# Generalized kappa-deformed spaces, star-products, and their realizations

###### Abstract.

In this work we investigate generalized kappa-deformed spaces. We develop a systematic method for constructing realizations of noncommutative (NC) coordinates as formal power series in the Weyl algebra. All realizations are related by a group of similarity transformations, and to each realization we associate a unique ordering prescription. Generalized derivatives, the Leibniz rule and coproduct, as well as the star-product are found in all realizations. The star-product and Drinfel’d twist operator are given in terms of the coproduct, and the twist operator is derived explicitly in special realizations. The theory is applied to a Nappi-Witten type of NC space.

PACS numbers: 02.20.Sv, 02.20.Uw, 02.40.Gh

## 1. Introduction

Recently there has been a growing interest in the formulation of physical theories on noncommutative (NC) spaces. The structures of such theories and their physical consequences were studied in [1]-[7]. Classification of NC spaces and investigation of their properties, in particular the development of a unifying approach to a generalized theory suitable for physical applications, is an important problem. In order to make a contribution in this direction we analyze a Lie algebra type NC space which is a generalized version of the kappa-deformed space.

Kappa-deformed spaces were studied by different groups, from both the mathematical and physical point of view [8]-[33]. There is also an interesting connection between the kappa-deformed spaces and Doubly Special Relativity program [17], [18]. In a kappa-deformed space the noncommutative coordinates satisfy Lie algebra type relations depending on a deformation parameter . The parameter is on a very small length scale and yields the undeformed space when . Other types of NC spaces frequently studied in the literature are the canonical theta-deformed spaces where the corresponding commutation relations are given by a second rank antisymmetric tensor , see [3], [4] and references therein.

A simple unification of kappa and theta-deformed spaces was first used in the study of the Wess-Zumino-Witten model [34]. Unification of these spaces was also the starting point in the algebraic study of the time-dependent NC geometry of a six dimensional Cahen-Wallach pp-wave string backgroud [35]-[37]. In this approach the unification is achieved by adding a central element to the NC coordinates whose commutation relations are parametrized by real-valued parameters and which are assumed to be equal.

The motivation for the present work is twofold. First, we want to generalize the unified kappa and theta-deformed spaces to arbitrary dimensions, and for arbitrary values of the parameters and . Second, we want to develop a unifying approach to constructing realizations of such spaces in terms of ordinary commutative coordinates and derivatives which are convenient for physical applications. In the present work we assume that are coordinates in a Euclidean space, but the analysis can be easily extended to Minkowski space. We shall be mainly concerned with the Nappi-Witten type of NC space which arises in the study of pp-wave string background [37]. This space is made up of two copies of kappa-deformed space and one copy of theta-deformed space. Our analysis is based on the methods developed for algebras of deformed oscillators and the corresponding creation and annihilation operators [38]-[47]. The realization of a general Lie algebra type NC space in symmetric Weyl ordering was given in [48].

The outline of the paper and the summary of the main results is as follows. In Sec. 2 we introduce a generalized kappa-deformed space of Nappi-Witten type, . We study realizations of the generators of as formal power series with coefficients in the Weyl algebra. We show that there exist an infinite family of such realizations parameterized by two functions and . For special choices of and we obtain three important realizations: the right, symmetric left-right, and Weyl realization. In Sec. 3 we construct a group of similarity transformations acting transitively on the realizations. Sec. 4 deals with ordering prescriptions for . We show that to each realization one can associate an ordering prescription, and we find the prescriptions explicitly in terms of the parameter functions and . In our approach the right, symmetric left-right and Weyl realizations correspond to the time, symmetric time and Weyl orderings defined in [37], respectively. Thus the orderings found in [37] are only special cases of an infinite family of ordering prescriptions for constructed here. Furthermore, the time and symmetric time orderings can be viewed as limiting cases of an ordering prescription which interpolates between the two orderings.

In Sec. 5 we consider the problem of extending the NC space by generalized derivatives such that the extended space is a deformed Heisenberg algebra. We also define rotation operators on the extended space which generate the undeformed algebra. The generalized derivatives and rotation operators are found in all realizations of . Sec. 6 deals with Leibniz rule and coproduct for the deformed Heisenberg algebra introduced in Sec. 5. We find explicitly the coproduct depending on the parameter functions and , and we give a relation between the coproducts in different realizations. Furthermore, star-products and Drinfel’d twist operators are considered in Sec. 7. A general formula for the star-product in terms of the coproduct is given, and an expression depending on and is derived. Also, the corresponding twist operator is found explicitly for a wide class of realizations of . Finally, we describe how the obtained results generalize to higher dimensions.

## 2. Realizations of the Nappi-Witten space

Let us consider a unification of the canonical theta-deformed NC space and a Lie algebra type NC space with generators and structure constansts . Throughout the paper capital letters will be used consistently to denote NC coordinates. In order to include the theta-deformation given by a constant anytisymmetric tensor , we introduce a central element such that

(1) |

The NC space defined by the commutation relations (1) is also of Lie algebra type provided all the Jacobi identitities are satisfied:

(2) | |||

(3) |

When we obtain a Lie algebra type NC space with structure constants . Similarly, when the space reduces to the canonical theta-deformed space with the additional central element .

As an example consider a NC space with coordinates , , satisfying the commutation relations

(4) | ||||

(5) | ||||

(6) |

Here is the central element and . We shall refer to the NC space defined by (4)-(6) as the generalized Nappi-Witten space . In the special case when and , this space was recently studied by Halliday and Szabo in [37].

Without loss of generality we may assume that since all the results are easily extended to . Thus, we shall consider the NC space generated by and satisfying

(7) | ||||

(8) | ||||

(9) |

The space may be considered a generalized kappa-deformed space since (9) defines a theta-deformation while (7) and (8) define two kappa-deformations. Since is a Lie algebra, in future reference it will be denoted .

For notational ease let , and let be the ordinary commutative coordinates with the corresponding derivatives . We seek a realization of the generators of as formal power series with coefficients in the Weyl algebra generated by and . Let us consider realizations of the form

(10) |

that are linear in and is a formal power series in . We assume that there exists a dual relation

(11) |

where is also a formal power series in . A realization characterized by the functions will be called a -realization. The generators of belong to , the formal completion of . One may also consider realizations in which is placed to the right of , or any linear combination of the two types. This is convenient when one requires Hermitian realizations. Indeed, let be the Hermitian operator defined by , and . If (10) is a realization, then

(12) |

is a Hermitian realization since . Although such realizations are interesing in their own right, in this paper we shall restrict our attention to realizations of the type (10).

Let us assume the Ansatz

(13) | ||||

(14) | ||||

(15) | ||||

(16) |

where , and the functions and satisfy the boundary conditions and , finite. The Ansatz (13)-(16) is of a fairly general nature leading to a number of interesting realizations discussed below. The boundary conditions ensure that in the limit the NC coordinates become the commutative coordinates .

Let us analyze the realization (13)-(16). Define to be the antilinear map given by , , and which also preserves the Lie bracket, . Then acts as a formal conjugation, and . The action of on the generators of is defined in the obvious way: , , and , , , and . The condition holds if and only if is an even function, as seen from Eq. (14). The commutation relations (7)-(9) imply that the functions and are constrained by the system of equations

(17) | ||||

(18) | ||||

(19) |

where the prime denotes the differentiation with respect to . It is convenient to introduce the auxiliary function

(20) |

Then Eq. (18) implies that is odd and, furthermore, if and only if . For a given choice of and one can uniquely determine the remaining functions and . Therefore, the Lie algebra admits infinitely many realizations parameterized by and satisfying and .

Now we turn our attention to special realizations of : the right realization, symmetric left-right and Weyl realization. As noted in the introduction to every realization one can associate an ordering prescription on the universal enveloping algebra . The aforementioned realizations correspond to the time ordering, symmetric time ordering and Weyl symmetric ordering discussed in [37], respectively.

### 2.1. Special realizations

#### 2.1.1. Special realization

This subsection deals with the realization (13)-(16) when is a constant function, , and . For this choice of the parameters Eqs. (17)-(20) imply that and . Hence, the realization is given by

(21) | ||||

(22) | ||||

(23) | ||||

(24) |

Of particular interest are the realizations with , and :

Right realization:

(25) | ||||

(26) | ||||

(27) | ||||

(28) |

Symmetric left-right realization:

(29) | ||||

(30) | ||||

(31) | ||||

(32) |

Left realization:

(33) | ||||

(34) | ||||

(35) | ||||

(36) |

These realizations will be considered later in more detail when we establish a connection between realizations and ordering prescriptions.

#### 2.1.2. Weyl Realization

The Ansatz (13)-(16) also includes the so-called Weyl realization of which corresponds to the symmetric Weyl ordering on . In this ordering all monomials in the basis of are completely symmetrized over all generators of .

To this end we recall the following general result proved in [48]. Consider a Lie algebra over with generators and structure constants satisfying

(37) |

The Lie algebra (37) can be given a universal realization in terms of the commutative coordinates and derivatives , , as follows. Let denote the matrix of differential operators with elements

(38) |

and let be the generating function for the Bernoulli numbers. Then, one can show that the generators of the Lie algebra (37) admit the realization

(39) |

This is called the Weyl realization since it gives rise to the symmetric Weyl ordering on the enveloping algebra of (37).

We shall use the result (39) in order to obtain the Weyl realization of the Lie algebra . Recall that the generators of are arranged as ; hence the structure constants can be gleamed off from Eqs. (7)-(9). Then Eq. (38) yields

(40) |

The explicit form of the matrix can be found from the identity

(41) |

One can show by induction that , , where , and

(42) |

Expanding Eq. (41) into Taylor series and collecting the terms with even powers of we obtain

(43) |

where we have defined

(44) |

(45) |

Substituting Eq. (45) into Eq. (39) and simplifying, we obtain the Weyl realization

(46) | ||||

(47) | ||||

(48) | ||||

(49) | ||||

(50) |

It is readily seen that the above realization is a special case of the original Ansatz (13)-(16) where

(51) | ||||

(52) | ||||

(53) | ||||

(54) | ||||

(55) |

As required, these functions can be shown to satisfy the compatibility conditions (17)-(19).

## 3. Similarity transformations

In this section we discuss similarity transformations which connect different realizations

(56) |

The transformations act in a covariant way in the sense that the transformed realization is of the same type. These transformations can be used to generate new realizations of and new ordering prescriptions on . They also relate the star-products and coproducts in different realizations, as discussed in Secs. 6 and 7.

Let denote the Weyl algebra generated by and , . Consider a differential operator of the form

(57) |

where is a formal power series of . We assume that . Since the commutator of any two elements of the form is again of the same form, it follows from the Baker-Campbell-Hausdorff (BCH) formula that the family of operators is a group under multiplication, with identity when for all . To each operator we associate a similarity transformation , . The transformations form a subgroup of the group of inner automorphisms of .

Let us examine the action of on the generators of . If we denote , then

(58) |

By induction one can show that

(59) |

where the functions are defined recursively by

(60) | ||||

(61) |

Hence, we obtain

(62) |

where

(63) |

Similarly, the transformation of yields

(64) |

where the operator is defined by

(65) |

We note that the transformation of is given only in terms of , which write symbolically as

(66) |

The inverse transformations of and are of the same type,

(67) |

The functions and are related through the commutation relations for and . Substituting Eqs. (62) and (66) into the commutator we find

(68) |

Define the vector and matrix . Then Eq. (68) implies that

(69) |

where

(70) |

is the Jacobian of .

To prove the covariance of the realization (56) under the action of consider

(71) |

Using Eq. (67) the above expression becomes

(72) |

where

(73) |

Let us introduce the new variables and (which also generate the Weyl algebra ). Then Eq. (72) yields

(74) |

proving that the realization (56) is covariant under the change of variables and . Thus, the similarity transformation maps the -realization (56) to -realization (74).

As an example, consider the right realization (25)-(28). It can be shown that the operator mapping the right realization to the general Ansatz (13)-(16) (parameterized by and ) is given by

(75) |

Direct calculation yields

(76) | ||||

(77) | ||||

(78) | ||||

(79) |

Now, the functions can be calculated from Eq. (69). The group of transformations acts transitively since any two realizations are related by where maps the right realization to the -realization.

## 4. Generalized orderings

When considering the NC space only three ordering prescriptions have been used in [37] for the construction of the corresponding star-products: time ordering, symmetric time ordering and symmetric Weyl ordering. The time ordering is defined by

(80) |

where we have denoted and is the Euclidean space scalar product . Since is the central element the position of is irrelevant. Here we consider only the Euclidean space, but the theory can be easily generalized to spaces with other signatures, e.g. the Minkowski space. The symmetric time and symmetric Weyl orderings are defined by

(81) |

and

(82) |

respectively. We note that the orderings are determined by the position of in the monomial basis of . For illustration, the monomials of order three (modulo ) in the time ordering are

(83) |

The corresponding monomials in the symmetric time ordering and symmetric Weyl ordering are given by

(84) |

and

(85) |

respectively. In future reference the time ordering and symmetric time ordering will be called the right ordering and symmetric left-right ordering, respectively, as implied by the position of in the monomial basis.

In this section we show that to each realization (13)-(16) of the generators of one can associate an ordering prescription on . This leads to an infinite family of ordering prescriptions parameterized by the functions and . In our approach the orderings used in [37] appear as special cases corresponding to the right, symmetric left-right and Weyl realization found in Sec. 2.

Let us begin by defining the “vacuum” state

(86) |

Let denote the generator in -realization (56), and let denote the Weyl realization of . It has been shown in [25] and [27] that for kappa-deformed spaces a simple relation holds,

(87) |

Eq. (87) can be generalized to any -realization by requiring

(88) |

for some function . Let be the similarity transformation mapping the -realization (56) to the symmetric Weyl realization

(89) |

where the variables and are given by Eqs. (62) and (66). In this realization we have

(90) |

hence . Since , it follows that which implies

(91) |

Thus, is completely determined by the similarity transformation mapping the -realization to the Weyl realization.

For each -realization we define the -ordering by

(92) |

where is given by Eq. (91). If is represented in -realization, then

(93) |

The above expression gives a simple relation between a -realization and -ordering. The monomial basis for in -ordering can be explicitly derived from Eq. (92). Let be a multi-index with , and let

(94) |

A basis element of order is given by

(95) |

Since , is a polynomial of degree . In the Weyl realization when , Eq. (95) leads to the Weyl ordering whereby the polynomials are completely symmetrized over the generators of .

Let us illustrate the above ideas by computing an ordering prescription for the NC space . For a general realization parameterized by and we have

(96) |

One can use Eqs. (76)-(79) to find the similarity transformation mapping the -realization to Weyl realization. Then, Eq. (91) yields

(97) | ||||

(98) | ||||

(99) | ||||

(100) |

where and are the parameter functions defined by Eqs. (51) and (55). Thus, Eqs. (96)-(100) define an infinite family of orderings on depending on the parameter functions and .

Of particular interest is the realization , in which case

(101) |

In this realization the function becomes

(102) |

It can be shown that the ordering induced by this realization can be written in exponential form as

(103) |

where the central element has been left out. The above ordering has three interesting cases: the right ordering for , left ordering for