Quantum Algebra Uq Research Articles

Q-Difference intertwining operators and q-conformal invariant equations

We first recall a canonical procedure for the construction of the invariant differential operators and equations for arbitrary complex or real noncompact semisimple Lie groups. Then we present the application of this procedure to the case of quantum groups. In detail is given the construction of representations of the quantum algebra U q (sl(n)) labelled by n−1 complex numbers and acting in the spaces of functions of n(n−1)/2 noncommuting variables, which generate a q-deformed SL(4) flag manifold. The conditions for reducibility of these representations and the procedure for the construction of the q-difference intertwining operators are given. Using these results for the case n=4 we propose infinite hierarchies of q-difference equations which are q-conformal invariant. The lowest member of one of these hierarchies are new q-Maxwell equations. We propose also new q-Minkowski spacetime which is part of a q-deformed SU(2,2) flag manifold.

Three-dimensional representation of the double quantum algebrasu q(η(J)) and theq-deformation of the double complex ernst equation

A three-dimensional representation of the double quantum algebrasuq(η(J)) is given. By the use of this representation and a Lax pair, we obtain a nonlinear Ernst equation system. By the harmonic function method, a solution of theq-deformed double complex Ernst equation is given.

Q-Weyl coefficients forU q (3) andq-Racah coefficients forSU q (2)

We show that theq-Weyl coefficients of the quantum algebraSUq(3) are equal to theq-Racah coefficients of the quantum algebraSUq(2) (up to a simple phase factor). Using aq-analog of the resummation procedure we obtain also theq-analogues of all known general analytical expressions for the 6j-symbols (or the Racah coefficients) of the quantum algebraSUq(2) starting from one such formula.

Coherent States and Some Realizations of the Quantum Algebra SUq(1,1)

For the quantum algebra SUq(1,1), a q-analogue of the SU(1,1) coherent states is constructed. These coherent states are shown to satisfy a unity resolution relation. With the help of coherent states the q-deformed Dyson and Holstein–Primakoff realizations of the quantum algebra SUq(1,1) are also obtained.

The Relations Between q-Deformed Oscillator and Fermionic Oscillator11The project supported partially by the Grant LWTZ-1298 of the Chinese Academy of Sciences.

We study the relevance between the statistical property of q-deformed oscillator and the value of q. It is shown that there is the consistency condition |q| = 1;. When q is not a real number. The q-Fock space is reconstructed so that it is applicable to the case in which q is extended to the complex number. And the general expression of q-deformed oscillator bq and in the new q-Fock space is given. It is found that q = ±i correspond to fermionic oscillator (the mix of the ordinary proper and improper one). This conclusion implies that q-deformed oscillator relates naturally and inherently the fermionic oscillator with the bosonic oscillator by choosing the value of q. Actually, the bosonization structure of the single mode of fermion can also be obtained in the way. Of course, the solution of the fermionic oscillator given in this paper is a representation of quantum algebra SUq(2).

Biorthogonal coupling coefficients of uq(n)

The coupling (Wigner-Clebsch-Gordan) coefficients and their isofactors for the unitary quantum algebras uq(n) with the repeating irreducible representations in the coproduct decomposition are considered. Generalizing the U(n) case, the biorthogonal systems of the uq(n) isofactors with the dual multiplicity labels are constructed by means of the recoupling technique in terms of the isofactors with simpler multiplicity structure. A first construction, correlated with the inverted Littlewood-Richardson rules, gives the bilinear combinations of isofactors after applying the proportionality of the q-recoupling (Racah) coefficients to the boundary q-isofactors. An alternative recursive construction gives the nonorthogonal q-isofactors satisfying the most elementary boundary conditions and proportional to the uq(n-1) recoupling coefficients for some less restricted values of parameters. Some multiplicity-free and more general uq(n) recoupling coefficients are found, the blocks (bilinear combinations) of which (equal to the resubducing coefficients of the complementary chains of q-algebras) are proposed to use for the orthonormalization of some uq(n) biorthogonal isofactors, including the general uq(3) case.

Racah coefficients and 6−j symbols for the quantum superalgebra Uq (osp(1‖2))

The tensor product of three irreducible representations of the quantum superalgebra Uq(osp(1‖2)) is studied and the super-q analogs of Racah coefficients and 6−j symbols for the quantum superalgebra Uq(osp(1‖2)) are defined. Racah coefficients and 6−j symbols depend on the superspin l and the parity λ which characterize the irreducible representations of Uq(osp(1‖2)) but the dependence on parities can be factored out so that one can define parity independent 6−j symbols. It is shown that the 6−j symbols for the quantum superalgebra Uq(osp(1‖2)) satisfy symmetry properties and orthogonality relations similar to those of the 6−j symbols for quantum algebra Uq(su(2)).

Irreducible Representations and Reduced Coefficients of Quantum Algebra SUq(4) in the SUq(4) ⊃ SUq(2) ⊕ SUq(2) Basis11The project partially supported by National Natural Science Foundation of China.

In this paper, the tensor-like method is presented, and the applications to quantum algebra SUq(4) are given. The irreducible representation of SUq(4) and a recurrent formula to calculate reduced coefficients of SUq(4) ⊃ SUq(2) ⊕ SUq(2) is obtained. Some of reduced matrix elements and q-reduced scalar factors (qRF) are tabulated.

A New Realization of Quantum Algebra SUq(2) in Terms of the Basis of Lie Algebra su(2)11The project supported in part by National Natural Science Foundation of China.

In this letter, we present a set of new formulas of three q-generators of quantum algebra SUq(2), which are the nonlinear combination of the basis elements of Lie algebra su(2). In the limit q → 1, these q-generators contract to the quantities that satisfy the commutation relation of Lie algebra su(2).

Multiplicity-free uq(n) coupling coefficients

The coupling (Wigner-Clebsch-Gordan) coefficients of the unitary quantum algebras uq(n) are considered. The tensorial properties of the generator powers and their ordered products, used in the explicit projectors and weight lowering procedures, are established. Different expressions for the multiplicity-free isofactors of uq(n) coupling coefficients (those coupling an arbitrary and symmetric representations) are derived. Explicit expression of the arbitrary isofactors in terms of their boundary values are proposed. Proportionality of the semistretched isofactors to the stretched q-recoupling coefficients of uq(n-1) (q-analogues of 9j-symbols) is demonstrated. The stretched isofactors of uq(3) are expressed in terms of Clebsch-Gordan coefficients of uq(2).

Reduced coefficients of quantum algebra SUq(4)

Irreducible representation and a recurrent formula to calculate reduced coefficients of quantum algebra SUq(4)⊇SUq(2)⊕SUq(2) has been obtained through a tensorlike method.

The massless representations of the conformal quantum algebra

The massless representations of the conformal quantum algebra Uq(su(2,2)) for complex q such that mod q mod =1 are studied in detail. By factorizing out all singular vectors of the corresponding Verma modules, a simple basis for these representations is explicitly constructed. This basis is new also for the usual conformal algebra su(2,2). This construction allows a straightforward treatment of the case q a root of unity, when the representations are unitary and finite-dimensional.

Representations of the q-deformed algebras U q(so2,1) and U q(so3,1)

Representations of the algebra Uq(so2,1) are studied. This algebra is a q-deformation of the universal enveloping algebra U(so2,1) of the Lie algebra of the group SO0(2,1) and differs from the quantum algebra Uq(su1,1). Classifications of irreducible representations and of infinitesimally unitary irreducible representations of Uq(so2,1), with simple and discrete spectrum of one of generators are derived. The set of infinitesimally unitary representations of Uq(so2,1) does not coincide with that for the algebra Uq(su1,1). The sets of irreducible representations and of infinitesimally unitary irreducible representations of the algebra Uq(so3,1) are given. We also consider representations of Uq(son,1) which are of class 1 with respect to the subalgebra Uq(son).

Q-Serre relations in Uq(un) and q-deformed meson mass sum rules

The q-Serre relations are shown to be necessary for fixing uniquely the value of deformation parameters within recently proposed applications of quantum algebras Uq(Un) in obtaining q-analogues of hadron mass sum rules. Coefficients in these q-analogues are expressed through Alexander polynomials of certain knots.

Finite dimensional irreducible representations of the quantum algebra Uq(C2)

In this article, the communtation relations of the generators of quantum algebra Uq(C2) are analyzed and irreducible q-tensor operators of rank (1/2) of quantum algebra SUq(2) are constructed. By means of the property of a q-tensor operator, finite irreducible representations of Uq(C2) can be easily obtained. Some low dimensional representations are tabulated.

Representations of the quantum algebra suq(2) on a real two-dimensional sphere

Representations of the quantum algebra suq(2) are constructed in terms of q-special functions, defined on real O(3) spheres and real planes. They include q-Vilenkin functions, related to little q-Jacobi functions, q-spherical functions, and q-Legendre polynomials.

Tensor-like technique and quantum algebra suq(3)

A new method is presented for constructing the irreducible representation and for calculating the Clebsch-Gordan coefficient of the quantum algebra suq(3). The method is different from others and does not depend on any concrete realization of suq(3). This means that the result holds in general. A set of recurrence formula has been obtained for calculating the Clebsch-Gordan coefficient. In fact the recurrence formula is aimed at the so-called scalar factor of suq(3), and means that the Racah factorization lemma also holds for the quantum algebra suq(3).

A q-deformed harmonic oscillator in a finite-dimensional Hilbert space

A q-deformed harmonic oscillator (q-HO) in a finite-dimensional Hilbert space (FDHS) is presented. Some basic characteristics of the q-HO in the FDHS is discussed. It is found that the quantum algebra suq (1,1) can be realized in terms of the q-HO in the FDHS on the root of unit.

Centralizer algebras of the mixed tensor representations of quantum group Uq(gl(n,C))*

Let G=GLn(C) be the group of linear transformations of the w-dimensional C-vector space FM. Let V* be the dual space of Vn and let Vi ':=Vf® (F?) Θ M be the (AT, M)-mixed tensor power of Vn. We denote the representation of G on Vψ' by φ(>K The decomposition of φ(* ) into a sum of irreducible representations of G is given in [8] and [13]. This result also indicates the structure of the centralizer algebra of Φ'(G). On the other hand, Jimbo [5, 6] showed that the centralizer algebra of the natural representation of the quantum algebra Uq(gl(n, C)) on Vψ' Q) is isomorphic to the Iwahori Hecke algebra if n>N and q^C is generic. In the present paper we introduce a generalization HnNM{q) of the Iwahori Hecke algebra of type A, which is defined by generators and relations. (See Section 2.) Our main result says that the algebra HnNfM{q) is semisimple and is isomorphic to the centralizer algebra Cψ-\q) of the natural representation of Uq(gl(n, C)) on Vi ' if n>N+M and q is generic. (See Theorem 6.7). To prove the above fact, we construct all the irreducible representations of HN.M(q) in Section 4, using the Bratteli diagram of the inclusions C^Hιt0(q)d #Ϊ.Ό(ϊ)C ••• aHnNt0(q)c:H n Ntl(q)c::. aH n NtM(q) as in [3,16]. We also give Markov traces of these algebras, which are related to the HOMFLY polynomial of knots and links as in [14]. (See [9].) In the special case N=M or Λf±l, the algebra H(n \q) is also studied in [2] from a different view point. Some results of this paper have been announced in [9]. The authors would line to thank Professor Noriaki Kawanaka for his useful advice and kind encouragement. They also thank Professor Kazuhiko Koike who informed results of [8] and suggested us to construct actual representations oiCψ'\q).

All real forms of Uq(sl(4;C)) and D = 4 conformal quantum algebras

The star operations and reality conditions for the complex quantum algebra Uq(sl(4;C)) providing real quantum algebras Uq(o(6-k,k)) k = 0, 1, 2, 3 and Uq(su(3,1)) are classified. Standard and non-standard star operations are considered. It appears that only four choices of real forms (one with mod q mod =1, three with q real) provide real Hopf algebra Uq(su(2,2)) approximately=Uq(o(4,2)) describing D = 4 conformal quantum algebras. The authors show that only the antipode-extended Cartan-Weyl basis of Uq(sl(4;C)) permits one to define real q-deformed D = 4 conformal algebra generators. In order to obtain the real D = 4 Weyl algebra as the Hopf subalgebra of Uq (su(2,2)) only the non-standard real forms can be employed.