Quantum line bundles via Cayley-Hamilton identity

D. Gurevich,

ISTV, Université de Valenciennes, 59304 Valenciennes, France

P. Saponov

Theory Division, Institute for High Energy Physics, 142284 Protvino, Russia

November 12, 1999

Abstract

As was shown in [GPS] the matrix L = ||l_{i}^{j}|| whose entries l_{i}^{j} are generators of the so-called reflection equation algebra is subject to some polynomial identity looking like the Cayley-Hamilton identity for a numeric matrix. Here a similar statement is presented for a matrix whose entries are generators of a filtered algebra being a ”noncommutative analogue” of the reflection equation algebra. In an appropriate limit we get a similar statement for the matrix formed by the generators of the algebra U(gl(n)). This property is used to introduce the notion of line bundles over quantum orbits in the spirit of the Serre-Swan approach. The quantum orbits in question are presented explicitly as some quotients of one of the mentioned above algebras both in the quasiclassical case (i.e. that related to the quantum group U_{q}(sl(n))) and a non-quasiclassical one (i.e. that arising from a Hecke symmetry with non-standard Poincaré series of the corresponding symmetric and skewsymmetric algebras).

AMS classification: 17B37; 81R50 Key words: Hecke symmetry, quantum group, quantum orbit, Cayley-Hamilton iden-

tity, line bundle

1 Introduction

Let G = GL(n), g = Lie (G) over the field k = R or k = C^{1}. As usual, let us identify g and g^{∗} and consider a matrix A ∈ g^{∗} with pairwise distinct eigenvalues μ_{1}, μ_{2}, ...μ_{n}. Denote M_{μ}, μ = (μ_{1}, μ_{2}, ...μ_{n}) a G-orbit of the matrix A w.r.t. coadjoint action of g. Then the orbit M_{μ} being an affine algebraic variety is defined by the following system of equations

Tr A^{k} = c_{k} = ∑ μ_{i}^{k}, k = 1, 2, 3, ..., n.  (1.1)

1The choice of the field is similar to that in the classical case. If k = R the entries of the quantum R-matrix R are assumed to be real.

Let k(M_{μ}) be its coordinate ring.

A typical example of a line bundle over M_{μ} is an eigenspace corresponding to an eigenvalue μ_{i}, i.e. the space of the vectors v ∈ V^{∗} such that

v A = μ_{i} v, v ∈ V^{∗} ,  (1.2)

V^{∗} being the right g^{∗}-module.

Then this line bundle itself is an algebraic variety: it is defined in the space g^{∗} × V^{∗} by system (1.1)—(1.2). This variety (i.e. the total space of the line bundle in question) will be denoted E_{μ_{i}}. The coordinate ring of this variety k(E_{μ_{i}}) has the structure of a k(M_{μ})-module.

This example is a particular case of a one-to-one correspondence between algebraic vector bundle over an affine algebraic variety and finitely generated projective modules over its coordinate ring realized in [Se] (a similar correspondence on compact smooth varieties was established in [Sw]).

Let us remark that the projectivity of the k(M_{μ})-module k(E_{μ_{i}}) can be shown by means of the projector

P_{i} = ∏_{j≠i}^{n} \frac{A − μ_{j} id}{μ_{i} − μ_{j}}  (1.3)

since this k(M_{μ})-module can be identified with Im P_{i}.

Our main purpose is to generalize the construction of considered (and some ”derived”) line bundles to the quantum case.

Up to our knowledge the first attempt to realize a line bundle over a quantum sphere in terms of projective modules featured in [HM]. Constructed there was a quantum analogue of projector (1.3) over a quantum sphere (or what is the same, a quantum hyperboloid if we ignore the involution operation).

In this paper we suggest a regular way of constructing projective modules for a generic μ, which are the quantum deformations of k(M_{μ})-modules k(E_{μ_{i}}). More precisely, we construct quantum two parameter deformation of the algebra k(M_{μ}) and line bundles k(E_{μ_{i}}) (see below).

The basic question arising from the very beginning is what are quantum analogues of the orbits in g^{∗}. An habitual way to introduce such quantum objects makes use of so-called Hopf-Galois extension (cf. [Sh], [HM] ). This approach allows one to generalize the notion of an ordinary orbit in terms of a couple of Hopf algebras. The famous ”RTT” algebra and some its Hopf subalgebra are usually employed as such couples. However, such an approach does not enables one either to control the flatness of deformation^{2} in the quasiclassical case or to generalize construction of quantum orbits to a non-quasiclassical case.

Let us note that we use the term ”quasiclassical” for objects arising from deformations of classical ones. In this paper all quasiclassical objects in question are endowed with a structure of U_{q}(sl(n))-module (and as usual the products in all quantum algebras involved are supposed to be U_{q}(sl(n))-covariant). By ”non-quasiclassical” objects we mean those arising from the solutions R of the quantum Yang-Baxter (YB) equation (2.1) whose symmetric and skewsymmetric algebras have non-quasiclassical Poincaré series. These algebras are well defined if we assume R to be a Hecke symmetry (i.e. a solution of the YB equation subject to the Hecke condition (2.2)). For details the reader is referred to [G] where a large family of non-quasiclassical Hecke symmetries was constructed.

2We refer the reader to [DGK] for the rigorous definition of this notion. Roughly speaking, this means that the supply of elements does not change under deformation of the initial object. Let us remark that under a flat deformation A_{h} of a commutative associative algebra A = A_{0} the linear in h skewsymmetrized term of the deformed product is a Poisson bracket.

We suggest another way of introducing quantum orbits. The central role in our approach is played by so-called reflection equation (RE) algebra L_{q}(R) (see Section 2 for definition) which can be associated with any (quasiclassical or not) Hecke symmetry R. If R is a quasiclassical Hecke symmetry then the algebra L_{q}(R) (similarly to the RTT algebra) is a flat deformation of the coordinate ring k(Mat(n, k)). However, its properties differ drastically from those of the RTT algebra.

The main difference is that the RE algebra possesses a big center Z. In particular, the quantum trace Tr_{q} well defined in this algebra belongs to Z. On quotienting the RE algebra over the ideal generated by Tr_{q} we get an algebra with n^{2} − 1 generators which we consider as a q-analogue of the algebra k(sl(n)^{∗}) = Sym(sl(n)). If instead of the mentioned ideal we take that generated by the elements

z − χ(z), z ∈ Z  (1.4)

where χ is a generic character of Z, we get the quotient algebra (denoted k(M_{μ}^{q})) which can be considered as a quantum analogue of semisimple orbits above (we call semisimple orbits those of semisimple elements). Thus, both in the quasiclassical and non-quasiclassical cases such type quantum orbits are defined by means of some ”quantum (or braided) algebraic equations” in the spirit of affine algebraic geometry. By abusing the language we call quantum orbits the corresponding ”coordinate rings” in both cases.

Another important difference between the RTT and RE algebras consists in the fact that the latter one (being a quadratic algebra) admits a further flat deformation giving rise to a quadratic-linear algebra which looks like the enveloping algebra U(gl(n)). The final object of such a deformation is an algebra L_{q,h̄}(R) (see Section 2) depending on two parameters which tends to the RE algebra as h̄ → 0 and to U(gl(n)_{h̄}) as q → 1, where the defining relations of U(gl(n)_{h̄}) are as follows:

a_{j_{1}}^{i_{1}}a_{j_{2}}^{i_{2}} − a_{j_{2}}^{i_{2}}a_{j_{1}}^{i_{1}} = h̄(δ_{j_{1}}^{i_{2}}a_{j_{2}}^{i_{1}} − δ_{j_{2}}^{i_{1}}a_{j_{1}}^{i_{2}}) .  (1.5)

Hereafter we use the notation U(g_{h̄}) for the enveloping algebra of a Lie algebra g_{h̄} which differs from g by the factor h̄ introduced into the Lie bracket. We prefer to use the Lie algebra g_{h̄} instead of g in order to represent its enveloping algebra U(g_{h̄}) as a deformation of the algebra k(g^{∗}) = Sym(g). In a similar way we treat quotients of the algebra U(g_{h̄}) as deformations of the corresponding orbits^{3}.

Similarly to L_{q}(R) the algebra L_{q,h̄}(R) has a big center. On quotienting the algebra L_{q,h̄}(R) over an ideal looking like (1.4) we get ”quantum non-commutative” analogue of the orbits above. We treat the specialization of this quotient at the point q = 1 as a ”classical non-commutative” orbit. Thus, this specialization is just an appropriate quotient of the algebra U(gl(n)_{h̄}) (or U(sl(n)_{h̄})).

3As for the RTT algebra, it does not have any non-trivial quadratic-linear deformation which could be considered as a q-analogue of U(gl(n)) (cf. [GR]).

Note that we consider the RE algebra L_{q}(R) and its quotients as ” quantum commutative” algebras. Their non-commutative counterparts are the algebra L_{q,h̄}(R) and its quotients (all these algebras are well defined in the non-quasiclassical case as well). The term ”classical” means that the product in the algebra in question is G-covariant where G is a usual group. By contrast, ”quantum” means that the product in the algebra in question is covariant w.r.t. a Hopf algebra. In the quasiclassical case this Hopf algebra is just U_{q}(sl(n)). In a non-quasiclassical case an explicit description of a similar Hopf algebra is more complicated (cf. [AG] where an attempt to describe such an algebra featured). This is the reason why in a generic case it is more convenient to use the RTT algebra in order to define ”symmetries” of the objects in terms of its coaction (see Section 2).

Now let us explain what we understand by quantum analogues of the line bundles above. Replace the matrix A in (1.2) by matrix L = ||l_{i}^{j}|| subject to (2.4) (this means that the matrix L is formed by the elements l_{i}^{j} satisfying the quadratic relations (2.4)). Thus, we have the following system

v_{i} l_{j}^{i} − ν v_{j} = 0,  ν ∈ k,  (1.6)

where the summation over repeated indices is assumed. Otherwise stated, we consider the free right L_{q}(R)-module

V ⊗_{k} L_{q}(R) where V = span (v_{i})

and its submodule R_{ν} generated by the l.h.s. of (1.6). Let us restrict ourselves to a ”quantum orbit” k(M_{μ}^{q}). This means that instead of the free L_{q}(R)-module V ⊗_{k} L_{q}(R) and its submodule R_{ν} we consider the free right k(M_{μ}^{q})-module

R(V, M_{μ}^{q}) = V ⊗ k(M_{μ}^{q})

and its submodule generated by the l.h.s. of (1.6) (we keep the notation R_{ν} for it). We call a quantum line bundle over the given quantum orbit k(M_{μ}^{q}) the quotient

R(V, M_{μ}^{q})/R_{ν}

if it is non-trivial. This definition can be extended to quantum line bundle over ”noncommutative quantum orbits”. For that its suffices to replace ”commutative quantum orbit” in this definition by its ”non-commutative” counterpart (denoted k(M_{μ}^{qh̄})). The problem is when the quotient R(V, M_{μ}^{q})/R_{ν} (or its non-commutative analogue R(V, M_{μ}^{qh̄})/R_{ν}) is non-trivial. In the classical commutative case (q = 1, h̄ = 0) it is so iff ν is a root of the characteristic polynomial of A (i.e., ν = μ_{i} for some i).

In this paper we give a criterion on ν which yields non-triviality of these quotients. This criterion is based on the quantum version of the Cayley-Hamilton (CH) identity for the matrix L found in [GPS]. This version of the CH theorem states that there exists a polynomial P whose coefficients belongs to Z and such that P (L) = 0. When we restrict ourselves to a quantum orbit the coefficients of P become numerical. So, we get a polynomial \overline{P} with numerical coefficients such that \overline{P} (L) = 0 . Our main statement says that the quotient module R(V, M_{μ}^{q})/R_{ν} is non-trivial iff ν is a root of \overline{P} (we assume that the roots μ_{i} of \overline{P} are pairwise distinct). Moreover, this quotient is projective and in the quasiclassical case it is a flat deformation of its classical counterpart.

Besides, we present here a version of the CH identity valid for two parameter algebra L_{q,h̄}(R) and by passing to the limit q → 1 we get such an identity for the algebra U(gl(n)_{h̄}). This allows us to get a similar description for ”non-commutative orbits” both in the classical and quantum cases. Let us remark that a version of the CH identity for the algebra U(gl(n)) is already known from the late sixties due to works [BL]. But in the cited works the identity was established for any finite representation of U(gl(n)) and looks like

∏_{i}(A − μ_{i}) = 0 ,

where μ_{i} are integer numbers, depending on a given representation. In a sense, our result is more general, since the CH identity is realized with coefficients being elements of the RE algebra itself without using any representation. In particular, this allows us to consider the orbits of general form, where the coefficients μ_{i} are not obligatory integer numbers. Also a non-commutative version of the CH identity is presented in [G-T]. However it is rather useless for our aims since the coefficients of the CH polynomial are scalar matrices.

In the classical case besides the above line bundles related to the fundamental vector sl(n)-module V (called in the sequel basic) there exist other line bundles which can be obtained via the tensor products of the basic ones. Moreover, the family of all line bundles over a regular algebraic (or a smooth) variety forms a ring w.r.t. the tensor product. Then a natural question arises: what is a regular way to construct quantum line bundles over the ”quantum orbits” which would be different from the basic ones (we will refer to them as derived line bundles). If we want to realize the tensor product of two or more basic line bundles in terms of projective modules we should construct the corresponding projector. In fact the problem of constructing such a projector reduces to the problem of finding the CH identity for the matrix L extended to the tensor product of two (or more) copies of the space V. In the classical case this CH identity can be easily found.

However, it is not so in the quantum case. It is not even clear what is a reasonable way to extend the action of the matrix L to a tensor power of the space V. Remark that any ”reasonable” at first glance way leads to an extension of the matrix L for which we are not able to find any polynomial identity which would be a deformation of the classical one (see Section 5). Nevertheless, there exists a ”canonical” way to extend the action of the matrix L to the symmetric part of V^{⊗l}. Hopefully, for such an extension of the matrix L the CH identity can be found and it is a flat deformation of its classical counterpart. At least, it is so in a particular case when l = 2 and rank (R) = 2 (this means that the Poincaré series of ”skewsymmetric algebra” of the space V is of the form P_{−}(t) = 1 + n t + t^{2}).

In subsequent publications we will apply our approach to a quantum version of Ktheory which on one hand would enable us to control the flatness of deformation in the quasiclassical case and on the other hand would be valid in non-quasiclassical case.

The paper is organized as follows. In Section 2 we give a description of the RE algebra in comparison with RTT algebra and introduce ”quantum orbits” as some quotients of the former algebra. ”Non-commutative” counterparts of these orbits are introduced as well. In Section 3 we present quasiclassical counterparts of these quantum orbits assuming them to be deformations of generic semisimple ordinary orbits. Section 4 is devoted to construction of ”basic line bundles” over quantum orbits in terms of projective modules. And in the last Section we discuss a way to define some derived line bundles related to the symmetric product of the basic modules and calculate the corresponding CH identity in the simplest case mentioned above. In Appendix we present calculations of coefficients of the CH identity for ”non-commutative” cases.

Acknowledgment The authors are supported by the grant PICS-608. One of the authors (P.S.) is recognized to Laboratoire de Mathématiques de l’Université de Valenciennes for hospitality, during his stay at this laboratory the paper was started.

2 Reflection equation algebra and quantum orbits

Consider a matrix solution R_{j_{1}j_{2}}^{i_{1}i_{2}} ∈ Mat(n^{2}, k) of the Yang-Baxter equation^{4}

R_{12}R_{23}R_{12} = R_{23}R_{12}R_{23} ,  (2.1)

satisfying the additional Hecke condition

R^{2} = id + λR where λ = q − q^{−1} ,  (2.2)

the value of nonzero number q ∈ k being generic: q^{r} ≠ 1 for any integer r. Such solutions will be refered to as Hecke symmetries, and following to [G] we will also suppose, that the Hecke symmetry R is an even symmetry of finite rank p ≤ n. This means that

P_{−}^{(p+1)} ≡ 0 and dim P_{−}^{(p)} = 1 ,

where P_{−}^{(l)} stands for the projector of V^{⊗l} onto its subspace of totally skewsymmetric tensors. It is possible to show that such a Hecke symmetry is closed in the sense of [G], i.e., the matrix R^{t_{1}} is invertible. More detailed treatment can be found in [G, GPS].

With any Hecke symmetry R (quasiclassical or not) we can associate two matrix algebras. One of them, denoted below as T_{q}(R) and called RTT algebra is generated by n^{2} quantities t_{j}^{i} which can be considered as entries of some matrix T = ||t_{j}^{i}|| subject to the following quadratic relations [FRT]:

R_{12}T_{1}T_{2} = T_{1}T_{2}R_{12} .  (2.3)

It is well known that the algebra T_{q}(R) possesses the bialgebra structure w.r.t. to the comultiplication:

Δ(t_{j}^{i}) = t_{p}^{i} ⊗ t_{j}^{p} .

In the case of an even Hecke symmetry (quasiclassical or not) and under the assumption that so-called quantum determinant is central (cf. [G]) we can extend this bialgebra structure to the Hopf algebra by introducing the antipodal mapping

S : T_{q}(R) → T_{q}(R).

4The standard matrix conventions of [FRT] are used throughout the paper.

Another of mentioned algebras is so-called RE algebra L_{q}(R)^{5} The corresponding n × n matrix L = ||l_{j}^{i}|| obeys the relation

R_{12}L_{1}R_{12}L_{1} = L_{1}R_{12}L_{1}R_{12} .  (2.4)

This algebra can be given a structure of the adjoint comodule w.r.t. the coaction δ of the algebra T_{q}(R):

δ(l_{j}^{i}) = t_{p}^{i} S(t_{j}^{k}) ⊗ l_{k}^{p}.  (2.5)

Remark 1 As we have said in Introduction we prefer using the RTT algebra as a substitute of the symmetry group since it is well defined in the both quasiclassical and nonquasiclassical cases. As for the dual object, its explicit description in a non-quasiclassical case is not easy (cf. [AG]). However, in fact we can do without the RTT algebra at all. Let us also point out that the RE algebra has the structure of a braided Hopf algebra. This property was discovered by S.Majid (cf. [M2]). However, we do not use this property either.

An important feature of the both algebras mentioned above is the existence of polynomial identities on the quantum matrices L and T [GPS, IOP], which generalize the well-known CH identity of the classical matrix analysis.

For the RE algebra L_{q}(R) this identity looks as follows [GPS]:

(−L)^{p} + ∑_{k=0}^{p−1}(−L)^{k}σ_{p−k}(L) ≡ 0 .  (2.6)

Note, that the upper limit of summation in the identity is defined by the number p = rank(R), not by n — the dimension of the space V. The coefficients σ_{k}(L) in the above relation are polynomial combinations of generators l_{j}^{i}:

\begin{matrix}σ_{k}(L) = α_{k} Tr_{(12...p)}P_{−}^{(p)}(L_{1}R_{12} … R_{k−1,k})^{k} \\ α_{k} = q^{−k(p−k)}[C_{p}^{k}]_{q} \end{matrix}  (2.7)

where [C_{p}^{k}]_{q} are q-binomial coefficients. Let us note that in the quasiclassical case the set {σ_{k}(L)} generates the center Z of the algebra L_{q}(R). We will conjecturally suppose the same property to be true in the non-quasiclassical case as well.

The CH identity for the algebra T_{q}(R) is completely different from that above. As was shown in [IOP] the matrix T of generators of T_{q}(R) algebra satisfies the following identity:

(−T)p̄ + ∑_{k=1}^{p−1}(−T)k̄σ_{p−k}(T) + σ_{p}(T)D ≡ 0 ,  (2.8)

where D is a numeric matrix and

T k̄ = Tr_{(2...k)}R_{12}R_{23} … R_{k−1,k}T_{1}T_{2} … T_{k}  (2.9)
σ_{k}(T) = α_{k} Tr_{(12...p)}P_{−}^{(p)}T_{1}T_{2} … T_{k} .  (2.10)

5There are known different versions of the RE algebra, cf. [KSk, KSa]. We use that introduced in [M1] in terms of braided matrix algebra.

In contrast to the algebra L_{q}(R), the quantities (2.10) are not central, they only form a commutative subalgebra of T_{q}(R). It is this property that prevents us from defining a quantum orbit in T_{q}(R) algebra as a quotient algebra over an ideal generated by the elements σ_{k}(T) − c_{k} since due to the non-centrality of σ_{k}(T) the corresponding quotient would not be a flat deformation of its classical counterpart.

Now we consider a special case of the quasiclassical Hecke symmetry related to the QG U_{q}(sl(n)) (see Introduction). In this case the R-matrix is a deformation of the usual permutation P : P_{12}(v_{1} ⊗ v_{2}) = v_{2} ⊗ v_{1}. This means that lim_{q→1} R = P. Therefore, at the limit q → 1 the quadratic quantum algebra (2.4) turns into the commutative algebra L(P) = lim_{q→1} L_{q}(R):

P_{12}L_{1}P_{12}L_{1} − L_{1}P_{12}L_{1}P_{12} ≡ L_{1}L_{2} − L_{2}L_{1} = 0

where as usual, L_{2} = P L_{1} P. Let us pass from the quadratic algebra L_{q}(R) to a quadraticlinear algebra L_{q,h̄}(R) with two parameters q and h̄ which can be treated as deformation of L_{q}(R). Note, that this deformation is also well defined in a non-quasiclassical case. The algebra L_{q,h̄}(R) will be introduced by the following simple procedure. On shifting the generators of L_{q}(R) l_{j}^{i} = l̄_{j}^{i} − hδ_{j}^{i} we come to the equivalent algebra:

R_{12}L̄_{1}R_{12}L̄_{1} − L̄_{1}R_{12}L̄_{1}R_{12} = λh(R_{12}L̄_{1} − L̄_{1}R_{12}) .

Now redefining the combination λh as a new parameter h̄ and treating it as independent on q, we get the two-parameter quadratic-linear algebra L_{q,h̄}(R)

R_{12}L̄_{1}R_{12}L̄_{1} − L̄_{1}R_{12}L̄_{1}R_{12} = h̄(R_{12}L̄_{1} − L̄_{1}R_{12}) .  (2.11)

We keep the notation L̄ for the quantum matrix formed by the generators of algebra L_{q,h̄}(R) in order to distinguish it from the matrix L. In the quasiclassical case the algebra L_{q,h̄}(R) can be considered as a two parameter deformation of the commutative algebra L(P) = k(gl(n)^{∗}) = Sym(gl(n)) since as is evident from (2.11)

L(P) = lim_{{q→1 \atop h̄→0}} L_{q,h̄}(R) L_{q}(R) = lim_{{h̄→0 \atop q=const}} L_{q,h̄}(R) .  (2.12)

It is important, that all these deformations are flat. Otherwise stated, the Poincaré series of L(P), L_{q}(R) and that of the graded algebra associated to L_{q,h̄}(R) are equal to each other. This statement is also valid in the non-quasiclassical case if limits (2.12) exist. Besides, the algebra L_{q,h̄}(R) admits a nontrivial classical limit — the noncommutative algebra

L_{h̄} = U(gl(n)_{h̄}) = lim_{{q→1 \atop h̄=const}} L_{q,h̄}(R).  (2.13)

Indeed, as q → 1 the quadratic-linear relations (2.11) turns into the following ones

A_{1}A_{2} − A_{2}A_{1} = h̄(A_{1}P_{12} − P_{12}A_{1}) ,  (2.14)

which are just relations (1.5). Moreover, we have

L(P) = lim_{h̄→0} L_{h̄} .

Accordingly to what we said in Introduction in the quasiclassical case we treat the algebras L_{q}(R),L_{q,h̄}(R), and U(gl(n)_{h̄}) respectively as ”quantum commutative”, ”quantum non-commutative” and ”classical non-commutative” counterparts of the commutative algebra L(P).

The crucial point is that for the algebras L_{q,h̄}(R) and U(gl(n)_{h̄}) there also exists some version of the CH identity. For the matrix L̄ formed by the generators of the algebra L_{q,h̄}(R) the CH identity is similar to (2.6):

(−L̄)^{p} + ∑_{k=0}^{p−1}(−L̄)^{k}σ_{p−k}^{(h̄)}(L̄) ≡ 0  (2.15)

(this relation is valid in a non-quasiclassical case as well). By passing to the limit q → 1 we get a version of the CH identity for the matrix A formed by the generators of U(gl(n)_{h̄}):

(−A)^{p} + ∑_{k=0}^{p−1}(−A)^{k}τ_{p−k}^{(h̄)}(A) ≡ 0  (2.16)

(see Introduction on other versions of the CH identity). The coefficients σ_{k}^{(h̄)} and τ_{k}^{(h̄)} are central elements of the corresponding algebras. The explicit form of coefficients σ_{k}^{(h̄)} and the proof of the existence of their limits (denoted τ_{k}^{(h̄)}) as q → 1 are presented in Appendix. Let us pass now to quantum analogues of generic orbits in question. We introduce such a ”quantum orbit” (both in the quasiclassical and non-quasiclassical cases) as the quotient of the RE algebra over the ideal generated by the elements (1.4). Keeping in mind the fact that the center Z is generated by the elements σ_{k}(L) we can define the character χ by imposing χ(σ_{k}(L)) = c_{k}. In other words, we define the quantum orbit in question by the system of polynomial equations:

σ_{k}(L) − c_{k} = 0 k = 1 … p .  (2.17)

Then the polynomial \overline{P} mentioned in Introduction becomes

\overline{P} = (−L)^{p} + ∑_{k=1}^{p}(−L)^{k}c_{p−k} .  (2.18)

We consider the quotient of the RE algebra over the ideal generated by the l.h.s. of (2.17) as a quantum commutative orbit. Let us denote this quotient by k(M_{μ}^{q}), μ = (μ_{1}, ...μ_{p}), μ_{i} being roots of the polynomial \overline{P}. Let us remark that in the quasiclassical case the quotient k(M_{μ}^{q}) is a flat deformation of its classical counterpart k(M_{μ}) being the coordinate ring of a semisimple orbit. It can be shown by the methods of [D1]. In a similar way we can introduce non-commutative quantum orbit k(M_{μ}^{qh̄}) as the quotient of the algebra L_{q,h̄}(R) over the ideal generated by σ_{k}^{(h̄)}−c_{k} (with a similar meaning of μ). The corresponding polynomial (2.18) can be obtained if we replace the coefficients σ_{k}^{(h̄)}(L̄) in (2.15) by c_{k}. In the quasiclassical case by passing to the limit q → 1 we get a classical noncommutative orbit (its ”coordinate ring” will be denoted k(M_{μ}^{h̄})) and the corresponding polynomial (2.18). In all cases we suppose the roots μ_{1}, ...μ_{n} of corresponding polynomials to be pairwise distinct.

3 Quasiclassical case: related Poisson structures

In this section we will briefly describe quasiclassical counterparts of the algebras L_{q}(R) and L_{q,h̄}(R) and their restrictions to the orbits in question assuming R to be quasiclassical Hecke symmetry (see Introduction). Let

r = ∑ X_{α} ∧ X_{−α} ∈ ∧^{2}(g)  (3.1)

be the classical r-matrix related to a simple classical Lie algebra g. The quasiclassical counterpart of the RTT algebra is well known. It is so-called Sklyanin bracket defined as the difference between the left-invariant and right-invariant brackets on the corresponding Lie group G associated to r-matrix (3.1), these invariant brackets are not Poisson separately. Sklyanin bracket can be reduced to any semisimple orbit in g^{∗}. It can be quantized in the sense of deformation quantization (cf. [DG]). The resulting algebra is U_{q}(g)-covariant. This quantum algebra can be also treated in terms of the Hopf-Galois extension mentioned in Introduction.

However, the reduced Sklyanin bracket well defined on any semisimple orbit is not defined on the whole of g^{∗}. Roughly speaking, we say that the Poisson brackets defined on each semisimple orbit separately cannot be ”glued” into a global Poisson bracket.

By contrast, the quasiclassical counterpart of the RE algebra is well defined on the whole gl(n)^{∗}. Let us describe it. Let us put g = sl(n) and associate to r-matrix (3.1) a bi-vector field arising from the representation

ad^{∗} : g → Vect (g^{∗}).

By applying this bi-vector field to functions f and g we get a bracket {f, g}_{r} which is not Poisson. Nevertheless, by adding some invariant summand we can convert it into a Poisson bracket. This summand can be constructed as follows. It is well known that in the decomposition of g^{⊗2} into a direct sum of irreducible g-modules the component isomorphic to g itself occurs twice: once in the symmetric part Sym^{2}(g) of g^{⊗2} and once in the skewsymmetric part ∧^{2}(g) (as usual, we assume that g acts onto itself by the adjoint action and this action is extended to g^{⊗2} via the Leibniz rule). Denote these components by g_{s} and g_{a} respectively:

\begin{matrix}g_{s} ⊂ Sym^{2}(g) \\ g_{a} ⊂ ∧^{2}(g) \end{matrix} \} , g_{a,s} ∼ g .

Let us consider a non-trivial g-morphism sending the component g_{a} to g_{s} (it is unique up to a factor). Let us extend this map to other components of ∧^{2}(g) by 0. Let {f, g}_{inv} be the extension of this map from ∧^{2}(k(g^{∗})) to k(g^{∗}) via the Leibniz rule. Then there exist two values of a such that the sum

{f, g} = {f, g}_{r} + a{f, g}_{inv}  (3.2)

is a Poisson bracket (cf. [DGS]). One of this two Poisson brackets is a quasiclassical counterpart of the RE algebra (the other one corresponds to a modified form of the RE algebra). In the sequel the appropriate a is assumed to be fixed. By this the corresponding Poisson bracket is defined on sl(n)^{∗} (note that it is quadratic). In order to pass to a Poisson bracket defined on gl(n)^{∗} we should add one more generator, which Poisson commutes with all other generators. Let us observe that the bracket (3.2) (or its extension to gl(n)^{∗}) is compatible with the corresponding linear Poisson-Lie bracket. The Poisson pencil generated by these two brackets on gl(n)^{∗} is just the quasiclassical counterpart of the two parameter family L_{q,h̄}(R).

As for other simple Lie algebras g any invariant correction to the bracket { , }_{r} converting it into a quadratic Poisson bracket (cf. [DGS]) does not exist.

Now let us pass to the quasiclassical counterparts of the quantum orbits above. Observe that the Poisson bracket (3.2) can be restricted to any orbit in sl(n)^{∗} (or in gl(n)^{∗} if we take its extension), cf. [D2] for a proof. In particular, it is so for semisimple generic orbits. Moreover, this restricted bracket is compatible with the Kirillov-Kostant-Souriau one. The Poisson pencil generated by these two brackets is just the quasiclassical counterpart of the algebra k(M_{μ}^{qh̄}) above.

Let us remark that for symmetric orbits the reduced Sklyanin bracket becomes a particular case of this Poisson pencil. Thus, for such type orbits the quantum objects can be described in two ways: in terms of the Hopf-Galois extension or as a quotient of the RE algebra or its quadratic-linear counterpart L_{q,h̄}(R). However, the latter way is more explicit and leads to objects of ”quantum affine algebraic geometry” (cf. [DGK] where the orbits of CP^{n} type were quantized in the spirit of such a type geometry by means of an operator method). For non-symmetric orbits the quotients of the algebra L_{q}(R) (or L_{q,h̄}(R)) and algebras arising from the reduced Sklyanin bracket are completely different in spite of the fact that they both are U_{q}(g)-covariant. Also note that the family of U_{q}(g)-covariant algebras which are deformations of the coordinate ring of a semisimple orbit in gl(n)^{∗} (or sl(n)^{∗}) is large enough. The reduced Sklyanin bracket or the Poisson brackets corresponding to the algebra L_{q}(R) or L_{q,h̄}(R) represent only particular cases of Poisson structures corresponding to this family (cf. [DGS] where such Poisson structures are classified).

4 Basic line bundles over quantum orbits

In this section we introduce quantum line bundles over the quantum orbits in question associated to the fundamental vector gl(n)- (or what is the same sl(n)-) module V. Let us fix an ”orbit” which is a quotient of one of the algebras L_{q}(R), L_{q,h̄}(R) or L_{h̄} = U(gl(n)_{h̄}) over the ideal generated by (1.4) (in the case of the algebras L_{q}(R), L_{q,h̄}(R) both in the quasiclassical or non-quasiclassical cases).

Similarly to the above consideration we assume the roots μ_{1}, ..., μ_{n} of the corresponding polynomial (2.18) to be pairwise distinct. In the sequel we use the notation k(\overline{M}_{μ}) for such an orbit (of one of the types above). Let

R(V, \overline{M}_{μ}) = V ⊗ k(\overline{M}_{μ})

be a free right k(\overline{M}_{μ})-module and be R_{ν} its submodule generated by the r.h.s. of (1.6). Let us consider the quotient module R(V, \overline{M}_{μ})/R_{ν}.

Theorem 1 The k(\overline{M}_{μ})-module R(V, \overline{M}_{μ})/R_{ν} is non-trivial iff ν in (1.6) coincides with one of μ_{i}, that is iff ν = μ_{i} for some i. In this case the module R(V, \overline{M}_{μ})/R_{ν} is projective. More precisely,

R(V, \overline{M}_{μ})/R_{μ_{i}} = Im P_{i} ,  (4.1)

where

P_{i} = ∏_{j≠i}^{p} \frac{L − μ_{j} id}{μ_{i} − μ_{j}} .  (4.2)

is a projector acting on the free module R(V, \overline{M}_{μ}). Furthermore, in the quasiclassical case the module R(V, \overline{M}_{μ})/R_{μ_{i}} is a flat deformation of its classical counterpart. (In the case of the algebra U(gl(n)_{h̄}) we assume L = A.)

Remark 2 Let us note that we do not consider the k(\overline{M}_{μ})-module R(V, \overline{M}_{μ})/R_{μ_{i}} as a quantum variety since the corresponding ”coordinate ring” is not well defined. In order to introduce such a ring we should define a commutation rule between the space V and the algebra k(\overline{M}_{μ}). However, apparently there is no reasonable way to do it (if we want to preserve the flatness of the deformation). We are planing to return to this question in a future publication.

Proof The proof of the theorem is based on the CH identities (2.6), (2.15) and (2.16) and looks like that in the classical case because the main difficulty is hidden in the quantum version of the CH identity. The necessity of restriction ν = μ_{i} for some i follows from relation (1.6) and \overline{P} (L) = 0 where \overline{P} is defined by (2.18). Note, that the latter relation can be rewritten in the form:

∏_{i=1}^{p}(L − μ_{i}id ) = 0, p = rank (R) .

Now, by virtue of (1.6) we have

(ν − μ_{1}) … (ν − μ_{p})v = 0, v = (v_{1}, ..., v_{n})

and if ∀i ν ≠ μ_{i} we have v = 0, that is the module R(V, \overline{M}_{μ})/R_{ν} is trivial. In order to prove the non-triviality and projectivity of the module in the case ν = μ_{i} we consider the projectors (4.2). Note that the action of the projectors P_{i} on the k(\overline{M}_{μ})-module R(V, \overline{M}_{μ}) is given by that of the matrix L which is defined as follows

∑_{i=1}^{n} v_{i}g^{i}(l) ◁ L = ∑_{i,j=1}^{n} v_{j}l_{i}^{j}g^{i}(l).  (4.3)

Thus, relation (1.6) can be represented in the form

v ◁ L − νv, v ∈ V.  (4.4)

Let us remark that the action (4.3) of the matrix L on the space V is coordinated with the coaction of the Hopf algebra T_{q}(R) in general and therefore (in the quasiclassical case) with the action of the dual object, namely the QG U_{q}(sl(n)). Taking into account that μ_{i} are distinct we have the following Proposition 1 The operators (4.2) form the full set of orthonormal projective operators on the space R(V, \overline{M}_{μ}), that is the following properties hold

i) P_{i}P_{j} = δ_{ij}P_{i},

ii) ∑_{i=1}^{n} P_{i} = id.

Proof is left to reader as an easy exercise. Let us return to the proof of the theorem. As

follows from the proposition the quotient R(V, \overline{M}_{μ})/R_{μ_{i}} can be identified with Im P_{i}. This shows that the k(\overline{M}_{μ})-module R(V, \overline{M}_{μ})/R_{ν} is projective. Moreover, in the quasiclassical case it also implies that this module is a flat deformation of its classical counterpart since under a deformation the projectors are deformed smoothly. This completes the proof.

5 Derived line bundles

In this section we consider the problem of constructing the quantum line bundles different from the basic ones. We call them derived line bundles. First, consider the classical case. Let us fix a generic semisimple orbit M_{μ} and two line bundles E_{μ_{i}}, i = 1, 2 (see Introduction). Let us consider their tensor product. We want to represent its coordinate ring as k(M_{μ})-module as well. It can be done as follows.

Let again V be the fundamental vector sl(n)-module. Consider the free right k(M_{μ})- module

R(V^{⊗2}, M_{μ}^{q}) = V^{⊗2} ⊗ k(M_{μ})

and extend the action of the matrix L = A to the space V^{⊗2} by setting

(u ⊗ v) ◁ L = u ⊗ (v ◁ L) + (u ◁ L) ⊗ v u, v ∈ V

(some sort of the Leibniz rule). Thus, the extended matrix L which will be denoted L^{(2)} can be written as

L_{1} + L_{2}, L_{1} = id ⊗ L, L_{2} = P_{12}L_{1}P_{12} .

Let us consider the submodule R_{ν} defined as in (4.4) but with v ∈ V replaced by u ⊗ v. Then the same problem arises: for what value of ν the factor

R(V^{⊗2}, M_{μ}^{q})/R_{ν}

is not trivial. It is not difficult to see that it is so iff ν = μ_{i} + μ_{j} where μ_{i} are the roots of the polynomial \overline{P}, i.e., ”eingenvalues” of the matrix L corresponding to the orbit in question. This allows us to find the CH identity for the matrix L^{(2)}.

Also it can be found directly from the identity for the matrix L in the following way. Let \overline{P} (L) = 0 be the CH identity for the matrix L (in this section the coefficients of polynomial \overline{P} are supposed to be numerical). Then it is not difficult to find an analogous relation for the matrix L^{(2)}. For this it is sufficient to raise this matrix to the powers 1, 2, ..., l(l + 1)/2 where l is the degree of the polynomial \overline{P} and on expressing the powers L_{1}^{l}, L_{1}^{l+1}, ... through L_{1}^{1}, ..., L_{1}^{l−1} and similarly for the matrix L_{2} we get the CH identity for the matrix L^{(2)}. The crucial property used in the construction is the mutual commutativity of the matrices L_{1} and L_{2}:

L_{1} L_{2} = L_{2} L_{1}.

Moreover, by assuming the sums μ_{i} + μ_{j} to be pairwise distinct we can construct the projector analogous to (1.3). Nevertheless, we should restrict ourselves to the symmetric part of the space V^{⊗2} to eliminate the multiplicity of the quantity μ_{i} + μ_{j} since (if i ≠ j) it occurs once in the symmetric part of this space and once in its skewsymmetric part. Thus, the projector corresponding to the eigenvalue μ_{i} +μ_{j} is the product of the projector onto symmetric part of V^{⊗2} and that looking like (1.3). Let us point out that in a similar way it is possible to extend the matrix L to the higher tensor powers of the space V: V^{⊗l}, l = 3, 4, .... Thus, if l = 3 the extended matrix is defined as L_{1} + L_{2} + L_{3} with the obvious definition of the matrix L_{3}. The details are left to the reader.

Turn now to the quantum case, i.e., assume that the matrix L is subject to relations (2.4). If we considered the matrix L^{(2)} = L_{1} + L_{2} with L_{2} defined as above but with P_{12} replaced by R_{12} we would be unable to find the CH identity for such an extension of the matrix L since the matrix L_{2} does not satisfy the polynomial relation valid for L_{1} = L. The point is that the matrices L_{1} and L_{2} are not similar. (However, if we chose as L_{2} the matrix R_{12}L_{1}R_{12}^{−1} then the matrices L_{1} and L_{2} would become similar but the commutativity L_{1}L_{2} = L_{2}L_{1} valid in the previous case by virtue of the RE would be lost.)

Nevertheless, we are interested in an extension of the matrix L to the symmetric part of the space V^{⊗2}. Let us define such an extension as follows

L_{+} = P_{+}LP_{+} where P_{+} = \frac{q^{−1}id + R_{12}}{q + q^{−1}} .  (5.1)

Such a way to extend the matrix L to the symmetric part of V^{⊗2} is motivated by the following observation. In the classical case (q = 1) such an extension of the matrix L = A coincides (up to a factor, which does not matter for us) with the restriction of the matrix L_{1} + L_{2} to the symmetric part of V^{⊗2}.

In the sequel we will restrict ourselves to the case rank (R) = 2. This implies that the CH identity for the matrix L is quadratic:

L^{2} − aL + b id = 0 a = μ_{1} + μ_{2}, b = μ_{1}μ_{2} .  (5.2)

Proposition 2 If the CH identity for the matrix L is of the form (5.2) then the matrix L_{+} defined by (5.1) obeys the CH identity of the form:

L_{+}^{3} − a(1 + \frac{q^{−1}}{2_{q}}) L_{+}^{2} + (a^{2} \frac{q^{−1}}{2_{q}} − b) L_{+} + ab\frac{q^{−1}}{2_{q}} id = 0.  (5.3)

Proof Taking into account the formulas for the symmetrizer P_{+}^{(2)} we can express the R-matrix via P_{+} and rewrite the RE algebra in the equivalent form:

P_{+}LP_{+}L − LP_{+}LP_{+} + \frac{q^{−1}}{2_{q}} (L^{2}P_{+} − P_{+}L^{2}) = 0 .  (5.4)

If the matrix L obeys the CH identity (5.2) we then have

P_{+}LP_{+}L − LP_{+}LP_{+} + a\frac{q^{−1}}{2_{q}} (LP_{+} − P_{+}L) = 0 .  (5.5)

Now the CH identity for L_{+} is a consequence of direct calculations. Indeed, let us calculate successively the powers of matrix L_{+}. Below we use the abbreviation ξ = a\frac{q^{−1}}{2_{q}}. For L_{+}^{2} we get

L_{+}^{2} ≡ (P_{+}LP_{+}L)P_{+} = (use (5.5)) = LL_{+} − ξLP_{+} + ξL_{+} .

We have here unwanted terms LL_{+} and LP_{+} and therefore should calculate the next power of L_{+} in order to get rid of them. So

L_{+}^{3} = L_{+}^{2}L_{+} = (insert L_{+}^{2} above) = LL_{+}^{2} − ξLL_{+} + ξL_{+}^{2} =

(insert L_{+}^{2} again and use (5.2)) =

a(1 + \frac{q^{−1}}{2_{q}})LL_{+} − a(1 + \frac{q^{−1}}{2_{q}})ξLP_{+} + (ξ^{2} − b)L_{+} + bξP_{+}.

No new unwanted term besides LP_{+} and LL_{+} appears and excluding them from expressions for L_{+}^{2} and L_{+}^{3} we come to the following

L_{+}^{3} − a(1 + \frac{q^{−1}}{2_{q}})L_{+}^{2} + (a^{2} \frac{q^{−1}}{2_{q}} + b)L_{+} − ab\frac{q^{−1}}{2_{q}} P_{+} = 0.

It remains to observe that P_{+} = id on the symmetric part of V^{⊗2}. In a similar way we can extend the matrix L to the higher symmetric powers of the space V: it suffices to replace the projectors P_{+} in the formula (5.1) by the symmetrizer in a given power. However, the problem of finding the corresponding CH identity is much more complicated and is still open.

Appendix

In this appendix we present the explicit form of the coefficients entering (2.15) and show that in the quasiclassical case these coefficients have finite limit as q → 1. This fact gives rise to the formula (2.16). Let us recall that in general p = rank(R) ≠ n = dim V. Also recall that the degree of the polynomial \overline{P} and hence the number of the roots μ_{i} is equal to p. We will use the fact, that the L_{q,h̄}(R) algebra (2.11) can be formally obtained from L_{q}(R) by the shift of generators l_{j}^{i} = l̄_{j}^{i} − δ_{j}^{i}h and subsequent changing of the parameter: h̄ = hλ. So, if we realize these two operations in the CH identity (2.6) for the matrix L we get some polynomial identity on the matrix L̄ formed by the generators of the algebra L_{q,h̄}(R).

The problem is to find in which way the central elements σ_{k}(L) are transformed. This can be directly calculated from the definition of σ_{k}(L) (2.7). Indeed, on making the above mentioned shift in σ_{k}(L) we have to transform the number of arising terms, the typical form of which are

Tr_{(12...p)}P_{−}^{(p)}(L_{1}R_{1} … R_{k−1})^{s}R_{1} … R_{k−1}(L_{1}R_{1} … R_{k−1})^{k−s−1} .  (A.1)

In the above formula the concise notation R_{i} ≡ R_{i i+1} is used. Now one should ”draw out” the string of R-matrices and cancel them on the projector P_{−}^{(p)}. Then it is necessary to get rid of all the matrices R_{k−1} in monomials (L_{1}R_{1} … R_{k−1}). The basic formulas for such transformations are

(L̄_{1}R_{1} … R_{k})R_{i} = R_{i+1}(L̄_{1}R_{1} … R_{k}) ∀ i ≤ k − 1 ,

(this is a trivial consequence of the Yang-Baxter equation) and

(L̄_{1}R_{1} … R_{k})^{r} = (L̄_{1}R_{1} … R_{k−1})^{r}R_{k}R_{k−1} … R_{k−r+1}

which in turn is a direct consequence of the previous relation. The cancellation of Rmatrices in (A.1) is due to the cyclic property of trace and the defining property of the skewsymmetrizer P_{−}^{(r)}:

R_{i} P_{−}^{(r)} = P_{−}^{(r)} R_{i} = −\frac{1}{q} P_{−}^{(r)} ∀i ≤ r − 1 .

Now after straightforward calculations we come to the following transformation of coefficients:

σ_{k}(L) ⟶ ∑_{r=0}^{k} (−\frac{h̄}{λ})^{r} q^{−r(p−1)}C_{k}^{p} \frac{[C_{r}^{k}]_{q}}{[C_{p}^{k−r}]_{q}} σ_{k−r}(L̄) .  (A.2)

Here the symbol [C_{p}^{k}]_{q} stands for the q-binomial coefficient

[C_{p}^{k}]_{q} = \frac{p_{q}!}{k_{q}!(p − k)_{q}!}

where p_{q}! = (p − 1)_{q}!p_{q} and q-numbers r_{q} are defined as

r_{q} ≡ \frac{q^{r} − q^{−r}}{q − q^{−1}} .

The elements σ_{k}(L̄) are defined in (2.7) where matrix L should be changed for L̄. Obviously, these elements are central in L_{q,h̄}(R) algebra. Now given the rule (A.2) it is not so difficult to obtain from (2.6) the CH identity for the L_{q,h̄}(R) algebra

(−L̄)^{p} + ∑_{k=0}^{p−1}(−L̄)^{k}σ_{p−k}^{(h̄)}(L̄) ≡ 0 .  (2.15)

The quantities σ_{p−k}^{(h̄)}(L̄) are the following polynomials in h̄ with coefficients depending on σ_{p−k}(L̄):

σ_{p−k}^{(h̄)}(L̄) = σ_{p−k}(L̄) + ∑_{r=1}^{p−k} h̄^{r}ω_{r+k,k}^{(p)} σ_{p−k−r}(L̄) .  (A.3)

The numeric coefficients ω_{s,k}^{(p)} are as follows (s > k)

ω_{s,k}^{(p)} = \frac{λ^{k−s}}{[C_{p}^{s}]_{q}} ∑_{r=0}^{s−k}(−1)^{r}q^{−r(p−1)}C_{s−r}^{k}C_{p−s+r}^{r}[C_{p}^{s−r}]_{q} .  (A.4)

Now we calculate the sum in (A.4) and show that it is proportional to λ^{s−k} and therefore the whole coefficient h̄^{r}ω_{r−k,k}^{(p)} admits a non-singular classical limit as q → 1, h̄ =const. On taking this limit in CH identity (2.15) we get the corresponding identity (2.16) for the matrix A of U(gl(n)_{h̄}) algebra (2.14). Denote the sum in (A.4) by ξ_{s,k}^{(p)}:

ξ_{s,k}^{(p)} ≡ ∑_{r=0}^{s−k}(−1)^{r}q^{−r(p−1)}C_{s−r}^{k}C_{p−s+r}^{r}[C_{p}^{s−r}]_{q}

and calculate the generating function Φ_{ξ}(x, y) of the coefficients ξ_{s,k}^{(p)}

Φ_{ξ}(x, y) =^{def} ∑_{s=0}^{p} ∑_{k=0}^{s}(−x)^{p−s} (−y)^{k} ξ_{s,k}^{(p)} .

If one knew the function Φ_{ξ}(x, y) then the coefficients ξ_{s,k}^{(p)} could be found as

ξ_{s,k}^{(p)} = \frac{(−1)^{p−s+k}}{k!(p − s)!} [\frac{∂^{p−s}}{∂x^{p−s}} \frac{∂^{k}}{∂y^{k}} Φ_{ξ}(x, y)]_{x=y=0} .  (A.5)

The calculation of Φ_{ξ}(x, y) is rather simple where the only thing we need for is the Newton binomial formula and its q-analogue

∑_{k=0}^{p} x^{k}q^{−k(p−1)}[C_{p}^{k}]_{q} = od_{k=0}^{p−1} (1 + \frac{x}{q^{2k}} ) .

So we present the final result

Φ_{ξ}(x, y) = (−1)^{p}q^{−p(p−1)} ∏_{k=0}^{p}(xq^{p−1} + yq^{2k} − λ q^{k}k_{q}) .  (A.6)

At last, upon taking the partial derivatives in (A.5) we find the form of coefficients ξ_{s,k}^{(p)}:

ξ_{s,k}^{(p)} = λ^{s−k}q^{\frac{(p−1)(p−2s)}{2}} (p − 1)_{q}! (V_{k} + V_{s}) ,  (A.7)

where by V_{k} and V_{s} we denote the following sums

V_{k} ≡ \{ \begin{matrix}0 if k = 0 \\ ∑_{1≤l_{1}<...<l_{k−1}≤p−1 ; {1≤r_{1}<...<r_{p−s}≤p−1 \atop \{l_{i}\}∩\{r_{j} \}=∅}} & \frac{q^{(l_{1}+...+l_{k−1}−r_{1}−...−r_{p−s})}}{(l_{1})_{q} … (l_{k−1})_{q}(r_{1})_{q} … (r_{p−s})_{q}} \end{matrix} if k ≠ 0

V_{s} ≡ \{ \begin{matrix}0 if s = p \\ ∑_{1≤l_{1}<...<l_{k}≤p−1 ; {1≤r_{1}<...<r_{p−s−1}≤p−1 \atop \{l_{i}\}∩\{r_{j} \}=∅}} & \frac{q^{(l_{1}+...+l_{k}−r_{1}−...−r_{p−s−1})}}{(l_{1})_{q} … (l_{k})_{q}(r_{1})_{q} … (r_{p−s−1})_{q}} \end{matrix} if s ≠ p .

In particular,

ξ_{s,s}^{(p)} ≡ 1 ξ_{p,0}^{(p)} ≡ 0 .

Now we can find the coefficients ω_{s,k}^{(p)} entering the CH identity (2.15) of the L_{q,h̄}(R) algebra

ω_{s,k}^{(p)} ≡ \frac{λ^{k−s}}{[C_{p}^{s}]_{q}} ξ_{s,k}^{(p)} = q^{\frac{(p−1)(p−2s)}{2}} \frac{(p − 1)_{q}!}{[C_{p}^{s}]_{q}} (V_{k} + V_{s}) .

This gives the explicit form of σ_{k}^{(h̄)} and completes the proof of the CH identity for the algebra L_{q,h̄}(R).

From the above relation it is obvious that the coefficients ω_{s,k}^{(p)} admits a non-singular classical limit

lim_{q→1} ω_{s,k}^{(p)} ≡ ρ_{s,k}^{(p)} = \frac{(p − 1)!}{C_{p}^{s}} (V_{k}^{cl} + V_{s}^{cl}) ,  (A.8)

where V_{k,s}^{cl} are given by formulas for V_{k,s} with substitution q = 1 and all q-numbers changed for ordinary ones. Content-Length: 936 X-Lines: 22 Status: RO

Finally, on taking into account that U(gl(n)_{h̄}) = lim_{q→1} L_{q,h̄}(R) we deduce from (2.15) the CH identity (2.16) where coefficients τ_{p−k}^{(h̄)}(A) are the classical limit of σ_{p−k}^{(h̄)}(L̄) in (A.3):

τ_{p−k}^{(h̄)}(A) = lim_{q→1} σ_{p−k}^{(h̄)}(L̄) = τ_{p−k}(A) + ∑_{s=1}^{p−k} h̄^{s}ρ_{s+k,k}^{(p)} τ_{p−k−s}(A)  (A.9)

with ρ_{s,k}^{(p)} defined in (A.8). The central elements τ_{k}(A) have the form:

τ_{k}(A) = \frac{C_{p}^{k}}{p!} ε_{i_{1}...i_{k}a_{k+1}...a_{p}}A_{j_{1}}^{i_{1}} … A_{j_{k}}^{i_{k}}ε^{j_{1}...j_{k}a_{k+1}...a_{p}} ,

ε being the skew-symmetric Levi-Civita tensor. These elements are analogues of spectral invariants of the usual matrix with commutative entries: in that case each τ_{k} is the sum of all principal minors of k-th order.

From identity (A.9) one can see, that in the classical non-commutative case each coefficient in the CH identity is modified by adding a polynomial in h̄. In particular, the free term of the identity which can be treated as non-commutative analogue of the determinant is given by the following formula

detA + ∑_{k=1}^{p−1} h̄^{k}ρ_{k,0}^{(p)}τ_{p−k}(A)  (A.10)

In this formula it is taken into account that ρ_{p,0}^{(p)} = 0.

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