which is not
q q
possible in the approach of [EK-1], [EK-2].
Finally, a brief sketch of the main ideas of the paper. Our aim being to study the
”dual” of a quantum group UM (g) (M being a lattice of weights), we proceed as fol-
q,ϕ
lows. First, we select as operation of ”dualization” (of a Hopf algebra H ) the most
na¨ıve one, namely taking the full linear dual (i. e. H∗, rather than the usual Hopf
dual H◦), the latter being a formal Hopf algebra (rather than a common Hopf algebra).
Second, we remark that UM (g) is a quotient, as a Hopf algebra, of a quantum double
q,ϕ
DM (g) := D UQ (b ),UM (b ),πϕ (cf. §3), hence its linear dual UM (g)∗ embeds into
q,ϕ q,ϕ − q,ϕ + + q,ϕ
the formal Hopf algebra DM (g)∗. Third, since DM (g) ∼= UM (b )⊗UQ (b ) (as coal-
(cid:0) q,ϕ (cid:1) q,ϕ q,ϕ + q,ϕ −
gebras) we have DM (g)∗ ∼= UM (b )∗⊗UQ (b )∗ (as algebras), where ⊗ denotes topo-
q,ϕ q,ϕ + q,ϕ −
logical tensor product. Fourth, DRT pairings among quantum Borel algebras yield natural
embeddings UM (b ) ֒→ UM′(b )∗ (Mb′ being the dual latticeofM ). Fiftbh, from previous
q,ϕ ± q,ϕ ∓
remarks we locate a special formal Hopf subalgebra of UM′(b )⊗UP (b ) ⊆ DM (g)∗.
q,ϕ − q,ϕ + q,ϕ
(cid:16) (cid:17)
b
QUANTIZATION OF POISSON GROUPS 3
Seventh, we take out of this an axiomatic construction of a formal Hopf algebra UM′(h),
q,ϕ
by definition isomorphic to the one of step six and therefore naturally paired with UM (g):
q,ϕ
this is the object we were looking for, sort of completion of FM′[G], and from its very
q,ϕ
construction the whole series of results we stated above will be proved with no serious
trouble.
Sections 1 to 4 introduce the already known material; the original part of the paper is
sections 5 to 8. The Appendix describes in full detail the example G = SL(2).
Acknoledgements
The author wishes to thank C. Procesi for several fruitful conversations, and C. De
Concini for some useful talk; he is also endebted with M. Costantini, M. Varagnolo, and
I. Damiani for many helpful discussions.
§ 1 The classical objects
1.1 Cartan data. Let A := (a ) be a n×n symmetrizable Cartan matrix;
ij i,j=1,...,n
thus we have a ∈ Z with a = 2 and a ≤ 0 if i 6= j, and there exists a vector
ij ii ij
(d ,... ,d ) with relatively prime positive integral entries d such that (d a )
1 n i i ij i,j=1,...,n
is a symmetric positive definite matrix. Define the weight lattice P to be the lattice
with basis {ω ,...,ω } (the fundamental weights); let P = n Nω be the subset of
1 n + i=1 i
n
dominant integral weights, let α = a ω (j = 1,...,n) be the simple roots, and
Q = n Zα (⊂ P ) the root lajttice, iQ=1 =ij i n Nα thePpositive root lattice. Let W
j=1 j P + j=1 j
be the Weyl group associated to A, with generators s ,... ,s , and let Π := {α ,...,α }:
1 n 1 n
P P
then R := WΠ is the set of roots, R+ = R∩Q the set of positive roots; finally, we set
+
N := #(R+) (= |W|).
Define bilinear pairings h | i:Q×P → Z and ( | ):Q×P → Z by hα |ω i = δ and
i j ij
(α |ω ) = δ d . Then (α |α ) = d a , giving a symmetric Z-valued W-invariant bilinear
i j ij i i j i ij
formonQsuchthat(α|α) ∈ 2Z. WecanalsoextendtheZ-bilinearpairing (|):Q×P → Z
to a (non-degenerate) pairing ( | ):(Q⊗ Q)×(Q⊗ P) → Z of Q-vector spaces by scalar
Z Z
extension: restriction gives a pairing ( | ):P × P → Q (looking at P as a sublattice of
Q⊗ Q), which takesvalues in Z[d−1], where d := det (a )n . Given a pairof lattices
Z ij i,j=1
(M,M′), with Q ≤ M,M′ ≤ P , we say that they are(cid:16)dual of ea(cid:17)ch other if
M′ = {y ∈ P | (M,y) ⊆ Z} , M = {x ∈ P | (x,M′) ⊆ Z}
the two conditions being equivalent; then for any lattice M with Q ≤ M ≤ P there exists
a unique dual lattice M′ such that Q ≤ M′ ≤ P .
1.2 The Poisson groups G and H. Let G be a connected simply-connected semisim-
pleaffinealgebraicgroupoveranalgebraicallyclosedfieldk ofcharachteristic0. Fixamax-
imal torus T ≤ G and opposite Borel subgroups B , with unipotent subgroups U , such
± ±
that B ∩B = T , and let g := Lie(G), t := Lie(T), b := Lie(B ), n := Lie(U );
+ − ± ± ± ±
fix also τ := (τ ,... ,τ ) ∈ Qn such that (τ ,α ) = −(τ ,α ) for all i,j = 1,...,n: when
1 n i j j i
τ = (0,... ,0) we shall simply skip it throughout. Set K = G × G, define Gτ := G
embedded in K as the diagonal subgroup, and define a second subgroup
Hτ := {(u t ,t u ) | u ∈ U ,t ∈ T,t t ∈ exp(tτ)} (≤ B ×B ≤ K)
− − + + ± ± ± − + − +
4 F. GAVARINI
where tτ := n k ·h ⊕h ≤ t⊕t ≤ g⊕g = k := Lie(K); thus
i=1 −αi+2τi αi+2τi
P hτ := Lie(Hτ) = (n−,0)⊕tτ ⊕(0,n+).
Thus we have a triple (K,Gτ,Hτ); this is an algebraic Manin triple (i. e. its ”tangent
triple” (k,gτ,hτ) is a Manin triple), when a non-degenerate symmetric invariant bilinear
form on k is defined as follows: first rescale the Killing form ( , ):g⊗g → k so that short
roots of g have square length 2; then define the form on k = g⊕g by
1 1
hx ⊕y ,x ⊕y i := (y ,y )− (x ,x ).
1 1 2 2 1 2 1 2
2 2
In general, if (k′,g′,h′) is any Manin triple, the bilinear form on k′ gives by restriction
a non-degenerate pairing h , i:h′ ⊗g′ → k which is a pairing of Lie bialgebras, that is
hx,[y1,y2]i = hδh(x),y1 ∧y2i, h[x1,x2],yi = hx1 ∧x2,δg(y)i
for all x,x1,x2 ∈ h′, y,y1,y2 ∈ g′, where δh′, resp. δg′, is the Lie cobracket of h′, resp. g′;
we shall call such pairing also Poisson pairing, and denote it by π (h,g) := hh,gi for all
P
h ∈ h′, g ∈ g′. In the present case the Poisson pairing is described by formulas
hfτ,f i = 0 hfτ,h i = 0 hfτ,e i = −1δ d−1
i j i j i j 2 ij i
hhτ,f i = 0 hhτ,h i = a d−1 = a d−1 hhτ,e i = 0 (1.1−a)
i j i j ij j ji i i j
heτ,f i = 1δ d−1 heτ,h i = 0 heτ,e i = 0
i j 2 ij i i j i j
where the fτ, hτ, eτ (s = 1,...,n), resp. f , h , e (s = 1,...,n), are Chevalley-type
s s s s s s
generators of hτ, resp. gτ, embedded inside k = g ⊕ g, namely fτ = f ⊕ 0, hτ =
s s s
h ⊕ h , eτ = 0 ⊕ e , and f = f ⊕ f , h = h ⊕ h , e = e ⊕ e (see
−αs+2τs αs+2τs s s s s s s s s s s s
below); moreover we also record that
1 1
heτ,f i = δ d−1 , hfτ,e i = − δ d−1 (1.1−b)
α β 2 αβ α α β 2 αβ α
for all α,β ∈ R+, where eτ,f ,e ,f are Chevalley-type generators of (hτ) , (hτ) ,
α α β β α −α
(gτ) , (gτ) and d := (α,α) for all α ∈ R+ (in particular d = d ∀i = 1,...,n).1
β −β α 2 αi i
1.3 The Poisson Hopf coalgebra U(gτ). It is known that the universal enveloping
algebra U(gτ) = U(g) canbepresented astheassociativek-algebra with1generated by el-
ements,f , h , e (i = 1,...,n)(theChevalley generators)satisfyingthewell-knownSerre’s
i i i
relations (cf. for instance [Hu], ch. V). As an enveloping algebra of a Lie algebra, U(gτ) has
a canonical structure of Hopf algebra, given by ∆(x) = x⊗1+1⊗x, S(x) = −x, ε(x) = 0
for all x ∈ g; finally, the Poisson structure of Gτ reflects into a Lie coalgebra structure
δ = δgτ:gτ −→ gτ ⊗gτ of gτ, extending to a co-Poisson structure δ:U(gτ) −→ U(gτ)⊗
U(gτ) (compatible with the Hopf structure) given by (cf. [DL], §8, and [DKP], §7.7)
(α +2τ |α +2τ ) (α +2τ |α +2τ )
i i i i i i i i
δ(f ) = h ⊗f − f ⊗h
i 2 αi+2τ i 2 i αi+2τ
δ(h ) = 0 (1.2)
i
(α −2τ |α −2τ ) (α −2τ |α −2τ )
i i i i i i i i
δ(e ) = h ⊗e − e ⊗h .
i 2 αi−2τ i 2 i αi−2τ
1Warning: beware, in particular, of the normalization we chose for the symmetric form of k, which is
different (by a coefficient 1) from the one fixed in [DP].
2
QUANTIZATION OF POISSON GROUPS 5
1.4 The Poisson Hopf coalgebra U(hτ). We already remarked that hτ ∼= n ⊕t⊕
−
n ∼= n ⊕tτ ⊕n as vector spaces: then the Lie structure of hτ is uniquely determined by
+ − +
thefollowingconstraints: (a) n , tτ, n areLiesubalgebrasofhτ; (b) [n ,n ] = 0 ∀n ∈
− + + − +
n ,n ∈ n ; (c) [hτ,n ] = −hα − 2τ ,βin ∀i = 1,...,n,n ∈ (n ) ,β ∈ R−;
+ − − i − i i − − − β
(d) [hτ,n ] = hα +2τ ,βin ∀i = 1,...,n,n ∈ (n ) ,α ∈ R+.
i + i i + + + α
Thus if fτ, hτ, eτ (i = 1,...,n) are Chevalley-type generators of g (thought of as
i i i
elements of n , t ∼= tτ , n ), U(hτ) can be presented as the associative k-algebra with 1
− +
generated by fτ, hτ, eτ (i = 1,...,n) with relations
i i i
hτhτ −hτhτ = 0, eτfτ −fτeτ = 0
i j j i i j j i
hτfτ −fτhτ = hα −2τ ,α ifτ , hτeτ −eτhτ = hα +2τ ,α ieτ
i j j i i i j j i j j i i i j j
1−aij 1−a
(−1)k ij (fτ)1−aij−kfτ(fτ)k = 0 (i 6= j) (1.3)
k i j i
k=0 (cid:18) (cid:19)
X
1−aij 1−a
(−1)k ij (eτ)1−aij−keτ(eτ)k = 0 (i 6= j)
k i j i
k=0 (cid:18) (cid:19)
X
for all i,j = 1,...,n; its natural Hopf structure is given by
∆(fτ) = fτ ⊗1+1⊗fτ , S(fτ) = −fτ , ε(fτ) = 0
i i i i i i
∆(hτ) = hτ ⊗1+1⊗hτ , S(hτ) = −hτ , ε(hτ) = 0 (1.4)
i i i i i i
∆(eτ) = eτ ⊗1+1⊗eτ , S(eτ) = −eτ , ε(eτ) = 0 .
i i i i i i
and the co-Poisson structure δ = δ :U(hτ) −→ U(hτ)⊗U(hτ) by
hτ
δ(fτ) = d hτ ⊗fτ −d fτ ⊗hτ
i i i i i i i
δ(hτ) = 4d−1 · d (γ|α )·(eτ ⊗fτ −fτ ⊗eτ)
i i γ i γ γ γ γ (1.5)
γ∈R+
X
δ(eτ) = d eτ ⊗hτ −d hτ ⊗eτ .
i i i i i i i
§ 2 Quantum Borel algebras and DRT pairings
2.1 Notations. We shall introduce our quantum groups using the construction of
[DL] and [CV-1], [CV-2]. For all s,n ∈ N, let (n) := qn−1 (∈ k[q]), (n) ! := n (r) ,
q q−1 q r=1 q
(n) := (n)q! (∈ k[q]), and [n] := qn−q−n (∈ k[q,q−1]), [n] ! := n [rQ] , [n] :=
s q (s) !(n−s) ! q q−q−1 q r=1 q s q
q q
[n] !
:= q (∈ k[q,q−1]); let q := qdα for all α ∈ R+, and q :=Qq . Let Q, P
[s] ![n−s] ! α i αi
q q
be as in §1; we fix an endomorphism ϕ of the Q-vector space QP := Q ⊗ P which is
Z
antisymmetric — with respect to ( | ) — and satisfies the conditions
6 F. GAVARINI
ϕ(Q) ⊆ Q, 1(ϕ(P) | P) ⊆ Z, 2AYA−1 ∈ M (Z)
2 n
where, letting τ := 1ϕ(α ) = n y α , we set Y := (y ) , M (Z) is the set
i 2 i j=1 ji j ij i,j=1,...,n; n
of n×n-matrices with integer entries, and A is our Cartan matrix. For later use, we also
P
define
1
τ := ϕ(α), τ := τ (2.1)
α 2 i αi
for all α ∈ R, i = 1,...,n. It is proved in [CV-1] that (id + ϕ) and (id −
QP QP
ϕ) are isomorphisms, adjoint of each other with respect to ( | ): then we define r :=
−1 −1
(id +ϕ) , r := (id −ϕ) .
QP QP
2.2 Quantum Borel algebras. Let M be any lattice such that Q ≤ M ≤ P ; then
(cf. [DL] and [CV-1]) UM (b ) (resp. UM (b )) is the associative k(q)-algebra with 1
q,ϕ − q,ϕ +
generated by {L | µ ∈ M } ∪ {F | i = 1,...,n} (resp. {L | µ ∈ M } ∪ {E | i =
µ i µ i
1,...,n} ) with relations
L = 1, L L = L ,
0 µ ν µ+ν
1−a
L F = q−(µ|αj)F L , (−1)s ij FpF Fs = 0
µ j j µ s i j i
p+sX=1−aij (cid:20) (cid:21)qi
1−a
(resp. L E = q(µ|αj)E L , (−1)s ij EpE Es = 0)
µ j j µ s i j i
p+sX=1−aij (cid:20) (cid:21)qi
for all i,j = 1,...,n and µ,ν ∈ M ; furthermore, it is proved in [CV-1] that UM (b ) and
q,ϕ −
UM (b ) can be endowed with a Hopf algebra structure given by2
q,ϕ −
∆ (F ) = F ⊗L +L ⊗F , ε (F ) = 0, S (F ) = −F L
ϕ i i −αi−τi τi i ϕ i ϕ i i αi
∆ (L ) = L ⊗L , ε (L ) = 1, S (L ) = L
ϕ µ µ µ ϕ µ ϕ µ −µ
∆ (E ) = E ⊗L +L ⊗E , ε (E ) = 0, S (E ) = −L E
ϕ i i τi αi−τi i ϕ i ϕ i −αi i
for all i = 1,...,n, µ ∈ M . In particular we use notation K := L ∀α ∈ Q. We shall
α α
also consider the subalgebras (of quantum Borel algebras) UM (t) (generated by the L ’s),
q,ϕ µ
U (n ) (generated by the E ’s), and U (n ) (generated by the F ’s).
q,ϕ + i q,ϕ − i
When ϕ = 0 we could skip the superscript ϕ, our quantum algebras then conciding
with the one-parameter ones, i. e. those of, say, [Lu-1], [DP], etc.
2.3 DRT pairings. From now on, if H is any Hopf algebra, then Hop will denote
the same coalgebra as H with the opposite multiplication, while H will denote the same
op
algebra as H with the opposite comultiplication.
From [DL], §2 (for the 1-parameter case), and [CV-1], §3 (for the multiparameter case)
we recall the existence of perfect (i. e. non-degenerate) pairings of Hopf algebras among
quantum Borel algebras
πϕ:UP (b ) ⊗UQ (b ) → k(q), πϕ:UQ (b ) ⊗UP (b ) → k(q)
− q,ϕ − op q,ϕ + + q,ϕ − op q,ϕ +
πϕ:UP (b ) ⊗UQ (b )k(q), πϕ:UQ (b ) ⊗UP (b )k(q)
− q,ϕ + op q,ϕ − + q,ϕ + op q,ϕ −
2Actually, we use here a different normalization than in [CV-1]: the results therein coincide with the
ones we list below up to change q ↔q−1, Lλ ↔L−λ (or Kλ ↔K−λ).
QUANTIZATION OF POISSON GROUPS 7
which we select as given by
q−(r(τi)|τi)
πϕ(L ,K ) = q−(r(λ)|α), πϕ(L ,E ) = 0, πϕ(F ,K ) = 0, πϕ(F ,E ) = δ
− λ α − λ j − i α − i j ij (q−1 −q )
i i
q−(r(τi)|τi)
πϕ(K ,L ) = q−(r(α)|λ), πϕ(K ,E ) = 0, πϕ(F ,L ) = 0, πϕ(F ,E ) = δ
+ α λ + α j + i λ + i j ij (q−1 −q )
i i
q+(r(τi)|τi)
πϕ(L ,K ) = q(r(λ)|α), πϕ(E ,K ) = 0, πϕ(L ,F ) = 0, πϕ(E ,F ) = δ
− λ α − i α − λ j − i j ij (q −q−1)
i i
q+(r(τi)|τi)
πϕ(K ,L ) = q(r(α)|λ), πϕ(E ,K ) = 0, πϕ(L ,F ) = 0, πϕ(E ,F ) = δ
+ α λ + i α + λ j + i j ij(q −q−1)
i i
∀ i,j = 1,...,n, α ∈ Q, λ ∈ P .
Remark : Hopf pairings like those above were introduced by Drinfeld, Rosso, Tanisaki,
and others, whence we shall call them DRT pairings; here again some differences occur
with respect to [CV-1], [CV-2] (or even [DL] for the simplest case), because of different
definition of the Hopf structure: cf. also [Tn] and [DD].
In the sequel if π is any DRT pairing we will also set hx,yi for π(x,y).
π
2.4 PBW bases. It is known that both UQ (b ) and UP (b ) have bases of Poincar´e-
q,ϕ − q,ϕ +
Birkhoff-Witt type (in short ”PBW bases”): we fix (once and for all) any reduced ex-
pression of w (the longest element in the Weyl group W), namely w = s s ···s
0 0 i1 i2 iN
(with N := #(R+) = number of positive roots); thus we have a total convex order-
ing (cf. [Pa] and [DP], §8.2) α1,α2 := s (α ),...,αN := s s ···s (α ), hence
i1 i2 i1 i2 iN−1 iN
— following Lusztig and others: cf. e. g. [Lu-2] — we can construct root vectors E ,
αr
r = 1,...,N as in [DP] or [CV-1] and get PBW bases of increasing ordered monomi-
als L · N Ffr µ ∈ M;f ,...,f ∈ N for UM (b ) and L · N Eer µ ∈
µ r=1 αr 1 N q,ϕ − µ r=1 αr
M;e1n,...,eQN ∈ N (cid:12)for UqM,ϕ(b+) or similar PoBW bases of decreasin(cid:8)g ordeQred monom(cid:12)ials.
(cid:12) (cid:12)
Now, foreverymonomialE intheE ’s, define s(E) := 1ϕ(wt(E)),r(E) := 1r(ϕ(wt(E))),
(cid:9) i 2 2
r(E) := 1r(ϕ(wt(E))), where wt(E) denotes the weight of E (E having weight α ), and
2 i i
similarly for every monomial F in the F ’s, (F having weight −α ). Then the values of
i i i
DRT pairings on PBW monomials are given by
1 1
πϕ Ffr ·L , Eer ·K =
− αr λ αr α
!
r=N r=N
Y Y
= q− r(λ)−r( 1r=NFαfrr) α−s( 1r=NEαerr)! N δ [er]qαr! qα+r(e2r)
Q (cid:12) Q er,fr (q−1 −q )er
(cid:12) r=1 αr αr
(cid:12) Y (2.2−a)
1 1
πϕ Ffr ·K , Eer ·L =
+ αr α αr λ
!
r=N r=N
Y Y
= q− r(α)−r( 1r=NFαfrr) λ−s( 1r=NEαerr)! N δ [er]qαr! qα+r(e2r)
Q (cid:12) Q er,fr (q−1 −q )er
(cid:12) r=1 αr αr
(cid:12) Y
8 F. GAVARINI
1 1
πϕ L ·Eer,K · Ffr =
− λ αr α αr
!
r=N r=N
Y Y
= q r(α)−r( 1r=NFαfrr) λ−s( 1r=NEαerr)! N δ [er]qαr! qα−r(e2r)
Q (cid:12) Q er,fr (q −q−1)er
(cid:12) r=1 αr αr
(cid:12) Y (2.2−b)
1 1
πϕ K ·Eer,L · Ffr =
+ α αr λ αr
!
r=N r=N
Y Y
= q r(λ)−r( 1r=NFαfrr) α−s( 1r=NEαerr)! N δ [er]qαr! qα−r(e2r)
Q (cid:12) Q er,fr (q −q−1)er
(cid:12) r=1 αr αr
(cid:12) Y
(cf. [CV-2] §1, taking care of our different normalizations; see also [DD] and [DL]).
2.5 Integer forms and duality. Let UQ (b ) be the k[q,q−1]-subalgebra of UQ (b )
q,ϕ − q,ϕ −
generated by
K ;c b
F(m), i ,K−1 m,c,t ∈ N;i = 1,...,n
i t i
(cid:26) (cid:18) (cid:19) (cid:12) (cid:27)
(cid:12)
where F(m) := Fm [m] ! and Ki;c :=(cid:12)(cid:12) t Kiqic−s+1−1 are the so-called ”divided
i i qi t s=1 qis−1
powers”; it is know(cid:14)n (cf. [Lu-2],(cid:16)[DL](cid:17), [CVQ-2]) that UqQ,ϕ(b−) is a Hopf subalgebra of
UQ (b ), having a PBW basis (as a k[q,q−1]-module) of increasing ordered monomials
q,ϕ −
b
n N
K ;0
i K−Ent(ti/2) · F(nr) t ,...,t ,n ,...,n ∈ N
t i αr 1 n 1 N
(i=1(cid:18) i (cid:19) r=1 (cid:12) )
Y Y (cid:12)
(cid:12)
(cid:12)
and a similar PBW basis of decreasing ordered monomials; in particular UQ (b ) is a
q,ϕ −
k[q,q−1]-form of UQ (b ). A similar definition gives us the k[q,q−1]-subalgebra UP (b )
q,ϕ − q,ϕ +
of UP (b ) generated by divided powers in the E ’s and L ’s, which is an inbteger form of
q,ϕ + i i
UP (b )withaPBWbasisofdecreasingorderedmonomialsandaPBWbasisofinbcreasing
q,ϕ +
ordered monomials. With the same procedure one defines Hopf algebras UQ (b ) and
q,ϕ +
UP (b ) (over k[q,q−1]) and locates PBW bases for them.
q,ϕ −
b
Now let UP (b ) be the k[q,q−1]-subalgebra of UP (b ) generated by
q,ϕ + q,ϕ +
b
e {E ,...,E }∪{L±,...,L±}
α1 αN 1 n
where E := (q −q−1)E , ∀r = 1,...,N ; then (cf. [DKP], [DP]) UP (b ) is a Hopf
αr αr αr αr q,ϕ +
subalgebra of UP (b ), having a PBW basis (as a k[q,q−1]-module)
q,ϕ +
e
n N
Lti · Enr t ,...,t ∈ Z;n ,...,n ∈ N
i αr 1 n 1 N
( )
i=1 r=1 (cid:12)
Y Y (cid:12)
(cid:12)
(cid:12)
QUANTIZATION OF POISSON GROUPS 9
ofincreasing orderedmonomialsandasimilarPBWbasisofdecreasing orderedmonomials;
in particular UP (b ) is a k[q,q−1]-form of UP (b ). Similarly we define UQ (b ) to be
q,ϕ + q,ϕ + q,ϕ −
the k[q,q−1]-subalgebra of UQ (b ) generated by {F ,...,F }∪{K±,...,K±} (with
q,ϕ − α1 αN 1 n
F := (q −eq−1)F , ∀r = 1,...,N ), which isanintegerform ofUQ (b )e, having PBW
αr αr αr αr q,ϕ −
k[q,q−1]-bases of decreasing or increasing ordered monomials. With the same procedure
one defines Hopf algebras UQ (b ) and UP (b ) (over k[q,q−1]) and locates PBW bases
q,ϕ + q,ϕ −
for them.
Similar constructions (and corresponding notations) and results also hold for the alge-
e e
bras UM (t), U (n ), and U (n ).
q,ϕ q,ϕ + q,ϕ −
Finally, from the very definitions and from (2.2) one immediately gets (as in [DL] §3)
UQ (t) = {y ∈ UQ (t) | πϕ(UP (t),y) ≤ k[q,q−1]} =
q,ϕ q,ϕ − q,ϕ
= {x ∈ UQ (t) | πϕ(x,UP (t)) ≤ k[q,q−1]}
b q,ϕ + e q,ϕ
UP (t) = {y ∈ UP (t) | πϕ(UQ (t),y) ≤ k[q,q−1]} =
q,ϕ q,ϕ + q,eϕ
= {x ∈ UP (t) | πϕ(x,UQ (t)) ≤ k[q,q−1]}
e q,ϕ − b q,ϕ
UQ (b ) = {x ∈ UQ (b ) | πϕ(x,UP (b )) ≤ k[q,q−1]}
q,ϕ − op q,ϕ − op +b q,ϕ +
UP (b ) = {x ∈ UP (b ) | πϕ(x,UQ (b )) ≤ k[q,q−1]}
bq,ϕ − op q,ϕ − op − eq,ϕ +
(2.3)
UQ (b ) = {x ∈ UQ (b ) | πϕ(x,UP (b )) ≤ k[q,q−1]}
eq,ϕ + op q,ϕ + op + bq,ϕ −
UP (b ) = {x ∈ UP (b ) | πϕ(x,UQ (b )) ≤ k[q,q−1]}
bq,ϕ + op q,ϕ + op − eq,ϕ −
UP (b ) = {y ∈ UP (b ) | πϕ(UQ (b ) ,y) ≤ k[q,q−1]}
e q,ϕ + q,ϕ + + q,ϕb− op
UQ (b ) = {y ∈ UQ (b ) | πϕ(UP (b ) ,y) ≤ k[q,q−1]}
eq,ϕ + q,ϕ + − bq,ϕ − op
UP (b ) = {y ∈ UP (b ) | πϕ(UQ (b ) ,y) ≤ k[q,q−1]}
bq,ϕ − q,ϕ − + eq,ϕ + op
UQ (b ) = {y ∈ UQ (b ) | πϕ(UP (b ) ,y) ≤ k[q,q−1]}.
eq,ϕ − q,ϕ − − bq,ϕ + op
b e
§ 3 Quantization of U(gτ)
3.1 Drinfeld’s double. Let H , H be two arbitrary Hopf algebras on the ground
− +
field (or ring) F, and let π:(H ) ⊗ H → F be any arbitrary Hopf pairing. The
− op +
Drinfeld’s double D = D(H ,H ,π) is the algebra T(H ⊕H ) R, where R is the ideal
− + − +
of relations
(cid:14)
1 = 1 = 1 , x⊗y = xy ∀x,y ∈ H or x,y ∈ H
H− H+ + −
π(y ,x )x ⊗y = π(y ,x )y ⊗x ∀x ∈ H , y ∈ H .
(2) (2) (1) (1) (1) (1) (2) (2) + −
(x),(y) (x),(y)
X X
10 F. GAVARINI
Then (cf. [DL], Theorem 3.6) D has a canonical structure of Hopf algebra such that
H , H are Hopf subalgebras of it and multiplication yields isomorphisms of coalgebras
− +
m m
H ⊗H ֒→ D⊗D −→ D, H ⊗H ֒→ D⊗D −→ D. (3.1)
+ − − +
We apply this to get the Drinfeld’s double DM (g) := D(UQ (b ),UM (b ),πϕ) (for
q,ϕ q,ϕ − q,ϕ + +
any lattice M, with Q ≤ M ≤ P) which we call quantum double; from the very definition,
DM (g) is generated by K , L , F , E — identified with 1⊗K , L ⊗1, 1⊗F , E ⊗1
q,ϕ α µ i i α µ i i
when thinking at DM (g) ∼= UM (b )⊗UQ (b ) — (α ∈ Q, µ ∈ M, i = 1,...,n), while
q,ϕ q,ϕ + q,ϕ −
relations defining the ideal R clearly reduce to commutation relations between generators,
namely (for all i,j = 1,...,n)
K L = L K , K E =q(α|αj)E K , L F = q−(µ|αj)F L ,
α µ µ α α j i j α µ j i j µ
L −K (3.2)
E F −F E =δ αi −αi .
i j j i ij q −q−1
i i
For later use we also record the following (deduced from (3.2))
t≤r,s r s K ⊗;2t−r−s
ErFs = [t] !2 ·Fs−t · αi⊗ ·Er−t
i i Xt≥0 htiqihtiqi qi i (cid:20) t (cid:21) i (3.3)
t≤r,s r s K ⊗;2t−r−s
FsEr = [t] !2 ··Er−t · −αi⊗ ·Fs−t
i i Xt≥0 htiqihtiqi qi i (cid:20) t (cid:21) i
(∀i = 1,...,n) where Kαi⊗⊗;c := t qic−p+1·Lαi−K−αi·qi−c+p−1 and K−αi⊗⊗;c :=
t p=1 qp−q−p t
i i
:= t qic−p+1·K−αi−Lαi·hqi−c+p−1 ifor allQc ∈ Z, t ∈ N. h i
p=1 qp−q−p
i i
Finally, PBW bases of quantum Borel algebras clearly provide (tensor) PBW bases of
Q
DM (g) (identified with UM (b )⊗UQ (b ), as we shall always do in the sequel).
q,ϕ q,ϕ + q,ϕ −
3.3 The quantum algebra UM (g). Let KP be the ideal of DP (g) generated by
q,ϕ ϕ q,ϕ
the elements K⊗1−1⊗K, K ∈ UQ(t); KP is in fact a Hopf ideal, whence DP (g) KP
q ϕ q,ϕ ϕ
is a Hopf algebra; then the above presentation of DP (g) yields the following one of
q,ϕ (cid:14)
UP (g) := DP (g) KP : it is the associative k(q)-algebra with 1 given by generators
q,ϕ q,ϕ ϕ
.
F , L (or K ), E (λ ∈ P;i = 1,...,n)
i λ λ i
and relations
L = 1, L L = L = L L , L F = q−(αj|λ)F L , L E = q(αj|λ)E L
0 λ µ λ+µ µ λ λ i i λ λ i i λ
L −L
E F −F E = δ αi −αi
i h h i ih q −q−1
i i (3.4)
1−aij 1−a 1−aij 1−a
(−1)k ij E1−aij−kE Ek = 0, (−1)k ij F1−aij−kF Fk = 0
k i j i k i j i
k=0 (cid:20) (cid:21)qi k=0 (cid:20) (cid:21)qi
X X