A q-Analog of the Hua Equations
A necessary condition is established for a function to be in the image of a quantum Poisson integral operator associated to the Shilov boundary of the quantum matrix ball. A quantum analogue of the Hua equations is introduced.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2009
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irk-123456789-1065422016-10-01T03:01:42Z A q-Analog of the Hua Equations Bershtein, O. Sinel’shchikov, S. A necessary condition is established for a function to be in the image of a quantum Poisson integral operator associated to the Shilov boundary of the quantum matrix ball. A quantum analogue of the Hua equations is introduced. 2009 Article A q-Analog of the Hua Equations / O. Bershtein, S. Sinel’shchikov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 3. — С. 219-244. — Бібліогр.: 45 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106542 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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A necessary condition is established for a function to be in the image of a quantum Poisson integral operator associated to the Shilov boundary of the quantum matrix ball. A quantum analogue of the Hua equations is introduced. |
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Article |
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Bershtein, O. Sinel’shchikov, S. |
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Bershtein, O. Sinel’shchikov, S. A q-Analog of the Hua Equations Журнал математической физики, анализа, геометрии |
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Bershtein, O. Sinel’shchikov, S. |
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Bershtein, O. |
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A q-Analog of the Hua Equations |
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A q-Analog of the Hua Equations |
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A q-Analog of the Hua Equations |
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A q-Analog of the Hua Equations |
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A q-Analog of the Hua Equations |
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q-analog of the hua equations |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/106542 |
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A q-Analog of the Hua Equations / O. Bershtein, S. Sinel’shchikov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 3. — С. 219-244. — Бібліогр.: 45 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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2025-07-07T18:37:34Z |
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| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 3, pp. 219–244
A q-Analog of the Hua Equations
O. Bershtein∗ and S. Sinel’shchikov
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:bershtein@ilt.kharkov.ua
sinelshchikov@ilt.kharkov.ua
Received March 28, 2008
A necessary condition is established for a function to be in the image
of a quantum Poisson integral operator associated to the Shilov boundary
of the quantum matrix ball. A quantum analogue of the Hua equations is
introduced.
Key words: quantum matrix ball, Shilov boundary, Poisson integral ope-
rator, invariant kernel, Hua equations.
Mathematics Subject Classification 2000: 81R50, 17B37 (primary); 31B10
(secondary).
Dedicated to the memory of L.L. Vaksman
1. Introduction
In late 90s three groups of specialists became advanced in putting the basics
of quantum theory of bounded symmetric domains.
T. Tanisaki and his team introduced q-analogs for the prehomogeneous vector
spaces of commutative parabolic type and found an explicit form of the associated
Sato–Bernstein polynomials [16–18, 21].
On the other hand, H. Jakobsen suggested a less intricate method of producing
the above quantum vector spaces. Actually, he was on the way to quantum
Hermitian symmetric spaces of noncompact type [13, 12]. A similar approach
was used by W. Baldoni and P. Frajria [1] for q-analogs of algebras of invariant
differential operators and the Harish-Chandra homomorphism for these quantum
symmetric spaces. During the same period H. Jakobsen obtained a description
of all the unitarizable highest weight modules over the Drinfeld–Jimbo algebras
[11].
∗Partially supported by a grant of the President of Ukraine.
c© O. Bershtein and S. Sinel’shchikov, 2009
O. Bershtein and S. Sinel’shchikov
The authors named above made no use of the full symmetry of the quantum
prehomogeneous vector spaces in question [35, 31], what became an obstacle in
producing the quantum theory of bounded symmetric domains.
The paper [36] laid the foundations of this theory. The subsequent results
were obtained in the works by L. Vaksman, D. Proskurin, S. Sinel’shchikov,
A. Stolin, D. Shklyarov, L. Turowska, H. Zhang [39, 23, 34, 42, 27, 41, 32].
The compatibility of the approaches described above [38, 11, 36] was proved by
D. Shklyarov [28]. The study of quantum analogs of the Harish-Chandra modules
related to quantum bounded symmetric domains and their geometric realizations
has been started in [29, 37, 30]. The present work proceeds with this research.
Recall that a bounded domain D in a finite dimensional vector space is said to
be symmetric if every point p ∈ D is an isolated fixed point of the biholomorphic
involutive automorphism ϕp : D → D, ϕp ◦ ϕp = id.
Equip the vector space of linear maps in C
n and the canonically isomorphic
vector space Matn of complex n × n matrices with the operator norms. It is
known [8] that the unit ball D = {z ∈ Matn | zz∗ < 1} is a bounded symmetric
domain.
Denote by S(D) the Shilov boundary of D, S(D) = {z ∈ Matn |I−zz∗ = 0} ∼=
Un. It is well known that both D and S(D) are homogeneous spaces of the group
SUn,n. Consider a function on D × S(D) given by
P (z, ζ) =
det(1− zz∗)n
|det(1− ζ∗z)|2n
, z ∈ D, ζ ∈ S(D). (1.1)
It is called the Poisson kernel [10] associated to the Shilov boundary.
General concepts on the boundaries of Hermitian symmetric spaces of non-
compact type and the associated Poisson kernels are exposed in [20, 19].
The Poisson kernel, together with the S(Un ×Un)-invariant integral
∫
Un
·dν(ζ)
on Un, allows one to define the Poisson integral operator
P : f(ζ)
→
∫
Un
P (z, ζ)f(ζ)dν(ζ), z ∈ D, ζ ∈ S(D).
It intertwines the actions of SUn,n in the spaces of continuous functions on the
domain D and on the Shilov boundary S(D). However, not every continuous
function on D can be produced by applying the Poisson integral operator to
a continuous function on the Shilov boundary S(D).
In [10], L.K. Hua obtained the initial results on differential equations whose
solutions included the functions of the form P(f). A later result of K. Johnson
and A. Korányi [15] provided a system of differential equations by giving a com-
plete characterization of these functions. Their version of the Hua equations is
220 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
as follows:
n∑
i,j,k=1
(
δij −
n∑
c=1
zc
i z
c
j
)(
δkα −
n∑
c=1
zk
cz
α
c
)
∂2
∂zk
j∂z
a
i
u(z) = 0,
n∑
i,j,k=1
(
δai −
n∑
c=1
zc
az
c
i
)(
δjk −
n∑
c=1
zj
cz
k
c
)
∂2
∂zj
i∂z
k
α
u(z) = 0
for a, α = 1, 2, . . . , n.
This can be also represented in the form
n∑
c=1
∂2u(g · z)
∂zβ
c ∂zα
c
∣∣∣∣∣
z=0
= 0, g ∈ SUn,n, α, β ∈ {1, 2, . . . , n}. (1.2)
It is known that the Poisson kernel (1.1) as a function of z is a solution of the
above equation system. While proving this, one may stick to the special case of
g = 1, because P (gz, ζ) = const(g, ζ)P (z, g−1ζ), cf. [15, p. 597]. What remains
is to note that the relations
n∑
c=1
∂2P
∂zβ
c ∂zα
c
∣∣∣∣∣
z=0
= 0, ζ ∈ Un, α, β ∈ {1, 2, . . . , n} (1.3)
follow from (1.1).
Thus, in the classical case, the Poisson integral operator applied to a function
on the Shilov boundary is a solution of (1.2). We are going to obtain a quantum
analog of this well-known result.
The work suggests a quantum analog for the Hua equations. We thus get
a quantization of the necessary condition for a function to be a Poisson integral
of a function on the Shilov boundary.
The authors acknowledge that the ideas of this work were inspired by plentiful
long-term communications with L. Vaksman. Also, we would like to express our
gratitude to the referee for pointing out several inconsistencies in the previous
version of this work and encouraging us to improve the exposition.
2. A Background on Function Theory in Quantum Matrix Ball
In what follows we assume C to be a ground field and all algebras to be
associative and unital.
Recall a construction of the quantum universal enveloping algebra for the
Lie algebra slN . The quantum universal enveloping algebras were introduced by
V. Drinfeld and M. Jimbo in an essentially more general way than it is described
below. We follow the notation of [4, 5, 14, 25].
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 221
O. Bershtein and S. Sinel’shchikov
Let q ∈ (0, 1). The Hopf algebra UqslN is given by its generators Ki, K−1
i ,
Ei, Fi, i = 1, 2, . . . , N − 1, and the relations:
KiKj = KjKi, KiK
−1
i = K−1
i Ki = 1,
KiEj = qaijEjKi, KiFj = q−aijFjKi,
EiFj − FjEi = δij
Ki −K−1
i
q − q−1
,
E2
i Ej − (q + q−1)EiEjEi + EjE
2
i = 0, |i− j| = 1,
F 2
i Fj − (q + q−1)FiFjFi + FjF
2
i = 0, |i− j| = 1,
EiEj − EjEi = FiFj − FjFi = 0, |i− j| �= 1,
with aii = 2, aij = −1 for |i− j| = 1, aij = 0 otherwise, and the comultiplication
∆, the antipode S, and the counit ε being defined on the generators by
∆(Ei) = Ei ⊗ 1 +Ki ⊗ Ei, ∆(Fi) = Fi ⊗K−1
i + 1⊗ Fi, ∆(Ki) = Ki ⊗Ki,
S(Ei) = −K−1
i Ei, S(Fi) = −FiKi, S(Ki) = K−1
i ,
ε(Ei) = ε(Fi) = 0, ε(Ki) = 1,
(see also [14, Ch. 4]).
Consider the Hopf algebra Uqsl2n. Equip Uqsl2n with a structure of Hopf
∗-algebra determined by the involution
K∗
j = Kj , E∗
j =
{
KjFj , j �= n,
−KjFj , j = n,
F ∗
j =
{
EjK
−1
j , j �= n,
−EjK
−1
j , j = n.
This Hopf ∗-algebra (Uqsl2n, ∗) is denoted by Uqsun,n.
Denote by Uqk = Uqs(un × un) the Hopf ∗-subalgebra generated by
Ei, Fi, i �= n; K±1
j , j = 1, 2, . . . , 2n − 1.
Now we introduce the notation to be used in the sequel and recall some known
results on the quantum matrix ball D.
Consider a ∗-algebra Pol(Matn)q with generators {zα
a }a,α=1,2,..., n and defining
relations
zα
a z
β
b =
qzβ
b z
α
a , a = b & α < β or a < b & α = β,
zβ
b z
α
a , a < b & α > β,
zβ
b z
α
a + (q − q−1)zβ
a zα
b , a < b & α < β,
(2.1)
222 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
(zβ
b )
∗zα
a = q2
n∑
a′,b′=1
m∑
α′,β′=1
R(b, a, b′, a′)R(β, α, β′, α′)zα′
a′ (zβ′
b′ )
∗ + (1− q2)δabδ
αβ
with δab, δαβ being the Kronecker symbols, and
R(b, a, b′, a′) =
q−1, a �= b & b = b′ & a = a′,
1, a = b = a′ = b′,
−(q−2 − 1), a = b & a′ = b′ & a′ > a,
0, otherwise.
Denote by C[Matn]q the subalgebra generated by zα
a , a, α = 1, 2, . . . , n. It is
a very well-known quantum analog of the algebra of holomorphic polynomials on
Matn.
Consider an arbitrary Hopf algebra A and an A-module algebra F . Suppose
that A is a Hopf ∗-algebra. The ∗-algebra F is said to be an A-module algebra if
the involutions are compatible as follows:
(af)∗ = (S(a))∗f∗, a ∈ A, f ∈ F,
where S is an antipode of A.
It was shown in [33] (see Props. 8.12 and 10.1) that one has
Proposition 2.1. C[Matn]q carries a structure of Uqsl2n-module algebra given
by
K±1
n zα
a =
q±2zα
a , a = α = n,
q±1zα
a , a = n & α �= n or a �= n & α = n,
zα
a , otherwise,
Fnz
α
a = q1/2 ·
{
1, a = α = n,
0, otherwise,
Enz
α
a = −q1/2 ·
q−1zn
a z
α
n , a �= n & α �= n,
(zn
n)2, a = α = n,
zn
nz
α
a , otherwise,
and for k �= n
K±1
k zα
a =
q±1zα
a , k < n & a = k or k > n & α = 2n − k,
q∓1zα
a , k < n & a = k + 1 or k > n & α = 2n− k + 1,
zα
a , otherwise with k �= n;
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 223
O. Bershtein and S. Sinel’shchikov
Fkz
α
a = q1/2 ·
zα
a+1, k < n & a = k,
zα+1
a , k > n & α = 2n− k,
0, otherwise with k �= n;
Ekz
α
a = q−1/2 ·
zα
a−1, k < n & a = k + 1,
zα−1
a , k > n & α = 2n− k + 1,
0, otherwise with k �= n.
Also Pol(Matn)q is equipped this way with a structure of Uqsun,n-module algebra.
It is well known that in the classical case of q = 1 the Shilov boundary of
the matrix ball D is just the set S(D) of all unitary matrices. Our intention is
to produce a q-analogue of the Shilov boundary for the quantum matrix ball.
Introduce notation for the quantum minors of the matrix z = (zα
a ):
(z∧k){α1,α2,...,αk}
{a1,a2,...,ak}
def=
∑
s∈Sk
(−q)l(s)zαs(1)
a1 z
αs(2)
a2 · · · zαs(k)
ak ,
with α1 < α2 < . . . < αk, a1 < a2 < . . . < ak, and l(s) being the number of
inversions in s ∈ Sk.
As known, the quantum determinant
detq z = (z∧n){1,2,...,n}
{1,2,...,n}
is in the center of C[Matn]q. The localization of C[Matn]q with respect to the
multiplicative system (detq z)N is called the algebra of regular functions on the
quantum GLn and is denoted by C[GLn]q.
Lemma 2.2 (Lemma 2.1 of [41]). There exists a unique involution ∗ in
C[GLn]q such that
(zα
a )
∗ = (−q)a+α−2n(detq z)−1 detq zα
a ,
where zα
a is a matrix derived from z by deleting the row α and the column a.
The ∗-algebra C[S(D)]q = (C[GLn]q, ∗) is a q-analogue of the algebra of re-
gular functions on the Shilov boundary of matrix ball D. It can be verified easily
that C[S(D)]q is a Uqsun,n-module algebra (see [41, Th. 2.2 and Prop. 2.7] for the
proof).
There exists another definition of the algebra C[S(D)]q.
Consider the two-sided ideal J of the ∗-algebra Pol(Matn)q generated by the
relations
n∑
j=1
q2n−α−βzα
j (z
β
j )
∗ − δαβ = 0, α, β = 1, 2, . . . , n. (2.2)
224 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
One can prove that this ideal is Uqsun,n-invariant, what allows to introduce the
Uqsun,n-module algebra Pol(Matn)q/J . It was proved in [41, p. 381, Prop. 6.1]
that C[S(D)]q = Pol(Matn)q/J . The last equality also works as a definition for
the algebra of regular functions on the Shilov boundary of the quantum matrix
ball.
A module V over Uqsl2n is said to be a weight module if
V =
⊕
λ∈P
Vλ, Vλ =
{
v ∈ V
∣∣∣ Kiv = qλiv, i = 1, 2, . . . , 2n− 1
}
,
where λ = (λ1, λ2, . . . , λ2n−1), and P ∼= Z
2n−1 is a weight lattice of the Lie algebra
sl2n. A nonzero summand Vλ in this decomposition is called the weight subspace
for the weight λ.
To every weight Uqsl2n-module V we associate the linear maps Hi,
i = 1, 2, . . . , 2n − 1, in V such that
Hiv = λiv, iff v ∈ Vλ.
Fix the element
H0 =
n−1∑
j=1
j(Hj +H2n−j) + nHn.
Any weight Uqsl2n-module V can be equipped with a Z-grading V = ⊕
r
Vr by
setting v ∈ Vr if H0v = 2rv.
In what follows some more sophisticated spaces will be used. It is known that
Pol(Matn)q =
∞⊕
k,j=0
C[Matn]q,k · C[Matn]q,−j.
Here C[Matn]q is the subalgebra of Pol(Matn)q generated by (zα
a )
∗,
a, α = 1, 2, . . . , n, and C[Matn]q,−j, C[Matn]q,k are homogeneous components
related to the grading
deg(zα
a ) = 1, deg(zα
a )
∗ = −1, a, α = 1, 2, . . . , n.
To rephrase this, every f ∈ Pol(Matn)q is uniquely decomposable as a finite sum
f =
∑
k,j≥0
fk,j, fk,j ∈ C[Matn]q,k · C[Matn]q,−j. (2.3)
Note that dimC[Matn]q,k · C[Matn]q,−j <∞.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 225
O. Bershtein and S. Sinel’shchikov
Consider the vector space D(D)′q of formal series of the form (2.3) with the
termwise topology. The Uqsl2n-action and the involution ∗ admit an extension
by continuity from the dense linear subspace Pol(Matn)q to D(D)′q
∗ :
∞∑
k,j=0
fk,j
→
∞∑
k,j=0
f∗k,j.
Moreover, D(D)′q is a Uqsl2n-module bimodule over Pol(Matn)q. We call the
elements of D(D)′q distributions on a quantum bounded symmetric domain.
3. Statement of Main Result
We intend to determine the Poisson kernel (1.1) by listing some essential pro-
perties of the associated integral operator. For this purpose we use the normalized
S(Un × Un)-invariant measure on S(D) for integration on the Shilov boundary.
A principal property of the Poisson kernel is that the integral operator with this
kernel is a morphism of the SUn,n-module of functions on S(D) into the SUn,n-
module of functions on D which takes 1 to 1.
So, in the quantum case the required Poisson integral operator is a morphism
of the Uqsun,n-module C[S(D)]q into the Uqsun,n-module D(D)′q which takes 1
to 1. Recall that every u ∈ D(D)′q is of the form
u =
∞∑
j,k=0
uj,k, uj,k ∈ C[Matn]q,jC[Matn]q,−k,
and the set
{
zβ
b (z
α
a )∗
}
a,b,α,β=1,2,...,n
is a basis of the vector space
C[Matn]q,1C[Matn]q,−1. This allows one to introduce the mixed partial derivatives
at zero, the linear functionals
∂2
∂zβ
b ∂(zα
a )∗
∣∣∣∣∣
z=0
, such that
u1,1 =
n∑
a,b,α,β=1
(
∂2u
∂zβ
b ∂(zα
a )∗
∣∣∣∣∣
z=0
)
zβ
b (zα
a )
∗, u ∈ D(D)′q.
Now we are in position to produce a quantum analog of the Hua equations.
Theorem 3.1. If u ∈ D(D)′q belongs to the image of the Poisson integral
operator on the quantum n× n-matrix ball, then
n∑
c=1
q2c ∂2(ξu)
∂zβ
c ∂(zα
c )∗
∣∣∣∣∣
z=0
= 0 (3.1)
for all ξ ∈ Uqsl2n, α, β = 1, 2, . . . , n.
The equation system (3.1) is a q-analog of (1.2).
226 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
4. Invariant Generalized Kernels and Associated Integral
Operators
List some plausible but less known definitions and results on the invariant
integral and integral kernels (see [40]). Consider a Hopf algebra A and
an A-module algebra F . A linear functional ν on F is called the A-invariant
integral if ν is a morphism of A-modules:
ν(af) = ε(a)f, a ∈ A, f ∈ F,
with ε being the counit of A.
There exists a unique Uqs(gln × gln)-invariant integral
ν : C[S(D)]q → C, ν : ϕ
→
∫
S(D)q
ϕdν,
which is normalized by
∫
S(D)q
1dν = 1 (see [41, Ch. 3]). As shown in [41], C[S(D)]q
is isomorphic to the algebra of regular functions on the quantum Un
as a Uqs(gln × gln)-module ∗-algebra. This isomorphism can be used to consider
the transfer of ν on the latter algebra, where it is known to be positive [45].
Hence ν is itself positive:
∫
S(D)q
ϕ∗ϕdν > 0 for all nonzero ϕ ∈ C[S(D)]q. Our
subsequent results demonstrate how this Uqs(gln × gln)-invariant integral can be
used to produce integral operators which are the morphisms of Uqsl2n-modules.
Consider A-module algebras F1, F2. Given a linear functional ν : F2 → C
and K ∈ F1 ⊗ F2, we associate a linear integral operator
K : F2 → F1, K : f
→ (id⊗ν)(K(1⊗ f)).
In this context K is called the kernel of this integral operator. Assume that the
integral ν on F2 is invariant and the bilinear form
f ′ × f ′′
→ ν(f ′ f ′′), f ′, f ′′ ∈ F2,
is nondegenerate. It is easy to understand that the integral operator with the
kernel K is a morphism of A-modules if and only if this kernel is invariant [40].
Another statement from [40] that will be used essentially in a subsequent con-
struction of A-invariant kernels is as follows: A-invariant kernels form a subalge-
bra of F op
1 ⊗ F2, with F
op
1 being the algebra derived from F1 by replacement of
its multiplication by the opposite one.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 227
O. Bershtein and S. Sinel’shchikov
Additionally to the algebra of kernels Pol(Matn)
op
q ⊗C[S(D)]q, we will use the
bimodule of generalized kernels D(D×S(D))′q whose elements are just the formal
series ∑
i,j
fij ⊗ ϕij , fij ∈ C[Matn]
op
q,−jC[Matn]
op
q,i, ϕij ∈ C[S(D)]q.
It is a bimodule over the algebra Pol(Matn)
op
q ⊗ C[S(D)]q.
5. A Passage from Affine Coordinates
to Homogeneous Coordinates
This section, as well as Sections 2 and 4, contains some preliminary material
and known results obtained in [33, 41, 43].
Turn to the quantum group SLN . We follow the general idea by V.G. Drin-
feld [6] in considering the Hopf algebra C[SLN ]q of matrix elements of finite-
dimensional weight UqslN -modules. It is accustomed to call it the algebra of
regular functions on the quantum group SLN . The linear maps in (UqslN )∗
adjoint to the operators of left multiplication by the elements of UqslN equip
C[SLN ]q with a structure of UqslN -module algebra by duality.
Recall that C[SLN ]q can be defined by the generators tij, i, j = 1, 2, . . . , N ,
(the matrix elements of the vector representation in a weight basis) and by the
relations
tij′tij′′ = qtij′′tij′ , j′ < j′′,
ti′jti′′j = qti′′jti′j, i′ < i′′,
tijti′j′ = ti′j′tij , i < i′ & j > j′,
tijti′j′ = ti′j′tij + (q − q−1)tij′ti′j, i < i′ & j < j′,
which are tantamount to (2.1), together with one more relation
detq t = 1,
where detq t is a q-determinant of the matrix t = (tij)i,j=1,2,...,N :
detq t =
∑
s∈SN
(−q)l(s)t1s(1)t2s(2) . . . tNs(N),
with l(s) = card{(i, j)|i < j & s(i) > s(j)}.
It is well known that detq t commutes with all tij. Thus C[SLN ]q appears to
be a quotient algebra of C[MatN ]q by the two-sided ideal generated by detqt− 1.
Note that C[SLN ]q is a domain.
228 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
In the classical case of q = 1 the matrix ball admits a natural embedding into
the Grassmannian Grn,2n
(a contraction A ∈ EndC
n)
→ (the linear span of (v,Av), v ∈ C
n),
where the latter pair is an element of C
n ⊕ C
n � C
2n. We are going to describe
a q-analog for this embedding.
Let
I = {i1, i2, . . . , ik} ⊂ {1, 2, . . . , 2n}, i1 < i2 < . . . < ik;
J = {j1, j2, . . . , jk} ⊂ {1, 2, . . . , 2n}, j1 < j2 < . . . < jk.
The elements
t∧k
IJ =
∑
s∈Sk
(−q)l(s)tis(1)j1tis(2)j2 . . . tis(k)jk
of C[SL2n]q are called quantum minors, and it is easy to check that
t∧k
IJ =
∑
s∈Sk
(−q)l(s)ti1js(1)
ti2js(2)
. . . tikjs(k)
.
Consider the smallest unital subalgebra C[X]q ⊂ C[SL2n]q that contains the
quantum minors
t∧n
{1,2,...,n}J , t∧n
{n+1,n+2,...,2n}J , J = {j1, j2, . . . , jn} ⊂ {1, 2, . . . , 2n}.
It is a Uqsl2n-module subalgebra which substitutes the classical coordinate ring
of the Grassmannian.
The following results are easy modifications of those of [33].
Proposition 5.1. There exists a unique antilinear involution ∗ in C[X]q such
that (C[X]q, ∗) is a Uqsun,n-module algebra, and(
t∧n
{1,2,...,n}{n+1,n+2,...,2n}
)∗
= (−q)n2
t∧n
{n+1,n+2,...,2n}{1,2,...,n}.
Lemma 5.2 (Lemma 11.3 of [33]). Let J ⊂ {1, 2, . . . , 2n}, card(J) = n,
Jc = {1, 2, . . . , 2n} \ J , l(J, Jc) = card{(j′, j′′) ∈ J × Jc| j′ > j′′}. Then(
t∧n
{1,2,...,n}J
)∗
= (−1)card({1,2,...,n}∩J)(−q)l(J,Jc)t∧n
{n+1,n+2,...,2n}Jc . (5.1)
Impose the abbreviated notation
t = t∧n
{1,2,...,n}{n+1,n+2,...,2n}, x = tt∗.
Note that t, t∗ and x quasicommute with all generators tij of C[SL2n]q. Then
the localization C[X]q,x of the algebra C[X]q with respect to the multiplicative
set xZ+ is well defined. The structure of Uqsun,n-module algebra is uniquely
extendable from C[X]q to C[X]q,x [33].
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 229
O. Bershtein and S. Sinel’shchikov
Proposition 5.3. [cf. Prop. 3.2 of [33]] There exists a unique
embedding of Uqsun,n-module ∗-algebras I : Pol(Matn)q ↪→ C[X]q,x such that
Izα
a = t−1t∧n
{1,2,...,n}Jaα
, with Jaα = {a} ∪ {n+ 1, n + 2, . . . , 2n}\{2n + 1− α}.
Corollary 5.4. Iy = x−1, with
y = 1 +
m∑
k=1
(−1)k
∑
{J ′| card(J ′)=k}
∑
{J ′′| card(J ′′)=k}
z∧k J ′
J ′′
(
z∧k J ′
J ′′
)∗
. (5.2)
A formal passage to a limit as q → 1 leads to the relation y = det(1− zz∗).
Proposition 5.5 (see Ch. 11 of [33]). Let 1 ≤ α1 < α2 < . . . < αk ≤ n,
1 ≤ a1 < a2 <. . .< ak ≤ n, J = {n+1, n+2, . . . , 2n}\{n+α1, n+α2, . . . , n+αk}∪
{a1, a2, . . . , ak}. Then
Iz∧k{n+1−αk,n+1−αk−1,...,n+1−α1}
{a1,a2,...,ak} = t−1t∧n
{1,2,...,n}J . (5.3)
It is accustomed to identify the generators zα
a , a, α = 1, 2, . . . , n, with their
images under I.
Consider the subalgebra of C[X]q,x generated by Pol(Matn)q
I
↪→ C[X]q,x to-
gether with t±1, t∗±1.
The elements of this subalgebra admit a unique decomposition of the form∑
(i,j)/∈(−N)×(−N)
tit∗jfij, fij ∈ Pol(Matn)q (5.4)
(the choice of the set of pairs (i, j) is due to the fact that t−1t∗−1 ∈ Pol(Matn)q).
Equip C
n⊕C
n with the sesquilinear form (·, ·)1−(·, ·)2. In the classical case of
q = 1, the Shilov boundary of the matrix ball is just the group Un. The graphs of
unitary operators form the isotropic Grassmannian (its points are the subspaces
on which the above scalar product on C
2n vanishes). We are going to describe
a q-analog of it when the isotropic Grassmannian is replaced by its homogeneous
coordinate ring.
Consider an extension C[Ξ]q of the algebra C[S(D)]q in the class of Uqsun,n-
module ∗-algebras. This extension is produced by adding a generator t and the
relations
tt∗ = t∗t; tzα
a = q−1zα
a t; t∗zα
a = q−1zα
a t
∗, a, α = 1, 2, . . . , n. (5.5)
The Uqsun,n-action is extended to C[Ξ]q as follows:
Ejt = Fjt = (K±1
j − 1)t = 0, j �= n,
Fnt = (K±1
n − 1)t = 0, Ent = q−1/2tzn
n .
230 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
Let ξ = tt∗. One can introduce a localization C[Ξ]q,ξ of the algebra C[Ξ]q
with respect to the multiplicative set ξZ+ . The involution ∗ and the structure of
Uqsun,n-module algebra are uniquely extendable from C[Ξ]q to C[Ξ]q,ξ.
The algebra C[Ξ]q is equipped with a Uqsl2n-invariant bigrading
deg zα
a = deg(zα
a )
∗ = (0, 0), deg t = (1, 0), deg t∗ = (0, 1),
which extends to the localization C[Ξ]q,ξ. By [41], the homogeneous component
C[Ξ](−n,−n)
q,ξ carries a nonzero Uqsl2n-invariant integral η such that
η(t∗−nft−n) =
∫
S(D)q
fdν.
Since t, t∗ normalize every Uqs(un ×un)-isotypic component of C[S(D)]q, and the
Uqs(un × un)-action in C[S(D)]q is multiplicity-free [3, Prop. 4], it follows from
the definition of integral over the Shilov boundary that
η(t∗−nt−nf) = η(ft∗−nt−n) =
∫
S(D)q
fdν. (5.6)
6. The Poisson Kernel
In this section we construct a morphism of the Uqsun,n-module C[S(D)]q into
the Uqsun,n-module D(D)′q which takes 1 to 1. The introduced morphism will be
a Poisson integral operator, as mentioned in Section 3.
We will need a quantum analog C[Matn,2n]q for polynomial algebra on the
space of rectangular n× 2n-matrices. The algebra C[Matn,2n]q is determined by
the set of generators {tij}i=1,2,...,n; j=1,2,...,2n and the similar relations as in the
case of square matrices (see (2.1)). It is known that this algebra is a domain.
The structure of Uqsl2n-module algebra is given by
Kktij =
qtij , j = k,
q−1tij , j = k + 1,
tij , j /∈ {k, k + 1},
Ektij =
{
q−1/2ti,j−1, j = k + 1,
0, j �= k + 1,
Fktij =
{
q1/2ti,j+1, j = k,
0, j �= k,
with k = 1, 2, . . . , 2n − 1.
R e m a r k 6.1. Consider the Uqsl2n-module subalgebras in C[Matn,2n]q
and in C[Ξ]q generated by t∧n
{1,2,...,n}{n+1,n+2,...,2n} and t, respectively. The map
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 231
O. Bershtein and S. Sinel’shchikov
t∧n
{1,2,...,n}{n+1,n+2,...,2n}
→ t may be extended up to an isomorphism i of these
Uqsl2n-module subalgebras. In a similar way one can introduce Uqsl2n-module
subalgebras in C[Matn,2n]q and in C[Ξ]q, generated by t∧n
{1,2,...,n}{1,2,...,n} and t∗,
respectively. The map t∧n
{1,2,...,n}{1,2,...,n}
→ (−q)n2
t∗ extends up to the isomor-
phism i′ of these Uqsl2n-module subalgebras. The above isomorphisms can be
used together to embed the minors t∧n
{1,2,...,n}J and t∧n
{n+1,n+2,...,2n}J into C[Ξ]q.
Consider the map m : C[Matn,2n]q ⊗ C[Matn,2n]q → C[Mat2n]q defined as
follows. The tensor multipliers are embedded into C[Mat2n]q as subalgebras gen-
erated by the entries of, respectively, the upper n and the lower n rows of the
matrix t = (tij), i = 1, 2, . . . , 2n, j = 1, 2, . . . , 2n, and the map m is just a multi-
plication in the algebra C[Mat2n]q.
Lemma 6.2. The map m is an isomorphism of Uqsl2n-modules.
P r o o f. The map m takes the monomial basis
tj1111 . . . t
j1,2n
12n t
j21
21 . . . t
j2,2n
22n . . . tjn1
n1 . . . t
jn,2n
n2n ⊗
t
jn+1,1
n+11 . . . t
jn+1,2n
n+12n t
jn+2,1
n+21 . . . t
jn+2,2n
n+22n . . . t
j2n,1
2n1 . . . t
j2n,2n
2n2n
of the algebra C[Matn,2n]q ⊗ C[Matn,2n]q to the monomial basis
tj1111 . . . t
j1,2n
12n t
j21
21 . . . t
j2,2n
22n . . . tjn1
n1 . . . t
jn,2n
n2n t
jn+1,1
n+11 . . . t
jn+1,2n
n+12n
· tjn+2,1
n+21 . . . t
jn+2,2n
n+22n . . . t
j2n,1
2n1 . . . t
j2n,2n
2n2n
of the algebra C[Mat2n]q, jik ∈ Z+. Hence it is a bijective map and a morphism
of Uqsl2n-modules since C[Mat2n]q is a Uqsl2n-module algebra.
It is worthwhile to note that the definition of the Uqsl2n-module algebra C[X]q
allows a replacement of C[SL2n]q by C[Matn,2n]q.
Consider the elements of the Uqsl2n-module C[Matn,2n]q ⊗ C[Matn,2n]q given
by
L =
∑
J⊂{1,2,...,2n}& card(J)=n
(−q)l(J,Jc)t∧n
{1,2,...,n}J ⊗ t∧n
{n+1,n+2,...,2n}Jc ,
L =
∑
J⊂{1,2,...,2n}&card(J)=n
(−q)−l(J,Jc)t∧n
{n+1,n+2,...,2n}Jc ⊗ t∧n
{1,2,...,n}J .
Here Jc is the complement to J and l(I, J) = card{(i, j) ∈ I × J | i > j}.
232 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
Proposition 6.3. L and L are Uqsl2n-invariants.
P r o o f is expounded for L. In the case of L, similar arguments are
applicable.
Recall a q-analog for the Laplace formula of splitting the quantum determi-
nant of the 2n× 2n-matrix t = (tij) with respect to the upper n lines:
detq t =
∑
J⊂{1,2,...,2n}& card(J)=n
(−q)l(J,Jc)t∧n
{1,2,...,n}Jt
∧n
{n+1,n+2,...,2n}Jc
=
∑
J⊂{1,2,...,2n}&card(J)=n
(−q)−l(J,Jc)t∧n
{n+1,n+2,...,2n}Jct∧n
{1,2,...,n}J .
Our claim follows from Lemma 6.2, the relationmL = detq t and Uqsl2n-invariance
of the quantum determinant.
Note that, in view of Remark 6.1, one has L,L ∈ C[X]q ⊗ C[Ξ]q.
Introduce, firstly, a Uqsl2n-module of kernels D(D × Ξ)′q whose elements are
formal series with coefficients from C[Matn]q,−jC[Matn]q,i⊗C[Ξ]q,ξ, and, secondly,
a Uqsl2n-module of kernels D(X ⊗Ξ)′q whose elements are finite sums of the form∑
(i,j)/∈(−N)×(−N)
(tit∗j ⊗ 1)fij , fij ∈ D(D × Ξ)′q
(cf. (5.4)). Of course, D(D × Ξ)′q is a Pol(Matn)
op
q ⊗ C[Ξ]q,ξ-bimodule, and
D(X × Ξ)′q is a C[X]opq,x ⊗ C[Ξ]q,ξ-bimodule.
The kernel L ∈ C[X]opq ⊗ C[Ξ]q can be written in the form
L =
1 + ∑
J �={n+1,n+2,...,2n}
(−q)l(J,Jc)t∧n
{1,2,...,n}J t
−1 ⊗ t∧n
{n+1,n+2,...,2n}Jct∗−1
(t⊗t∗).
Note that t∧n
{1,2,...,n}J t
−1, t∧n
{n+1,n+2,...,2n}Jct∗−1 ∈ C[Matn]q (see (5.3)). This allows
one to write down explicitly such an element L−n of the space of generalized
kernels D(X ×Ξ)′q that Ln ·L−n = L−n ·Ln = 1, where · stands for the (left and
right) actions of Ln on the element L−n of the bimodule D(X × Ξ)′q.
A similar construction produces also a Uqsl2n-invariant generalized kernel
L
−n
.
Note that L−n =
∑
i
xi is a formal series with xi ∈ C[Matn]q,i⊗C[Ξ]q,ξ, i ∈ Z+,
and the terms of the formal series L
−n
=
∑
j
yj are such that yj∈C[Matn]q,−j⊗C[Ξ]q,ξ.
This allows one to define the ‘product’ L
−n
L−n as a double series
∑
i,j
yjxi which
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 233
O. Bershtein and S. Sinel’shchikov
is thus an element of the module of generalized kernels D(X ⊗ Ξ)′q. Clearly, one
has L
n ·
(
L
−n
L−n
)
·Ln = 1 in D(X×Ξ)′q, and this property determines uniquely
the generalized kernel L
−n
L−n. Furthermore, the above uniqueness allows one
to verify invariance of the generalized kernel L
−n
L−n. Of course, the argument
should apply the invariance of L, L.
Consider the Poisson kernel P ∈ D(D×S(D))′q for the matrix ball having the
following properties:
i) up to a constant multiplier, the Poisson kernel is just (1 ⊗ tt∗)nL
−n
L−n,
that is
P = const(q, n)(1⊗ tt∗)nL
−n
L−n;
ii) the integral operator with kernel P takes 1 ∈ C[S(D)]q to 1 ∈ D(D)′q.
Lemma 6.4. The integral operator C[S(D)]q → D(D)′q with kernel
(1⊗ tt∗)nL
−n
L−n is a morphism of Uqsl2n-modules.
P r o o f. It follows from the existence of an invariant integral
η : C[Ξ](−n,−n)
q → C and the invariance of L
−n
L−n that the integral operator
C[S(D)]q → D(D)′q with kernel L
−n
L−n is a morphism of Uqsl2n-modules. Now
it remains to move the multiplier (1⊗ t∗−nt−n) from L
−n
L−n to the left and to
apply (5.6).
R e m a r k 6.5. Since the integral operator C[S(D)]q → D(D)′q with kernel
(1 ⊗ tt∗)nL
−n
L−n is a morphism of Uqsl2n-modules, the image of 1 ∈ C[S(D)]q
is a Uqsl2n-invariant element of D(D)′q, that is just a constant. This proves the
existence of the Poisson kernel P .
E x a m p l e 6.6. Let us illustrate the Poisson kernel P determined above in
the simplest case when n = 1. The considered invariant kernels were obtained in
[26]. The elements
L = t11 ⊗ t22 − qt12 ⊗ t21, L = −q−1t21 ⊗ t12 + t22 ⊗ t11
of the algebra C[X]opq,x ⊗ C[Ξ]q,ξ are Uqsl2-invariant kernels. We present below
an easy computation, which uses, for the sake of brevity, the notation
z = qt−1
12 t11, z∗ = t22t−1
21 ,
instead of mentioning explicitly the embeddings I, I. One has
L = (1− z ⊗ z∗)(−qt12 ⊗ t21), L = (−q−1t21 ⊗ t12)(1 − q2z∗ ⊗ z),
234 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
and, as t = t12, t∗ = −qt21,
L = (1− z ⊗ z∗)(t⊗ t∗), L = (q−2t∗ ⊗ t)(1 − q2z∗ ⊗ z).
Hence,
L
−1
L−1 = q2(1⊗ t−1t∗−1)(1 − z∗ ⊗ z)−1((1− z∗z)⊗ 1)(1 − z ⊗ z∗)−1.
Omit ⊗ and in the second tensor multiplier z replace by ζ and z∗ by ζ∗ (which
is standard in function theory) to obtain
P = const(q)(1 − z∗ζ)−1(1− z∗z)(1 − zζ∗)−1.
What remains now is to find const(q), or, to be more precise, to prove that
const(q)=1. In fact, the integral operator with kernel (1−z∗ζ)−1(1−z∗z)(1−zζ∗)−1
takes 1 to 1. This is because ζ∗ = ζ−1, and integration of the product of the series
in ζ produces the constant term:
∞∑
k=0
zk(1− zz∗)z∗k = 1.
Note that the Poisson kernel is a formal series P =
∞∑
j,k=0
pjk, with
pjk ∈ C[Matn]q,−kC[Matn]q,j ⊗C[S(D)]q. In the sequel we will omit ⊗ and in the
second tensor multiplier replace z by ζ and z∗ by ζ∗ (that is standard in function
theory).
Lemma 6.7. The following relation is valid:
p11 = const(q, n)
n∑
a,b,α,β=1
(
1− q−2n
1− q−2
q2(2n−a−α)ζα
a (ζ
β
b )
∗ − δabδ
αβ
)
(zα
a )
∗zβ
b , (6.1)
with const(q, n) �= 0.
P r o o f. In the algebra of kernels one has
L =
1−
n∑
a,α=1
zα
a (ζ
α
a )
∗ + . . .
tτ∗, (6.2)
L = q−2n2
t∗τ
1− q2
n∑
a,α=1
q2(2n−a−α)(zα
a )
∗ζα
a + . . .
, (6.3)
which is easily deducible from (5.3), (5.1), and the fact that I is a homomorphism
of ∗-algebras. Also, it is seen from (5.2) that
y = (tt∗)−1 = 1−
n∑
a,α=1
(zα
a )
∗zα
a + . . . . (6.4)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 235
O. Bershtein and S. Sinel’shchikov
Here three dots replace the terms whose degree is above two, and the following
abbreviated notation is implicit:
t = t⊗ 1, τ = 1⊗ τ, zα
a = zα
a ⊗ 1, ζα
a = 1⊗ ζα
a ,
t∗ = t∗ ⊗ 1, τ∗ = 1⊗ τ∗, (zα
a )
∗ = (zα
a )
∗ ⊗ 1, (ζα
a )
∗ = 1⊗ (ζα
a )
∗.
Apply (6.2)–(6.4), together with the commutation relations (see also (5.5))
t∗τ
n∑
a,α=1
q2(2n−a−α)(zα
a )
∗ζα
a
= q−2
n∑
a,α=1
q2(2n−a−α)(zα
a )
∗ζα
a
t∗τ ;
tτ∗
n∑
a,α=1
zα
a (ζ
α
a )
∗
= q2
n∑
a,α=1
zα
a (ζ
α
a )
∗
tτ∗,
to obtain
L
−n
L−n =
n∏
j=1
1− q2j
n∑
a,α=1
q2(2n−a−α)(zα
a )
∗ζα
a + . . .
−1
× q2n3
(τ∗τ)−n(tt∗)−n
n−1∏
j=0
1− q2j
n∑
a,α=1
zα
a (ζ
α
a )
∗ + . . .
−1
.
Hence
P = const(q, n)
n−1∏
j=0
1− q−2j
n∑
a,α=1
q2(2n−a−α)(zα
a )
∗ζα
a + . . .
−1
×
1−
n∑
a,α=1
(zα
a )
∗zα
a + . . .
n
·
n−1∏
j=0
1− q2j
n∑
a,α=1
zα
a (ζ
α
a )
∗ + . . .
−1
, (6.5)
where three dots inside every parentheses denote the contribution of the terms
whose degree is above two, and the multipliers in the products are written in
order of decreasing of index j from left to right.
Now (6.1) follows from (6.5).
R e m a r k 6.8. A formal passage to a limit as q → 1 in (6.1) leads to
p11 = const(n)
n∑
a,b,α,β=1
(
n ζα
a ζ
β
b − δabδ
αβ
)
zα
a z
β
b .
This relation is well known (with const(n) = n); see, for example, [15, p. 597]
and is a consequence of (1.1).
236 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
R e m a r k 6.9. Using the definition of P , (6.5) and the definition of integral
over the Shilov boundary, one can compute const(q, n) explicitly. But we do not
need this value on the way to Hua equations.
7. Deducing the Hua Equations
Now we are about to produce a quantum analog of (1.3).
It follows from Lemma 6.7 and the definition of multiplication in the algebra
of kernels that
∂2P
∂zβ
b ∂(z
α
a )∗
∣∣∣∣∣
z=0
= const(q, n) ·
(
1− q−2n
1− q−2
q2(2n−a−α)ζα
a (ζ
β
b )
∗ − δabδ
αβ
)
.
Set here a = b = c to get
∂2P
∂zβ
c ∂(zα
c )∗
∣∣∣∣∣
z=0
= const(q, n) ·
(
1− q−2n
1− q−2
q2(2n−c−α) ζα
c (ζ
β
c )
∗ − δαβ
)
.
On the other hand, the generators of the function algebra on the Shilov boundary
are a subject to the relation
n∑
c=1
ζα
c (ζ
β
c )
∗ = q−2n+α+βδαβ , α, β = 1, 2, . . . , n,
see (2.2). Hence
1
const(q, n)
n∑
c=1
q2c ∂2P
∂zβ
c ∂(zα
c )∗
∣∣∣∣∣
z=0
=
1− q−2n
1− q−2
q2(2n−α)
n∑
c=1
ζα
c (ζ
β
c )
∗ − q2 1− q
2n
1− q2 δ
αβ
=
1− q−2n
1− q−2
q2(2n−α)q−2n+α+βδαβ − q2 1− q
2n
1− q2 δ
αβ = 0,
so that the following statement is valid.
Lemma 7.1. If u ∈ D(D)′q is a Poisson integral on the quantum n×n-matrix
ball, then
n∑
c=1
q2c ∂2u
∂zβ
c ∂(zα
c )∗
∣∣∣∣∣
z=0
= 0
for all α, β = 1, 2, . . . , n.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 237
O. Bershtein and S. Sinel’shchikov
Since the subspace of Poisson integrals
u =
∫
S(D)q
P (z, ζ)f(ζ)dν(ζ), f ∈ C[S(D)]q,
is a Uqsl2n-submodule, the above lemma implies Theorem 3.1.
It is known [9] that in the classical case of q = 1 the Poisson kernel P is
a solution of one more equation system
n∑
γ=1
∂2u(g · z)
∂zγ
b ∂z
γ
a
∣∣∣∣∣∣
z=0
= 0, g ∈ SUn,n, a, b ∈ {1, 2, . . . , n}.
An argument similar to the above allows one to obtain a q-analog of this result.
Proposition 7.2. If u ∈ D(D)′q is a Poisson integral on the quantum
n× n-matrix ball, then
n∑
γ=1
q2γ ∂2(ξu)
∂zγ
a∂(z
γ
b )
∗
∣∣∣∣∣∣
z=0
= 0
for all ξ ∈ Uqsl2n, a, b = 1, 2, . . . , n.
8. Addendum. Hint to a More General Case
Turn from the special case of n × n-matrix ball to a more general case of
bounded symmetric domain of tube type. We intend to introduce the Hua opera-
tor, which can be used in order to rewrite the Hua equations in a more habitual
form (see [15, p. 593]).
Let g be a simple complex Lie algebra, and (aij)i,j=1,2,...,l be an associated
Cartan matrix. We refer to a well known (see [14]) description of universal
enveloping algebra Ug in terms of its generators ei, fi, hi, i = 1, 2, . . . , l, and
standard relations. Consider also the linear span h of the set {hi|i = 1, 2, . . . , l}
(a Cartan subalgebra), and the simple roots {αi ∈ h∗|i = 1, 2, . . . , l} given by
αi(hj) = aji. Let δ be the maximal root, δ =
l∑
i=1
ciαi. Assume that it is possible
to choose l0 ∈ {1, 2, . . . , l} so that cl0 = 1. Fix an element h0 ∈ h with the
following properties:
αi(h0) = 0, i �= l0; αl0(h0) = 2.
In this case the Lie algebra g is equipped with the Z-grading as follows:
g = g−1 ⊕ g0 ⊕ g+1, gj = {ξ ∈ g| [h0, ξ] = 2jξ} (8.1)
(that is, gi = {0} for all i with |i| > 1).
238 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
Denote by k ⊂ g the Lie subalgebra generated by
ei, fi, i �= l0; hi, i = 1, 2, . . . , l.
If (8.1) is true, then g0 = k, and the pair (g, k) is called the Hermitian symmet-
ric pair. In what follows, we obey the conventions of the theory of Hermitian
symmetric spaces, where the notation p± is used instead of g±1 as in (8.1).
Harish-Chandra introduced a standard realization of an irreducible bounded
symmetric domain D, considered up to biholomorphic isomorphisms, as a unit
ball in the normed space p− [7, 44]. Let G be a simply connected complex linear
algebraic group with Lie(G) = g, and K ⊂ G such connected linear algebraic sub-
group that Lie(K) = k. In this context one has the well-known Harish-Chandra
embedding
i : K\G ↪→ p−.
Let W be the Weyl group of the root system R of g, and w0 ∈ W be the
longest element. The irreducible bounded symmetric domain D associated to the
pair (g, k) is a tube type domain if and only if =l0 = −w0=l0 .
Let Uqg be the quantum universal enveloping algebra of g. Recall that it is
a Hopf algebra and it can be described in terms of its generators Ei, Fi, K±1
i ,
i = 1, 2, . . . , l, and the standard Drinfeld–Jimbo relations.
Introduce quantum analogs of invariant differential operators to use them for
producing the Hua operator.
Let V be a finite-dimensional weight Uqg-module. In an obvious way,
V ⊗Uqk Uqg is equipped with a structure of right Uqg-module. It is easy to see
that in the category of left Uqg-modules
HomUqk(Uqg, V ) ∼= (V ∗ ⊗Uqk Uqg)∗,
f
→ f̃ , f̃(l ⊗ ξ) = l(f(ξ)), l ∈ V ∗, ξ ∈ Uqg.
The vector space HomUqk(Uqg, V ) is a quantum analog of the space of sections
of homogeneous vector bundle on the homogeneous space K\G. Suppose we are
given two finite-dimensional weight Uqg-modules V1, V2 and a morphism of right
Uqg-modules
A : V ∗
2 ⊗Uqk Uqg → V ∗
1 ⊗Uqk Uqg.
To the latter morphism associate the adjoint linear map
A∗ : HomUqk(Uqg, V1) → HomUqk(Uqg, V2),
which is also a morphism of Uqg-modules. These dual operators are treated as
quantum analogs of invariant differential operators.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 239
O. Bershtein and S. Sinel’shchikov
Thus the invariant differential operators are in one-to-one correspondence
with the elements of the space
HomUqg(V ∗
2 ⊗Uqk Uqg, V
∗
1 ⊗Uqk Uqg) ∼= HomUqk(V ∗
2 , V
∗
1 ⊗Uqk Uqg),
f
→ f̃ , f̃(l) = f(l ⊗ 1), l ∈ V ∗
2 .
Turn to a construction of the Hua operator. Set p+=UqkEl0 , p−=Uqk(Kl0Fl0),
both are finite-dimensional weight Uqk-modules [13]. The morphisms of right Uqk-
modules
p+ → C ⊗Uqk Uqg, El0
→ 1⊗ El0 ,
p− → C ⊗Uqk Uqg, Kl0Fl0
→ 1⊗Kl0Fl0
determine the invariant linear differential operators
HomUqk(Uqg,C) → HomUqk(Uqg, p
±).
Recall that Uqk-modules form a tensor category, and that the comultiplication
� : Uqg → Uqg ⊗ Uqg is a morphism of this category.
Consider the morphisms of Uqg-modules
HomUqk(Uqg,C) → HomUqk(Uqg ⊗ Uqg, p
+ ⊗ p−), (8.2)
HomUqk(Uqg ⊗ Uqg, p
+ ⊗ p−) → HomUqk(Uqg, p
+ ⊗ p−). (8.3)
Let kq be the finite-dimensional weight Uqk-module with the same weights and
weight multiplicities as the Uk-module k. There exists a unique Uqk-submodule
Hq ⊂ p+ ⊗ p− such that (p+ ⊗ p−)/Hq ≈ kq (because a similar fact is well known
in the classical case of q = 1 (see [2, Prop. 4.2])). Fix a surjective morphism
p+ ⊗ p− → kq and consider the associated invariant ‘formal’ differential operator
HomUqk(Uqg, p
+ ⊗ p−) → HomUqk(Uqg, kq). (8.4)
Denote by Dq the composition of the maps (8.2), (8.3), and (8.4). By definition,
Dq is an invariant differential operator.
Recall the standard definitions of quantum analogs for the algebras of re-
gular functions on the group G and on the homogeneous space K\G. Denote
by C[G]q ⊂ (Uqg)∗ the Hopf algebra of all matrix elements of weight finite di-
mensional Uqg-modules. C[G]q is equipped with the structure of Uop
q g ⊗ Uqg-
module algebra via quantum analogs of the standard right and left regular actions
(ξ′ ⊗ ξ′′)f = Lreg(ξ′)Rreg(ξ′′)f , where
Lreg(ξ′f)(η) = f(ξ′η), Rreg(ξ′′f)(η) = f(ηξ′′), ξ′, ξ′′, η ∈ Uqg, f ∈ C[G]q.
240 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
A q-Analog of the Hua Equations
(Uop
q g is the Hopf algebra with the opposite multiplication.) C[G]q is called the
algebra of regular functions on the quantum group G.
Introduce the notation
C[K\G]q = {ξ ∈ C[G]q| Lreg(η)ξ = 0, η ∈ Uqk}.
This Hopf subalgebra is a quantum analog for the algebra of regular functions on
the homogeneous spaceK\G. It is easy to prove that C[K\G]q ⊂ HomUqk(Uqg,C),
so one can consider the restriction of Dq to C[K\G]q.
Now introduce a localization C[K\G]q,x of the algebra C[K\G]q with respect
to the Ore set xZ+ . It can be proved that the extension of Dq up to C[K\G]q,x
is well-defined. Pass from C[K\G]q,x to Pol(p−)q (via the Harish-Chandra em-
bedding I, see Section 5 for the special case) and D(D)′q to get a q-analog for the
Hua operator.
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