sigma10-010. 271KB Jun 04 2011 12:10:36 AM
Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 6 (2010), 010, 13 pages
q-Analog of Gelfand–Graev Basis
for the Noncompact Quantum Algebra Uq (u(n, 1))⋆
‡,
ˇ
´ ‡ , Miloslav HAVLI´CEK
ˇ
Raisa M. ASHEROVA † , Cestm´
ır BURDIK
Yuri F. SMIRNOV †§ and Valeriy N. TOLSTOY †‡
†
Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia
E-mail: [email protected], [email protected]
‡
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering,
Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic
E-mail: [email protected], [email protected]
§
Deceased
Received November 05, 2009, in final form January 15, 2010; Published online January 26, 2010
doi:10.3842/SIGMA.2010.010
Abstract. For the quantum algebra Uq (gl(n + 1)) in its reduction on the subalgebra Uq (gl(n)) an explicit description of a Mickelsson–Zhelobenko reduction Z-algebra
Zq (gl(n+1), gl(n)) is given in terms of the generators and their defining relations. Using
this Z-algebra we describe Hermitian irreducible representations of a discrete series for the
noncompact quantum algebra Uq (u(n, 1)) which is a real form of Uq (gl(n + 1)), namely, an
orthonormal Gelfand–Graev basis is constructed in an explicit form.
Key words: quantum algebra; extremal projector; reduction algebra; Shapovalov form; noncompact quantum algebra; discrete series of representations; Gelfand–Graev basis
2010 Mathematics Subject Classification: 17B37; 81R50
1
Introduction
In 1950, I.M. Gelfand and M.L. Tsetlin [1] proposed a formal description of finite-dimensional
irreducible representations (IR) for the compact Lie algebra u(n). This description is a generalization of the results for u(2) and u(3) to the u(n) case. It is the following. In the IR space
of u(n) there is a orthonormal basis which is numerated by the following formal schemes:
m1n
m2n
...
mn−1,n
mnn
m1,n−1
m2,n−1
...
mn−1,n−1
,
.
.
.
.
.
.
.
.
.
m12
m22
m11
where all numbers mij (1 ≤ i ≤ j ≤ n) are nonnegative integers and they satisfy the standard
inequalities, “between conditions”:
mij+1 ≥ mij ≥ mi+1j+1
for 1 ≤ i ≤ j ≤ n − 1.
The first line of this scheme is defined by the components of the highest weight of u(n) IR, the
second line is defined by the components of the highest weight of u(n − 1) IR and so on.
⋆
This paper is a contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries (June 18–20, 2009, Prague, Czech Republic). The full collection is available at
http://www.emis.de/journals/SIGMA/ISQS2009.html
ˇ Burd´ık, M. Havl´ıˇcek, Yu.F. Smirnov and V.N. Tolstoy
R.M. Asherova, C.
2
Later this basis was constructed in many papers (see, e.g., [2, 3, 4]) by using one-step lowering
and raising operators.
In 1965, I.M. Gelfand and M.I. Graev [5], using analytic continuation of the results for u(n),
obtained some results for noncompact Lie algebra u(n, m). It was shown that some class of
Hermitian IR of u(n, m) is characterized by an “extremal weight” parametrized by a set of
integers mN = (m1N , m2N , . . . , mN N ) (N = n + m) such that m1N ≥ m2N ≥ · · · ≥ mN N , and
by a representation type which is defined by a partition of n in the sum of two nonnegative
integers α and β, n = α + β (also see [6]).
For simplicity we consider the case u(2, 1). In this case we have three types of schemes
m13
m23 m33
m12
m22
for (α, β) = (2, 0),
m11
m13 m23 m33
m12
m22
for (α, β) = (1, 1),
m11
m13 m23
m33
m12
m22
for (α, β) = (0, 2).
m11
The numbers mij of the first scheme satisfy the following inequalities
m12 ≥ m13 + 1,
m13 + 1 ≥ m22 ≥ m23 + 1,
m12 ≥ m11 ≥ m22 .
The numbers mij of the second scheme satisfy the following inequalities
m12 ≥ m13 + 1,
m33 − 1 ≥ m22 ,
m12 ≥ m11 ≥ m22 .
The numbers of the third scheme satisfy the following inequalities
m23 − 1 ≥ m12 ≥ m33 − 1,
m33 − 1 ≥ m22 ,
m12 ≥ m11 ≥ m22 .
Construction of the Gelfand–Graev basis for u(n, m) in terms of one-step lowering and raising
operators is more complicated than in the compact case u(n + m).
In 1975, T.J. Enright and V.S. Varadarajan [7] obtained a classification of discrete series of
noncompact Lie algebras. Later it was proved by A.I. Molev [8] that in the case of u(n, m) the
Gelfand–Graev modules are part of the Enright–Varadarajan modules and Molev constructed
the Gelfand–Graev basis for u(n, m) in terms of the Mickelsson S-algebra [9].
A goal of this work is to obtain analogous results for the noncompact quantum algebra
Uq (u(n,m)). Since the general case is very complicated we at first consider the case Uq (u(n,1)).
The special case Uq (u(2, 1)) was considered in [10, 11]. It should be noted that the principal
series representations of Uq (u(n, 1)) were studied in [12] and a classification of unitary highest
weight modules of Uq (u(n, 1)) was considered in [13].
2
Quantum algebra Uq (gl(N )) and its noncompact
real forms Uq (u(n, m)) (n + m = N )
The quantum algebra Uq (gl(N )) is generated by the Chevalley elements q ±eii (i = 1, . . . , N ),
ei,i+1 , ei+1,i (i = 1, 2, . . . , N − 1) with the defining relations [14, 15]:
q eii q −eii = q −eii q eii = 1,
(2.1)
q-Analog of Gelfand–Graev Basis for the Noncompact Quantum Algebra Uq (u(n, 1))
q eii q ejj = q ejj q eii ,
eii
q ejk q
−eii
=q
(2.2)
δij −δik
[ei,i+1 , ej+1,j ] = δij
3
ejk
(|j − k| = 1),
e
−e
q ii i+1,i+1 − q ei+1,i+1 −eii
[ei,i+1 , ej,j+1 ] = 0
q − q −1
for |i − j| ≥ 2,
[ei+1,i , ej+1,j ] = 0
for |i − j| ≥ 2,
(2.3)
,
(2.4)
(2.5)
(2.6)
[[ei,i+1 , ej,j+1 ]q , ej,j+1 ]q = 0
for |i − j| = 1,
(2.7)
[[ei+1,i , ej+1,j ]q , ej+1,j ]q = 0
for |i − j| = 1,
(2.8)
where [eβ , eγ ]q denotes the q-commutator:
[eβ , eγ ]q := eβ eγ − q (β,γ) eγ eβ .
The definition of a quantum algebra also includes operations of a comultiplication ∆q , an antipode Sq , and a co-unit ǫq . Explicit formulas of these operations will not be used in our later
calculations and they are not given here.
Let εi (i = 1, 2, . . . , N ) be a dual basis to the Cartan basis eii (i = 1, 2, . . . , N ), εi (ejj ) =
(εi , εj ) = δij . In terms of the orthonormal basis vectors εi the positive root system ∆+ of gl(N )
(Uq (gl(N ))) is presented as follows:
∆+ = {εi − εj | 1 ≤ i < j ≤ N },
where εi − εi+1 (i = 1, 2, . . . , N − 1) are the simple roots.
Since for construction of the composite root vectors eij := eεi −εj (|i − j| ≥ 2) of the quantum
algebra Uq (gl(N )) we need to use the notation of the normal ordering in the positive root system
∆+ , we recall this notation. We say that the system ∆+ is written in normal (convex) ordering,
~ + , if each positive composite root εi − εj = (εi − εk ) + (εk − εj ) (i ≤ k ≤ j) is located between
∆
~ + we have
its components εi − εk and εk − εj . It means that in the normal ordering system ∆
either
. . . , εi − εk , . . . , εi − εj , . . . , εk − εj , . . . ,
or
. . . , εk − εj , . . . , εi − εj , . . . , εi − εk , . . . .
There are many normal orderings in the root system ∆+ = ∆+ (gl(N )), more than (N − 1)! for
N > 3. To be definite, we fix the following normal ordering (see [14, 15])
ε1 − ε2 ≺ ε 1 − ε3 ≺ ε2 − ε3 ≺ ε1 − ε4 ≺ ε2 − ε4 ≺ ε3 − ε4 ≺ · · · ≺
ε1 − εk ≺ ε2 − εk ≺ · · · ≺ εk−1 − εk ≺ · · · ≺ ε1 − εN ≺ ε2 − εN ≺ · · · ≺ εN −1 − εN .
(2.9)
According to this ordering, we determine the composite root vectors eij for |i − j| ≥ 2 as follows:
eij := [eik , ekj ]q−1 ,
eji := [ejk , eki ]q ,
(2.10)
where 1 ≤ i < k < j ≤ N . It should be stressed that the structure of the composite root vectors
does not depend on the choice of the index k on the right-hand side of the definition (2.10). In
particular, we have
eij := [ei,i+1 , ei+1,j ]q−1 = [ei,j−1 , ej−1,j ]q−1 ,
eji := [ej,i+1 , ei+1,i ]q = [ej,j−1 , ej−1,i ]q ,
where 2 ≤ i + 1 < j ≤ N .
(2.11)
ˇ Burd´ık, M. Havl´ıˇcek, Yu.F. Smirnov and V.N. Tolstoy
R.M. Asherova, C.
4
Using these explicit constructions and defining relations (2.1)–(2.8) for the Chevalley basis it
is not hard to calculate the following relations between the Cartan–Weyl generators eij (i, j =
1, 2, . . . , N ):
q ekk eij q −ekk = q δki −δkj eij
[eij , eji ] =
[eij , ekl ]q−1
[eik , ejl ]q−1
q eii −ejj
(1 ≤ i, j, k ≤ N ),
(2.12)
q ejj −eii
−
(1 ≤ i < j ≤ N ),
q − q −1
= δjk eil
(1 ≤ i < j ≤ k < l ≤ N ),
−1
= q−q
ejk eil
(1 ≤ i < j < k < l ≤ N ),
(2.13)
(2.14)
(2.15)
[ejk , eil ]q−1 = 0
(1 ≤ i ≤ j < k < l ≤ N ),
(2.16)
[eik , ejk ]q−1 = 0
(1 ≤ i < j < k ≤ N ),
(2.17)
(1 ≤ i < j ≤ k < l ≤ N ),
(2.18)
(1 ≤ i < j < k < l ≤ N ),
(2.19)
[ekl , eji ] = 0
[eil , ekj ] = 0
[eji , eil ] = ejl q
eii −ejj
(1 ≤ i < j < l ≤ N ),
(2.20)
ekk −ell
(1 ≤ i < k < l ≤ N ),
(2.21)
[ekl , eli ] = eki q
[ejl , eki ] = q −1 − q ekl eji q ejj −ekk
(1 ≤ i < j < k < l ≤ N ).
(2.22)
If we apply the Cartan involution (e⋆ij = eji , q ⋆ = q −1 ) to the formulas (2.12)–(2.22), we get
all relations between the elements of the Cartan–Weyl basis.
The explicit formula for the extremal projector for Uq (gl(N )), corresponding to the fixed
normal ordering (2.9), has the form [14, 15]
p(Uq (gl(N )) = p(Uq (gl(N − 1))(p1N p2N · · · pN −2,N pN −1,N )
= p12 (p13 p23 ) · · · (p1k · · · pk−1,k ) · · · (p1N · · · pN −1,N ),
(2.23)
where the elements pij (1 ≤ i < j ≤ N ) are given by
pij =
∞
X
(−1)r
r=0
[r]!
ϕij,r erij erji ,
ϕij,r = q −(j−i−1)r
(
r
Y
)−1
[eii − ejj + j − i + s]
s=1
.
(2.24)
Here and elsewhere the symbol [x] is given as follows
[x] =
q x − q −x
.
q − q −1
The extremal projector p := p(Uq (gl(N )) satisfies the relations:
ei,i+1 p = pei+1,i = 0
(1 ≤ i ≤ N − 1),
p2 = p.
(2.25)
The extremal projector p belongs to the Taylor extension T Uq (gl(N )) of the quantum algebras
Uq (gl(N )). The Taylor extension T Uq (gl(N )) is an associative algebra generated by formal
Taylor series of the form
X
˜12 r˜13 r˜23
r˜N −1,N r12 r13 r23
rN −1,N
e31 e32 · · · eN,N
C{˜r},{r} q e11 , . . . , q eN N er21
−1 e12 e13 e23 · · · eN −1,N
{˜
r },{r}
provided that nonnegative integers r˜12 , r˜13 , r˜23 , . . . , r˜N −1,N and r12 , r13 , r23 , . . . , rN −1,N are subject to the constraints
X
X
r˜ij −
rij ≤ const
i = p for > = ⋆, ∗ (see
the formula (2.29)), it turns out that
zi> = pe>
i,n+1 p,
>
z−i
= pe>
n+1,i p
for i = 1, 2, . . . , n.
>
If q is circular, then e>
i,n+1 = ±en+1,i , en+1,i = ±ei,n+1 , where the plus belongs to the compact
case and the minus belongs to the noncompact case, and we obtain the formulas (4.1) and (4.2).
>
′
′
′
′
If q is real, then e>
n+1,i = ±ei,n+1 , ei,n+1 = ±en+1,i where the root vectors ei,n+1 and en+1,i
>
−zi =
are obtained from (2.10) by the replacement q ±1 → q ∓1 . Let us consider the difference z−i
′
p(ei,n+1 − ei,n+1 )p for 1 ≤ i ≤ n). Substituting here (see the formulas (2.11))
e′i,n+1 = e′in en,n+1 − q −1 en,n+1 e′in ,
ei,n+1 = ein en,n+1 − qen,n+1 ein ,
(4.3)
q-Analog of Gelfand–Graev Basis for the Noncompact Quantum Algebra Uq (u(n, 1))
9
>
and using the annihilation properties of the projector p (see (2.25)) we obtain z−i
− zi = p(e′in −
ein )en,n+1 p. In a similar way, using explicit formulas of type (4.3) for the generators e′in and ein ,
>
>
we obtain z−i
− zi = p(e′i,n−1 − ei,n−1 )en−1,n en,n+1 p. By proceeding as above, we have z−i
−
zi = p(ei,i+1 − ei,i+1 )ei+1,i+2 ei+2,i+3 · · · en−1,n en,n+1 p = 0. In a similar way, we prove that
zi> − z−i = 0.
The Z-algebra Zq (gl(n + 1), gl(n)) with the involution ⋆ is called the compact real form and is
(c)
denoted by the symbol Zq (gl(n + 1), gl(n)). The noncompact real form on Zq (gl(n + 1), gl(n))
(nc)
is defined by the involution ∗ and is denoted by the symbol Zq (gl(n + 1), gl(n)).
Let p(α) be an extremal projector for Zq (gl(n + 1), gl(n)) satisfying the relations
z−i p(α) = p(α) zi
zk p
(α)
=p
(α)
[eii , p(α) ] = 0
for i = 1, 2, . . . , α,
for k = α + 1, α + 2, . . . , n,
z−k
for i = 1, 2, . . . , n.
The extremal projector p(α) depends on the index α that defines what elements are considered as “raising” and what elements are considered as “lowering”, i.e. in our case the elements z−1 , z−2 , . . . , z−α , zα+1 , . . . , zn are raising and the elements z1 , z2 , . . . , zα , z−α−1 , . . . , z−n
are lowering. It should be stressed that the “raising” and “lowering” subsets generate disjoint
subalgebras in Zq (gl(n + 1), gl(n)). The operator p(α) can be constructed in an explicit form.
(nc)
Let us introduce on Zq (gl(n + 1), gl(n)) the following sesquilinear Shapovalov form [16].
(nc)
For any elements x, y ∈ Zq (gl(n + 1), gl(n)) we set
B (α) (x, y) = p(α) y ∗ xp(α) .
(4.4)
Therefore, the Shapovalov form also depends on the index α (α = 0, 1, 2, . . . , n). We fix α
(α = 0, 1, 2, . . . , n) and for each set of nonnegative integers {r} = (r1 , r2 , . . . , rn ) introduce
(α)
(nc)
a vector v{r} in the space Zq (gl(n + 1), gl(n)) by the formula
(α)
rn
rα +1
v{r} = zαrα · · · z1r1 z−α−1
.
· · · z−n
(4.5)
(α)
Theorem 2. For each fixed α (α = 0, 1, 2, . . . , n) the vectors {v{r} } are pairwise orthogonal with
respect to the Shapovalov form (4.4)
and
(α) (α)
(α) (α)
B (α) v{r} , v{r′ } = δ{r},{r′ } B (α) v{r} , v{r} .
α
n
Y
[ri ]![ϕi,n+1 − α + ri − 1]! Y [rl ]![ϕn+1,l + α + rl ]!
[ϕi,n+1 − α − 1]!
[ϕn+1,l + α]!
(α) (α)
B (α) v{r} , v{r} =
i=1
×
l=α+1
Y
1≤i
SIGMA 6 (2010), 010, 13 pages
q-Analog of Gelfand–Graev Basis
for the Noncompact Quantum Algebra Uq (u(n, 1))⋆
‡,
ˇ
´ ‡ , Miloslav HAVLI´CEK
ˇ
Raisa M. ASHEROVA † , Cestm´
ır BURDIK
Yuri F. SMIRNOV †§ and Valeriy N. TOLSTOY †‡
†
Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia
E-mail: [email protected], [email protected]
‡
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering,
Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic
E-mail: [email protected], [email protected]
§
Deceased
Received November 05, 2009, in final form January 15, 2010; Published online January 26, 2010
doi:10.3842/SIGMA.2010.010
Abstract. For the quantum algebra Uq (gl(n + 1)) in its reduction on the subalgebra Uq (gl(n)) an explicit description of a Mickelsson–Zhelobenko reduction Z-algebra
Zq (gl(n+1), gl(n)) is given in terms of the generators and their defining relations. Using
this Z-algebra we describe Hermitian irreducible representations of a discrete series for the
noncompact quantum algebra Uq (u(n, 1)) which is a real form of Uq (gl(n + 1)), namely, an
orthonormal Gelfand–Graev basis is constructed in an explicit form.
Key words: quantum algebra; extremal projector; reduction algebra; Shapovalov form; noncompact quantum algebra; discrete series of representations; Gelfand–Graev basis
2010 Mathematics Subject Classification: 17B37; 81R50
1
Introduction
In 1950, I.M. Gelfand and M.L. Tsetlin [1] proposed a formal description of finite-dimensional
irreducible representations (IR) for the compact Lie algebra u(n). This description is a generalization of the results for u(2) and u(3) to the u(n) case. It is the following. In the IR space
of u(n) there is a orthonormal basis which is numerated by the following formal schemes:
m1n
m2n
...
mn−1,n
mnn
m1,n−1
m2,n−1
...
mn−1,n−1
,
.
.
.
.
.
.
.
.
.
m12
m22
m11
where all numbers mij (1 ≤ i ≤ j ≤ n) are nonnegative integers and they satisfy the standard
inequalities, “between conditions”:
mij+1 ≥ mij ≥ mi+1j+1
for 1 ≤ i ≤ j ≤ n − 1.
The first line of this scheme is defined by the components of the highest weight of u(n) IR, the
second line is defined by the components of the highest weight of u(n − 1) IR and so on.
⋆
This paper is a contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries (June 18–20, 2009, Prague, Czech Republic). The full collection is available at
http://www.emis.de/journals/SIGMA/ISQS2009.html
ˇ Burd´ık, M. Havl´ıˇcek, Yu.F. Smirnov and V.N. Tolstoy
R.M. Asherova, C.
2
Later this basis was constructed in many papers (see, e.g., [2, 3, 4]) by using one-step lowering
and raising operators.
In 1965, I.M. Gelfand and M.I. Graev [5], using analytic continuation of the results for u(n),
obtained some results for noncompact Lie algebra u(n, m). It was shown that some class of
Hermitian IR of u(n, m) is characterized by an “extremal weight” parametrized by a set of
integers mN = (m1N , m2N , . . . , mN N ) (N = n + m) such that m1N ≥ m2N ≥ · · · ≥ mN N , and
by a representation type which is defined by a partition of n in the sum of two nonnegative
integers α and β, n = α + β (also see [6]).
For simplicity we consider the case u(2, 1). In this case we have three types of schemes
m13
m23 m33
m12
m22
for (α, β) = (2, 0),
m11
m13 m23 m33
m12
m22
for (α, β) = (1, 1),
m11
m13 m23
m33
m12
m22
for (α, β) = (0, 2).
m11
The numbers mij of the first scheme satisfy the following inequalities
m12 ≥ m13 + 1,
m13 + 1 ≥ m22 ≥ m23 + 1,
m12 ≥ m11 ≥ m22 .
The numbers mij of the second scheme satisfy the following inequalities
m12 ≥ m13 + 1,
m33 − 1 ≥ m22 ,
m12 ≥ m11 ≥ m22 .
The numbers of the third scheme satisfy the following inequalities
m23 − 1 ≥ m12 ≥ m33 − 1,
m33 − 1 ≥ m22 ,
m12 ≥ m11 ≥ m22 .
Construction of the Gelfand–Graev basis for u(n, m) in terms of one-step lowering and raising
operators is more complicated than in the compact case u(n + m).
In 1975, T.J. Enright and V.S. Varadarajan [7] obtained a classification of discrete series of
noncompact Lie algebras. Later it was proved by A.I. Molev [8] that in the case of u(n, m) the
Gelfand–Graev modules are part of the Enright–Varadarajan modules and Molev constructed
the Gelfand–Graev basis for u(n, m) in terms of the Mickelsson S-algebra [9].
A goal of this work is to obtain analogous results for the noncompact quantum algebra
Uq (u(n,m)). Since the general case is very complicated we at first consider the case Uq (u(n,1)).
The special case Uq (u(2, 1)) was considered in [10, 11]. It should be noted that the principal
series representations of Uq (u(n, 1)) were studied in [12] and a classification of unitary highest
weight modules of Uq (u(n, 1)) was considered in [13].
2
Quantum algebra Uq (gl(N )) and its noncompact
real forms Uq (u(n, m)) (n + m = N )
The quantum algebra Uq (gl(N )) is generated by the Chevalley elements q ±eii (i = 1, . . . , N ),
ei,i+1 , ei+1,i (i = 1, 2, . . . , N − 1) with the defining relations [14, 15]:
q eii q −eii = q −eii q eii = 1,
(2.1)
q-Analog of Gelfand–Graev Basis for the Noncompact Quantum Algebra Uq (u(n, 1))
q eii q ejj = q ejj q eii ,
eii
q ejk q
−eii
=q
(2.2)
δij −δik
[ei,i+1 , ej+1,j ] = δij
3
ejk
(|j − k| = 1),
e
−e
q ii i+1,i+1 − q ei+1,i+1 −eii
[ei,i+1 , ej,j+1 ] = 0
q − q −1
for |i − j| ≥ 2,
[ei+1,i , ej+1,j ] = 0
for |i − j| ≥ 2,
(2.3)
,
(2.4)
(2.5)
(2.6)
[[ei,i+1 , ej,j+1 ]q , ej,j+1 ]q = 0
for |i − j| = 1,
(2.7)
[[ei+1,i , ej+1,j ]q , ej+1,j ]q = 0
for |i − j| = 1,
(2.8)
where [eβ , eγ ]q denotes the q-commutator:
[eβ , eγ ]q := eβ eγ − q (β,γ) eγ eβ .
The definition of a quantum algebra also includes operations of a comultiplication ∆q , an antipode Sq , and a co-unit ǫq . Explicit formulas of these operations will not be used in our later
calculations and they are not given here.
Let εi (i = 1, 2, . . . , N ) be a dual basis to the Cartan basis eii (i = 1, 2, . . . , N ), εi (ejj ) =
(εi , εj ) = δij . In terms of the orthonormal basis vectors εi the positive root system ∆+ of gl(N )
(Uq (gl(N ))) is presented as follows:
∆+ = {εi − εj | 1 ≤ i < j ≤ N },
where εi − εi+1 (i = 1, 2, . . . , N − 1) are the simple roots.
Since for construction of the composite root vectors eij := eεi −εj (|i − j| ≥ 2) of the quantum
algebra Uq (gl(N )) we need to use the notation of the normal ordering in the positive root system
∆+ , we recall this notation. We say that the system ∆+ is written in normal (convex) ordering,
~ + , if each positive composite root εi − εj = (εi − εk ) + (εk − εj ) (i ≤ k ≤ j) is located between
∆
~ + we have
its components εi − εk and εk − εj . It means that in the normal ordering system ∆
either
. . . , εi − εk , . . . , εi − εj , . . . , εk − εj , . . . ,
or
. . . , εk − εj , . . . , εi − εj , . . . , εi − εk , . . . .
There are many normal orderings in the root system ∆+ = ∆+ (gl(N )), more than (N − 1)! for
N > 3. To be definite, we fix the following normal ordering (see [14, 15])
ε1 − ε2 ≺ ε 1 − ε3 ≺ ε2 − ε3 ≺ ε1 − ε4 ≺ ε2 − ε4 ≺ ε3 − ε4 ≺ · · · ≺
ε1 − εk ≺ ε2 − εk ≺ · · · ≺ εk−1 − εk ≺ · · · ≺ ε1 − εN ≺ ε2 − εN ≺ · · · ≺ εN −1 − εN .
(2.9)
According to this ordering, we determine the composite root vectors eij for |i − j| ≥ 2 as follows:
eij := [eik , ekj ]q−1 ,
eji := [ejk , eki ]q ,
(2.10)
where 1 ≤ i < k < j ≤ N . It should be stressed that the structure of the composite root vectors
does not depend on the choice of the index k on the right-hand side of the definition (2.10). In
particular, we have
eij := [ei,i+1 , ei+1,j ]q−1 = [ei,j−1 , ej−1,j ]q−1 ,
eji := [ej,i+1 , ei+1,i ]q = [ej,j−1 , ej−1,i ]q ,
where 2 ≤ i + 1 < j ≤ N .
(2.11)
ˇ Burd´ık, M. Havl´ıˇcek, Yu.F. Smirnov and V.N. Tolstoy
R.M. Asherova, C.
4
Using these explicit constructions and defining relations (2.1)–(2.8) for the Chevalley basis it
is not hard to calculate the following relations between the Cartan–Weyl generators eij (i, j =
1, 2, . . . , N ):
q ekk eij q −ekk = q δki −δkj eij
[eij , eji ] =
[eij , ekl ]q−1
[eik , ejl ]q−1
q eii −ejj
(1 ≤ i, j, k ≤ N ),
(2.12)
q ejj −eii
−
(1 ≤ i < j ≤ N ),
q − q −1
= δjk eil
(1 ≤ i < j ≤ k < l ≤ N ),
−1
= q−q
ejk eil
(1 ≤ i < j < k < l ≤ N ),
(2.13)
(2.14)
(2.15)
[ejk , eil ]q−1 = 0
(1 ≤ i ≤ j < k < l ≤ N ),
(2.16)
[eik , ejk ]q−1 = 0
(1 ≤ i < j < k ≤ N ),
(2.17)
(1 ≤ i < j ≤ k < l ≤ N ),
(2.18)
(1 ≤ i < j < k < l ≤ N ),
(2.19)
[ekl , eji ] = 0
[eil , ekj ] = 0
[eji , eil ] = ejl q
eii −ejj
(1 ≤ i < j < l ≤ N ),
(2.20)
ekk −ell
(1 ≤ i < k < l ≤ N ),
(2.21)
[ekl , eli ] = eki q
[ejl , eki ] = q −1 − q ekl eji q ejj −ekk
(1 ≤ i < j < k < l ≤ N ).
(2.22)
If we apply the Cartan involution (e⋆ij = eji , q ⋆ = q −1 ) to the formulas (2.12)–(2.22), we get
all relations between the elements of the Cartan–Weyl basis.
The explicit formula for the extremal projector for Uq (gl(N )), corresponding to the fixed
normal ordering (2.9), has the form [14, 15]
p(Uq (gl(N )) = p(Uq (gl(N − 1))(p1N p2N · · · pN −2,N pN −1,N )
= p12 (p13 p23 ) · · · (p1k · · · pk−1,k ) · · · (p1N · · · pN −1,N ),
(2.23)
where the elements pij (1 ≤ i < j ≤ N ) are given by
pij =
∞
X
(−1)r
r=0
[r]!
ϕij,r erij erji ,
ϕij,r = q −(j−i−1)r
(
r
Y
)−1
[eii − ejj + j − i + s]
s=1
.
(2.24)
Here and elsewhere the symbol [x] is given as follows
[x] =
q x − q −x
.
q − q −1
The extremal projector p := p(Uq (gl(N )) satisfies the relations:
ei,i+1 p = pei+1,i = 0
(1 ≤ i ≤ N − 1),
p2 = p.
(2.25)
The extremal projector p belongs to the Taylor extension T Uq (gl(N )) of the quantum algebras
Uq (gl(N )). The Taylor extension T Uq (gl(N )) is an associative algebra generated by formal
Taylor series of the form
X
˜12 r˜13 r˜23
r˜N −1,N r12 r13 r23
rN −1,N
e31 e32 · · · eN,N
C{˜r},{r} q e11 , . . . , q eN N er21
−1 e12 e13 e23 · · · eN −1,N
{˜
r },{r}
provided that nonnegative integers r˜12 , r˜13 , r˜23 , . . . , r˜N −1,N and r12 , r13 , r23 , . . . , rN −1,N are subject to the constraints
X
X
r˜ij −
rij ≤ const
i = p for > = ⋆, ∗ (see
the formula (2.29)), it turns out that
zi> = pe>
i,n+1 p,
>
z−i
= pe>
n+1,i p
for i = 1, 2, . . . , n.
>
If q is circular, then e>
i,n+1 = ±en+1,i , en+1,i = ±ei,n+1 , where the plus belongs to the compact
case and the minus belongs to the noncompact case, and we obtain the formulas (4.1) and (4.2).
>
′
′
′
′
If q is real, then e>
n+1,i = ±ei,n+1 , ei,n+1 = ±en+1,i where the root vectors ei,n+1 and en+1,i
>
−zi =
are obtained from (2.10) by the replacement q ±1 → q ∓1 . Let us consider the difference z−i
′
p(ei,n+1 − ei,n+1 )p for 1 ≤ i ≤ n). Substituting here (see the formulas (2.11))
e′i,n+1 = e′in en,n+1 − q −1 en,n+1 e′in ,
ei,n+1 = ein en,n+1 − qen,n+1 ein ,
(4.3)
q-Analog of Gelfand–Graev Basis for the Noncompact Quantum Algebra Uq (u(n, 1))
9
>
and using the annihilation properties of the projector p (see (2.25)) we obtain z−i
− zi = p(e′in −
ein )en,n+1 p. In a similar way, using explicit formulas of type (4.3) for the generators e′in and ein ,
>
>
we obtain z−i
− zi = p(e′i,n−1 − ei,n−1 )en−1,n en,n+1 p. By proceeding as above, we have z−i
−
zi = p(ei,i+1 − ei,i+1 )ei+1,i+2 ei+2,i+3 · · · en−1,n en,n+1 p = 0. In a similar way, we prove that
zi> − z−i = 0.
The Z-algebra Zq (gl(n + 1), gl(n)) with the involution ⋆ is called the compact real form and is
(c)
denoted by the symbol Zq (gl(n + 1), gl(n)). The noncompact real form on Zq (gl(n + 1), gl(n))
(nc)
is defined by the involution ∗ and is denoted by the symbol Zq (gl(n + 1), gl(n)).
Let p(α) be an extremal projector for Zq (gl(n + 1), gl(n)) satisfying the relations
z−i p(α) = p(α) zi
zk p
(α)
=p
(α)
[eii , p(α) ] = 0
for i = 1, 2, . . . , α,
for k = α + 1, α + 2, . . . , n,
z−k
for i = 1, 2, . . . , n.
The extremal projector p(α) depends on the index α that defines what elements are considered as “raising” and what elements are considered as “lowering”, i.e. in our case the elements z−1 , z−2 , . . . , z−α , zα+1 , . . . , zn are raising and the elements z1 , z2 , . . . , zα , z−α−1 , . . . , z−n
are lowering. It should be stressed that the “raising” and “lowering” subsets generate disjoint
subalgebras in Zq (gl(n + 1), gl(n)). The operator p(α) can be constructed in an explicit form.
(nc)
Let us introduce on Zq (gl(n + 1), gl(n)) the following sesquilinear Shapovalov form [16].
(nc)
For any elements x, y ∈ Zq (gl(n + 1), gl(n)) we set
B (α) (x, y) = p(α) y ∗ xp(α) .
(4.4)
Therefore, the Shapovalov form also depends on the index α (α = 0, 1, 2, . . . , n). We fix α
(α = 0, 1, 2, . . . , n) and for each set of nonnegative integers {r} = (r1 , r2 , . . . , rn ) introduce
(α)
(nc)
a vector v{r} in the space Zq (gl(n + 1), gl(n)) by the formula
(α)
rn
rα +1
v{r} = zαrα · · · z1r1 z−α−1
.
· · · z−n
(4.5)
(α)
Theorem 2. For each fixed α (α = 0, 1, 2, . . . , n) the vectors {v{r} } are pairwise orthogonal with
respect to the Shapovalov form (4.4)
and
(α) (α)
(α) (α)
B (α) v{r} , v{r′ } = δ{r},{r′ } B (α) v{r} , v{r} .
α
n
Y
[ri ]![ϕi,n+1 − α + ri − 1]! Y [rl ]![ϕn+1,l + α + rl ]!
[ϕi,n+1 − α − 1]!
[ϕn+1,l + α]!
(α) (α)
B (α) v{r} , v{r} =
i=1
×
l=α+1
Y
1≤i