Marek Mozrzymas

Quantum deformations of conformal algebras introducing fundamental mass parameters

Abstract We consider a new class of classical r -matrices for D = 3 and D = 4 conformal Lie algebras. There r -matrices do satisfy the classical Yang-Baxter equation and as two-tensors belong to the tensor product of Borel subalgebra. In such a way we generalize the lowest order of known nonstandard quantum deformation of sl (2) to the Lie algebras sp (4) ≅ so (5) and sl (4) ≅ so (6). As an exercise we interpret nonstandard deformation of sl (2) as describing quantum D = 1 conformal algebra with fundamental mass parameter. Further we describe the D = 3 and D = 4 conformal bialgebras with deformation parameters equal to the inverse of fundamental masses. It appears that for D = 4 the deformation of the Poincare algebra sector coincides with “null plane” quantum Poincare algebra.

The super‐rotation Racah–Wigner calculus revisited

It is shown that, since the finite dimensional representations of the super‐rotation algebra are characterized by the superspin j and the parity λ of the representation space, all features of the Racah–Wigner calculus: Clebsch–Gordan coefficients, recoupling coefficients as well as the Wigner and Racah symbols depend on both j and λ. However, it is noticed that the dependence on the parities of the Wigner and Racah symbols can be factorized out into phases so that one can define parity‐independent super S3−j and S6−j symbols. The properties of these symbols are analyzed, in particular, it is shown that the S6−j symbols possess a symmetry similar to the Regge symmetry satisfied by the rotation 6−j symbols. Analytical and numerical tables of the symbols are given for the lowest values of their arguments.

Racah–Wigner calculus for the super‐rotation algebra. I

The symmetry properties and the pseudo‐orthogonality relations of the super‐rotation Clebsh–Gordan coefficients for the tensor product of two irreducible representations of the super‐rotation algebra are derived. The symmetric super‐rotation 3‐j symbol and the symmetric and invariant super‐rotation 6‐j symbol are defined, their basic properties are described, and their relations to the usual 3‐j and 6‐j symbols are given.

Clebsch–Gordan coefficients for the quantum superalgebra Uq(osp(1‖2))

The structure of the finite‐dimensional representations of the quantum superalgebra Uq(osp(1‖2)) is considered and the projection operator for this quantum superalgebra is derived. Application of the projection operator permits one to obtain an explicit analytical formula for the q analog of the Clebsch–Gordan coefficients. Pseudo‐orthogonality relations and some other properties of Clebsch–Gordan coefficients are given.

Algebraic structure of tensor superoperators for the super-rotation algebra. II

It is shown that the sets of tensor superoperators for the super‐rotation algebra can be used to build explicit bases for the representations of several superalgebras. The representations built in this way are the fundamental representations of the special linear superalgebras sl(2j+1‖2j) and of the orthosymplectic superalgebras osp(2j+1‖2j) and the (4j+1)‐dimensional representations of osp(1‖2) and sl(1‖2) (Stavraki) superalgebras. It is shown that the chain osp(1‖2)⊆osp(2j+1‖2j) or osp(2j‖2j+1) explains the existence of a series of nontrival zeros for the super‐rotation 6‐j symbol (SR6‐j symbols).

Analytical formulas for Racah coefficients and 6-j symbols of the quantum superalgebra Uq(osp(1|2))

Using the method of projection operators, analytical formulas for Racah coefficients and 6-j symbols of the quantum superalgebra Uq(osp(1|2)) are derived. The formulas obtained by this method are transformed by means of algebraic identities into symmetrical analytical formulas, the form of which are very similar to the classical formulas obtained by Racah and Regge for su(2) Racah coefficients and 6-j symbols. Symmetry properties of Uq(osp(1|2)) Racah coefficients and 6-j symbols following from these analytical formulas are studied. In particular, it is shown that, similarly to the su(2) classical case, in addition to the usual tetrahedral symmetry, 6-j symbols of the quantum superalgebra Uq(osp(1|2)) satisfy a Regge type symmetry.

Quantum deformations of conformal algebras with mass-like deformation parameters

We recall the mathematical apparatus necessary for the quantum deformation of Lie algebras, namely the notions of coboundary Lie algebras, classical r-matrices, classical Yang-Baxter equations (CYBE), Frobenius algebras and parabolic subalgebras. Then we construct the quantum deformation of D=1, D=2 and D=3 conformal algebras, showing that this quantization introduce fundamental mass parameters. Finally we consider with more details the quantization of D=4 conformal algebra. We build three classes of sl(4,C) classical r-matrices, satisfying CYBE and depending respectively on 8, 10 and 12 generators of parabolic subalgebras. We show that only the 8-dimensional r-matrices allow to impose the D=4 conformal o(4,2)≃su(2,2) reality conditions. Weyl reflections and Dynkin diagram automorphisms for o(4,2) define the class of admissible bases for given classical r-matrices.

On the superalgebras built from SO(3) tensor operators

A study is made of the superalgebra structures of the sets of SO(3) tensor operators TLM(j,j) and TLM(j,j’). It is shown that for each set of operators there exist natural splittings in two subsets that form, respectively, the even and odd parts of a superalgebra. The superalgebras built in this way are: gl(j‖j+1) for integer j, gl(j+1/2‖j+1/2) for half‐odd integer j, gl(2j+1‖2j’+1) for all j and j’, the orthosymplectic superalgebras, P(2j+1) and Q(2j+1) for all j, gl(j+j’+1‖j+j’+1) [or gl(j+j’‖j+j’+2)] for j and j’ both integers with j+j’ odd (or even) and gl(j+j’+1‖j+j’+1) for j and j’ both half‐odd integers.

Representation properties, Racah sum rule, and Biedenharn{endash}Elliott identity for U{sub q}{bold (}osp{bold (}1{vert_bar}2{bold ))}

It is shown that the universal R matrix in the tensor product of two irreducible representation spaces of the quantum superalgebra Uq(osp(1|2)) can be expressed by Clebsch–Gordan coefficients. The Racah sum rule satisfied by Uq(osp(1|2)) Racah coefficients and 6−j symbols is derived from the properties of the universal R matrix in the tensor product of three representation spaces. Considering the tensor product of four irreducible representations, it is shown that Biedenharn–Elliott identity holds for Uq(osp(1|2)) Racah coefficients and 6−j symbols. A recursion relation for Uq(osp(1|2)) 6−j symbols is derived from the Biedenharn–Elliott identity.It is shown that the universal R matrix in the tensor product of two irreducible representation spaces of the quantum superalgebra Uq(osp(1|2)) can be expressed by Clebsch–Gordan coefficients. The Racah sum rule satisfied by Uq(osp(1|2)) Racah coefficients and 6−j symbols is derived from the properties of the universal R matrix in the tensor product of three representation spaces. Considering the tensor product of four irreducible representations, it is shown that Biedenharn–Elliott identity holds for Uq(osp(1|2)) Racah coefficients and 6−j symbols. A recursion relation for Uq(osp(1|2)) 6−j symbols is derived from the Biedenharn–Elliott identity.

Representation properties, Racah sum rule, and Biedenharn–Elliott identity for Uq(osp(1|2))

It is shown that the universal R matrix in the tensor product of two irreducible representation spaces of the quantum superalgebra Uq(osp(1|2)) can be expressed by Clebsch–Gordan coefficients. The Racah sum rule satisfied by Uq(osp(1|2)) Racah coefficients and 6−j symbols is derived from the properties of the universal R matrix in the tensor product of three representation spaces. Considering the tensor product of four irreducible representations, it is shown that Biedenharn–Elliott identity holds for Uq(osp(1|2)) Racah coefficients and 6−j symbols. A recursion relation for Uq(osp(1|2)) 6−j symbols is derived from the Biedenharn–Elliott identity.It is shown that the universal R matrix in the tensor product of two irreducible representation spaces of the quantum superalgebra Uq(osp(1|2)) can be expressed by Clebsch–Gordan coefficients. The Racah sum rule satisfied by Uq(osp(1|2)) Racah coefficients and 6−j symbols is derived from the properties of the universal R matrix in the tensor product of three representation spaces. Considering the tensor product of four irreducible representations, it is shown that Biedenharn–Elliott identity holds for Uq(osp(1|2)) Racah coefficients and 6−j symbols. A recursion relation for Uq(osp(1|2)) 6−j symbols is derived from the Biedenharn–Elliott identity.