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Reflection equation algebra in braided geometry 1 | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Reflection equation algebra in braided geometry 1

Dimitri GUREVICH *1, Pavel PYATOV 2, and Pavel SAPONOV 3

1USTV, Universit´e de Valenciennes, 59313 Valenciennes, France E-mail: [email protected]

2Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow region, Russia E-mail: [email protected]

3Division of Theoretical Physics, IHEP, 142281 Protvino, Moscow region, Russia E-mail: [email protected]

*Corresponding Author:
Dimitri GUREVICH
USTV,
Universit´e de Valenciennes,
59313 Valenciennes, France
E-mail:
[email protected]

Received date: December 20, 2007

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Abstract

Braided geometry is a sort of the noncommutative geometry related to a braiding. The central role in this geometry is played by the reflection equation algebra associated with a braiding of the Hecke type. Using this algebra, we introduce braided versions of the Lie algebras gl(n) and sl(n). We further define braided analogs of the coadjoint orbits and the vector fields on a q-hyperboloid which is the simplest example of a ”braided orbit”. Besides, we present a braided version of the Cayley-Hamilton identity generalizing the result of Kantor and Trishin on the super-matrix characteristic identities.

Key words

(modified) reflection equation algebra, braiding, Hecke symmetry, Hilbert- Poincar´e series, bi-rank, Cayley-Hamilton identity, q-hyperboloid, braided vector fields

Introduction

By braided geometry we understand a sort of the noncommutative geometry related to a braiding R, which is an invertible operator R : equation obeying to the Yang-Baxter equation

equation

Here V is a vector space over the ground field K = C (or R).

In what follows we are dealing with braidings satisfying a second order equality

(q I − R)(q−1 I + R) = 0

where q ∈ K is assumed to be generic (in particular, this means qk ≠ 1 for any integer k ≥ 2). Such a braiding is called a Hecke symmetry. It becomes an involutive symmetry in case q = 1.

Let Uq(sl(n)) → End(V) be the basic representation of the quantum group Uq(sl(n)), dimV = n. Then the product of the image of the quantum universal R-matrix in the space equationand the flip (transposition operator) is a Hecke symmetry. In what follows this Hecke symmetry will be referred to as a standard one. Note that its limit at q → 1 is the usual flip. In this sense we can treat the standard Hecke symmetry as a deformation of the flip. In a similar way one can construct deformation of a super-flip. Besides, there are known Hecke symmetries which are neither deformations of a flip, nor of a super-flip (see the next section).

With any Hecke symmetry we can associate (at least) two matrix algebras. One of them is called RTT (or Reshetikhin-Takhtajan-Faddeev) algebra (cf [17]), the other one is called Reflection Equation Algebra (REA)2.

Both, the RTT and the RE algebras (supplied with a spectral parameter) are employed in constructing integrable dynamical models. Besides, the latter algebra possesses a number of remarkable properties which make it a very interesting object of the braided geometry.

First, the REA admits some quotients which can be regarded as noncommutative (braided) analogs of the coadjoint orbits. Moreover, it enables one to define a braided analog of the Lie bracket.

Second, for certain matrices with entries belonging to REA one can find the Cayley-Hamilton (CH) identities with central coefficients. These identities allow one to define a whole family of projective modules over the braided orbits. An attempt to develop a sort of related K-theory based on a ”braided trace” instead of the usual one, which is traditionally employed in the classical K-theory, was made in [8]. Most of these objects are defined and studied for the REA related to the so called even Hecke symmetries (see the next section).

For the general linear type Hecke symmetries (further referred to as the GL(p|r) type symmetries) we have established the basic CH identity. This identity is a q-extension of the characteristic identity for super-matrices discovered by I.Kantor and I.Trishin in [13]. Note that it is this remarkable paper which stimulated our interest in the REAs related to the GL(p|r) type Hecke symmetries.

The present paper is a brief review of the main properties of the REA and their applications in braided geometry. In particular, we present a classification of the Hecke symmetries (section 2), define and compare the corresponding quantum matrix algebras (section 3), introduce braided analogs of the Lie algebras gl(n) and sl(n) (section 4), and consider braided version of the Cayley- Hamilton identity (section 5). We conclude the paper with an example of a q-hyperboloid for which we describe a braided analog of vector fields and describe their applications.

Classification of Hecke symmetries

Definition 2.1. Let V be a finite dimensional vector space over the field K, dimKV = n. An invertible operator R : equation is called a braiding if it satisfies the quantum Yang-Baxter equation

equation

If, in addition, a braiding R obeys the relation

(qI − R)(q−1I + R) = 0, q ∈ K \ 0

it is called a Hecke symmetry. If q = 1 a braiding R is called an involutive symmetry.

For any Hecke symmetry the braided analogs of the symmetric and skew-symmetric algebras of the space V can be constructed in the following way. Define the ”R-symmetric” I+ and ”R-skew-symmetric” I subspaces of equation by the relations

equation

Let T(V ) denotes the free tensor algebra of the space V , and I± stands for a two-sided ideal in T(V ) generated by the space {I±}. The quotients

equationare called R-symmetric and R-skew-symmetric algebras of the space V respectively.

Noticing that equation are quadratic graded algebras, we consider their Hilbert-Poincar´e (HP) series

equation

where equation are homogeneous components of the degree k.

Theorem 2.2. For generic values of q the fo llowing identity holds

P+(t)P−(−t) = 1

Example 1. If R = P where P is the usual flip then P−(t) = (1 + t)n where n = dimV .

Example 2. If V = V0 equation V1 is a super-space, sdim V = (p|r) and R is the corresponding super-flip then equation

Definition 2.3. If P−(t) (respectively P+(t)) is a monic polynomial then the space V and the braiding R are called even (respectively odd).

Theorem 2.4. If R is even then the polynomial P−(t) is reciprocal

Proposition 2.5. For any V with dimV = n ≥ 2 there exists a Hecke symmetry equation such that P−(t) = 1 + nt + t2.

Note that if n > 2 such a Hecke symmetry cannot be a deformation of the usual flip since the HP series are stable under the deformation.

Definition 2.6. Let R be an even Hecke symmetry. The degree p = deg P−(t) is called the rank of R.

Proposition 2.7. Let p be an integer constrained by 2 ≤ p ≤ n = dimV . Then there exists an even Hecke symmetry R : equation of the rank p.

All even Hecke symmetries of rank ∈ are classified (cf [4] and references therein).

Example 3. If p = 3, then P−(t) = 1 + nt + n t2 + t3.

Example 4. If p = 4, then P−(t) = 1 + n t + r t2 + n t3 + t4.

Definition 2.8. Let P−(t) be a rational function represented as a ratio of two coprime polynomials, the degrees of the numerator and denominator being p and r respectively. We shall say that the corresponding Hecke symmetry has the bi-rank (p|r).

The bi-rank of a super-flip coincides with its super-dimension.

Theorem 2.9. [10] Any Hecke symmetry has a certain bi-rank. In other words, its HP series P−(t) is a rational function.

Definition 2.10. A braiding R is called skew-invertible if there exists an operator equation equationsuch that

equation

where P13 is the flip transposing the first and the third spaces.

On fixing a basis {xi} 2 V we get the corresponding basis {xi equation xj} in the space equation. With respect to this basis the braiding R and the operator equation are represented by their matrices

equation

where the upper pair of indices enumerates columns of the matrix and the lower one — rows. In terms of matrices the relation in Definition 2.10 takes the form

equation

Theorem 2.11 ([2]). If R is a skew-invertible symmetry then the numerator N(t) of the rational function P−(t) is a reciprocal polynomial, while the denominator D(t) is a skew-reciprocal one (i.e D(−t) is reciprocal).

For any skew-invertible braiding the operators

equation

are well defined. They are analogs of the parity operators in super-spaces. They are used for the definition of R-traces associated with R in the spaces of endomorphisms. If End(V) = equation is the space of left endomorphisms and the set equation is its natural basis (i.e equation, then equation (for the space of right endomorphisms there is a similar formula with B instead of C). Note that if R is a super-flip then TrR is nothing but the usual super-trace.

Quantum matrix algebras

The most famous quantum matrix algebra related to a Hecke symmetry R is the RTT-algebra. It is defined in following way. Let T =equation be a matrix with entries equation

Then by definition an RTT algebra is generated by the unit and by the elements equation subject to the relations

equation

or in a coordinate form

equation

where the summation over repeated indices is always understood. Being equipped with the coproduct

equation

and a counit it becomes a bi-algebra. If in addition R is an even skew-invertible Hecke symmetry a group-like element detq(T) (called the quantum determinant) can be defined. If it is central, the quotient of the RTT algebra over the ideal generated by detq(T) − 1 is a Hopf algebra. For the standard Hecke symmetry (”the standard case” in what follows) this quotient is denoted Kq[sl(n)] and treated to be a quantum analog of the function space on the group sl(n). Its restricted dual is just the quantum group (QG) Uq(sl(n)).

Another important quantum matrix algebra associated with R is the so-called Reflection Equation Algebra (REA). It is defined as follows. Let equation be a matrix with entries equation (the lower index enumerates rows). Then the REA is generated by the unit and by the elements equation subject to the relations

RL1 RL1 − L1 RL1 R = 0

or, in coordinates

equation

This algebra has a braided bi-algebra structure and can be equipped with the coaction of the RTT algebra (in the standard case it can be also equipped with an action of the QG Uq(sl(n))). The term ”braided” means that

equation

where equation is an extension of the initial Hecke symmetry R to the REA (see the next section and [7] for more detail).

If R is a skew-invertible even Hecke symmetry a group-like element equation can be defined.

Then the quotient of the REA over the ideal generated by equation has a braided Hopf structure [15]. It is a braided analog of function space on sl(n). Denote it equation

In the standard case equation arise from the quantization of two different Poisson structures on the group sl(n). The algebra Kq[sl(n)] arises from the Sklyanin bracket, while equation originates from the Semenov-Tian-Shansky (S-T-S) one. A universal description for both these algebras is given in [11,5]. It is based on the use of a pair of compatible braidings, one of them being the Hecke symmetry.

However, the properties of Kq[sl(n)] and equation differ drastically.

1. In the REA the elements3 TrR Lk := Tr (C ≤ Lk) are central for any integer k ≥ 0. In the RTT algebra their analogs are not central bur form a commutative subalgebra (i.e they are in involution).

2. For the REA there is a Cayley-Hamilton identity in the classical form (see section 5). For the RTT algebra the ”powers” of the matrix T coming in such a relation are defined in a more complicated way.

3. The REA has a further deformation which plays the role of the enveloping algebra of a ”braided Lie algebra”. We call this algebra the modified REA (mREA).

Definition 3.1. The mREA is an associative unital algebra, generated by the elements equation subject to the following quadratic-linear relations

equation

or, in components,

equation

Theorem 3.2 ([7]). Let R = Rq be a Hecke symmetry depending on q ∈ K (for q = 1 it becomes involutive). Then for a generic q the dimensions of homogeneous components of the RTT algebra and of the REA equal to those at q = 1, that is the dimensions are stable under the q-deformation.

Moreover, there exists an analog of the PBW theorem for the mREA. Thus, in the standard case the mREA is a deformation algebra depending on 2 parameters equation and q. Its Poisson counterpart is a Poisson pencil generated by the linear Poisson-Lie bracket on gl(n)+ and by a quadratic bracket which is an extension of the S-T-S one to the whole space gl(n)+. We refer the reader to the paper [7] for description of this Poisson pencil and to [16] for details on S-T-S bracket on the group.

In the sequel we use the notation L(q, equation) for the mREA and L(q) for the REA.

Note that the element equation is central in the both algebras L(q, equation) and L(q).

So, it is natural to introduce the quotients SL(q, equation) = L(q, equation)/hli and SL(q) = L(q)/hli over an ideal, generated by l.

In the standard case all these algebras can be endowed with an action of the QG Uq(sl(n)) so that

equation

All the operators possessing this property are called equivariant.

mREA and braided Lie bracket

Let R be a skew-invertible braiding. We want to equip the space End(V) (say, the left endomorphisms for the definiteness) with a structure of a generalized Lie algebra.

Let us first assume R to be involutive (R2 = 1). Then there exists an extension of R up to

equation

such that it is involutive and coordinated with the natural product

equation

as follows

equation

Here the both sides of the equality should be applied to an element from End equation This means that the result of applying the product μ does not depend on the position. The operators satisfying this property will be called R-invariant.

Let us set by definition

equation

Theorem 4.1. The following properties hold true:

equation

2. [ , ]REnd(V) = −[ , ] (the R-skew-symmetry of the bracket);

equation

Definition 4.2. A generalized Lie algebra, or R-Lie algebra, is a data

equation

satisfying the properties 1 − 3 above where REnd(V) should be replaced by R.

The generalized Lie algebra defined above in the space End(V) is denoted gl(VR).

Let TrR : End(V) → K be the R-trace associated with a given skew-invertible involutive symmetry R (see section 2). The family of TrR-less elements of gl(VR) forms a generalized Lie subalgebra denoted sl(VR). For any generalized Lie algebra g its enveloping algebra can be defined in the natural way as the following quotient

equation

It is a braided Hopf algebra, its coproduct being additive on the generators:

equation

We would like to extend this construction to the non-involutive case. However, in contrast to the involutive case, here we first define the ”enveloping algebra” of a braided Lie bracket, and then the bracket itself. We consider the mREA L(q, equation) corresponding to a skew-invertible Hecke symmetry R as a proper analog of the enveloping algebra U(gl(n)).

In order to argue this point of view we restrict ourselves to the standard Hecke symmetries. The following statement is a corollary of the theorem 3.2.

Proposition 4.3. The mREA corresponding to a standard Hecke symmetry is a two-parameter deformation of the commutative algebra Sym(gl(n)) and a one-parameter deformation of U(gl(n)equation) (the subscript equation means that the parameter equation stands as a multiplier at the usual gl(n) Lie bracket).

Any finite dimensional representation of U(gl(n)) can be deformed into an equivariant representation of the mREA L(q, equation).

Besides, a sort of the PBW theorem is valid for the mREA (cf [7]). Concerning simple algebras of the series Bn, Cn, Dn there exists no similar deformation of their enveloping algebras.

Now, we are able to define a braided analog of the Lie bracket arising from L(q, equation). Below we put equation = 1. The commutative relations among the generators equation can be rewritten as follows (cf [7]):

equation

Denoting the right hand side of the above relation as equation we claim the following.

Theorem 4.4. Consider the map L(q, 1) → End(L(q, 1)) : equation is a linear operator acting on generators by the rule equation This map is a representation of the algebra L(q, 1).

We shall call the map equation the adjoint representation,equation the adjoint operator and the operation [ , ] the braided Lie bracket. Also, we denote the vector space Spanequation endowed with the braided Lie bracket as gl(VR). Unfortunately, the properties listed in the theorem 4.1 (where R is assumed to be an involutive symmetry) are not valid in general case. Their analogs for the braided Lie bracket in question are more complicated (cf [7, 9] for detail).

In fact, we have identified the space Span . with End(V). The R-trace in the basisequation ∈ End(V) is equation (up to a factor). It is not difficult to see that traceless elements of the braided Lie algebra gl(VR) form a subalgebra sl(VR). Moreover, if TrC ≠ 0 there exists a natural projector gl(VR) → sl(VR) similar to the classical one.

However, the restriction of the adjoint representation of the algebra gl(VR) to the subalgebra sl(VR) is not in general a representation of the latter algebra. However, by a slight modification of this restriction (described in [7], section 6) we can get an analog of the adjoint representation of the algebra sl(n).

Note that in the standard case the braided analogs of the gl(n) and sl(n) adjoint representations are equivariant. Also, note that by using the methods of [14] a map from SL(q, 1) to the QG Uq(sl(n)) (localized by the quantum Casimir element) can be constructed. Basing on this map it is possible to develop a representation theory of the algebra SL(q, 1) in the standard case.

Cayley-Hamilton identity

Let R be a skew-invertible Hecke symmetry and ≥ be a partition. Let us consider the corresponding REA or mREA and define a Schur function s≥(L) as follows

equation

Here

equation

equation is a primitive idempotent of the Hecke algebra Hn(q) corresponding to a partition λ. The symbol ρR stands for the ”local” representation of Hn(q) associating the braidings Rk k+1 with the standard generators σk ∈ Hn(q).

Theorem 5.1. The elements sλ(L) are central in the algebra L(q).

Theorem 5.2. [Cayley-Hamilton theorem] Assume that R is an even Hecke symmetry of rank p. Then the matrix equation composed of the REA generators satisfies the following Cayley- Hamilton (CH) identity

equation

Here the partition (1k) is represented by the one-column Young diagram of the height k.

Remark 5.3. In fact, this CH identity is the first one in a family of CH identities. It is called the basic identity. The other identities of the family deals with some extensions of the basic matrix L and they are called the higher identities.

Let us introduce formal elements μ1, μ2, . . . , μp such that

equation

Otherwise stated, μi are the roots of the equation

equation

So, μi belong to an algebraic extension of the center of the algebra L(q). Then the CH identity factorizes into the product

equation

Let us fix values of the elements μi and let Lμ(q) denotes the quotient of the REA over the ideal generated by the set of elements

equation

We assume the family {μ1, . . . , μp} to be generic (in particular, μi are pairwise distinct). Then the algebra Lμ(q) is a braided analog of the coordinate algebra of a generic coadjoint orbit in gl(n)+. This means that in the standard case this algebra arises from the quantization of a generic orbit in gl(n)+.

Proposition 5.4. The elements

equation

are pairwise orthogonal idempotents (projectors)

equation

Recall that the elements TrR Lk are central in the algebra L(q).

Theorem 5.5. In the algebra Lμ(q) there is a spectral decomposition

equation

Here TrR is normalized by he condition equation The quantities di are called the quantum dimensions.

Using the idempotents ei(L) we can define quantum analogs of line bundles over the quantum orbits via projective modules in the spirit of the Serre-Swan’s approach.

Note that a CH identity is also valid for the algebra L(q, equation). For this algebra the formula similar to (5.1) takes place too, but μi become now the roots of the CH identity for L(q, equation) and the quantum dimensions should be modified as follows

equation

In order to get the latter formula it suffices to replace the roots μi in (5.1) by equation

Formula (5.1) can be also considered as a parameterized relation between two families of central elements, namely, {TrR Lk}, 1 ≤ k ≤ p, and the set of coefficients of the corresponding CH identity in the algebras L(q) or L(q, equation).

Now let us pass to the Hecke symmetries of GL(p|r) type.

Theorem 5.6 ([5]). Let now R be a skew-invertible Hecke symmetry of the bi-rank (p|r). The matrix equation composed of the REA generators satisfies the following CH identity

equation where equation denotes the partition with the following Young diagram

equation

This is a q-generalization of the Kantor-Trishin’s result [13].

Theorem 5.7 ([6]). Being multiplied by s[p|r], the CH of Theorem 5.6 factorizes into the product

equation

Denoting the roots of the first and of the second factors as equation respectively, we get the following parameterizations of the normalized coefficients of these factors

equation

In terms of ”even” roots μi and ”odd” roots ºj the CH identity factorizes as follows

equation

Recently a formula similar to (5.1) has been obtained (the proof will be given in our forthcoming paper).

Theorem 5.8. Let L(q)μ,º be the quotient algebra of the GL(p|r) type REA by the relations (5.2), (5.3). In L(q)μ,º the following relations hold true

equation

where

equation

Similarly to the even case, if we replace μi (resp., ºj) by equation we get the ”quantum dimensions” di equation valid for the spectral decomposition in the algebra equationIn the modified formula μi and ºj are regarded to be the roots of the CH identity for the matrix L composed of the generators of the algebra equation

Example: q-Minkowski space and q-hyperboloid algebras

Let R be the standard Hecke symmetry in the case n = 2. In an appropriate basis {x, y} of the two dimensional space V we get the following matrix

equation

which represents this Hecke symmetry in the basis equation

Also, we put equation Then the system defining the mREA becomes equation

If equation = 0 this algebra is called the q-Minkowski space algebra [12]. For equation ≠ 0 it is regarded to be a braided or q-analog of the enveloping algebra equation

Changing the set of generators {a, d, b, c} for {l, h, b, c} where l = q−1a + qd, h = a − d

we come to the relations

equation

In this basis of generators it is seen explicitly that the element l = q−1a+qd is central. Therefore, we can introduce the quotient algebra

equation

The commutation relations among the independent generators of SL(q, equation) read

equation

Thus, we get a braided or q-counterpart of the algebra equation But in contrast with the classical case, the algebra equation is not a subalgebra of equation

Consider now the central element equation Its explicit form in the basis {l, h, b, c} reads

equation

Its image C in the algebra equation is

equation

This element is central in the algebra equation The quotient of this algebra over the ideal equation is called the quantum (or braided or q-)hyperboloid algebra

equation

On the next step we introduce the braided Lie algebra sl(VR). But in the low-dimensional case in question we can simplify the construction of the corresponding braided Lie bracket. We only use the fact that this bracket is Uq(sl(2))-covariant. We put SL = Span(b, h, c). Let us equip this space with an action of the QG Uq(sl(2)) and extend this action to the space equation by using the coproduct of the QG. Then for a generic q the space equation can be decomposed into a direct sum of irreducible Uq(sl(2)) submodules equation where the subscript stands for the spin. Then the operator

equation

is a Uq(sl(2)) morphism iff it is trivial on the components V0 and V2 and is an isomorphism between V1 and SL. By this property the bracket is uniquely defined up to a nonzero factor w. Let us exhibit the multiplication table of this braided bracket:

equation

As can be easily seen, the bracket [ , ] defines a representation of the algebra SL(q, 1) iff w = equation

Consider now q-analogs of adjoint operators. Being represented by matrices in the basis {b, h, c} they have the form

equation

Theorem 6.1. The operators Bq, Hq, Cq satisfy the relation

equation

Moreover, there exist extensions of these operators to the higher homogeneous components of the algebra SL(q) such that on each component we get a representation of the algebra SL(q, 1) and this relation is still valid for the extended operators Bq, Hq, Cq (we keep for them the same notations).

Now, we are able to define a braided analog of the space of vector fields on the classical hyperboloid. Let us consider the left A®-module generated by the operators Bq, Hq, Cq. It is possible to show that this A®-module is projective. It is natural to call it the ”tangent module” on the q-hyperboloid in question. In a similar way the ”cotangent module” on this q-hyperboloid can be introduced (it is isomorphic to the tangent one). Such braided geometrical structures on the quantum hyperboloid were first considered by one of the authors (D.G.) and P.Akueson (cf [1] and the references therein).

Note that the above vector fields are very useful for defining braided analogs of some operators of mathematical physics on the q-hyperboloid. Thus, the Laplace operator on a q-hyperboloid can be defined via the element (6.1) where the generators b, h, c should be replaced by braided vector fields Bq, Hq, Cq respectively. As a result, we get the following braided Laplace operator on the q-hyperboloid

equation

A braided version of the Dirac operator on the q-hyperboloid can be be defined a similar way. Recently a braided analog of the Maxwell operator on the q-hyperboloid was constructed. Its construction will be published in [3].

It would be very interesting to generalize these constructions and results on other ”braided varieties” of general type. The most intriguing problem is whether the module of the braided vector fields defined on these ”varieties” possesses the properties analogous to those considered above.

Acknowledgement

The work of D. G. was partially supported by the grant ANR-05-BLAN-0029-01, the work of P. P. and P. S. was partially supported by the RFBR grant 05-01-01086.

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