Riemann surface, 29 addition theorem for sn(u), 28 additive number theory, 15 algebraic function, 29 balanced hypergeometric series, 18 Bell number, 17, 107 Bernoulli counting scheme, 111 Bernoulli number, 17 Bernoulli polynomial, 17 beta integral, 19 binomial series, 18 branched covering map, 28 branchpoint, 29 Catalan number, 16 Chu-Vandermonde summation formula, 19 commutative ordinary differential operators, 117 complete symmetric polynomial, 12 completely multiplicative, 15 complex structure, 29 conformally equivalent, 29 conjugate partition, 5 contiguous relation, 20 Dedekind sum, 107 digamma function, 21 Dirichlet region, 26 double gamma function, 27 Eisenstein series, 26 elementary symmetric polynomial, 12 elliptic function, 26 elliptic integral, 25 Euler number, 22 Euler product, 15 Euler’s dilogarithm, 19 Euler’s transformation formula, 19 Euler-Maclaurin summation formula, 15, 21 Eulerian number, 17, 107 Fermat measure, 53 Fermat’s last theorem, 27 fractional differentiation, 18 function element, 28 fundamental region, 26 Galois field, 65, 121 Gauss second summation theorem, 21 Gauss summation formula, 20 Gegenbauer polynomial, 51 generalised Stirling number, 16 generalised Stirling polynomial, 16 generalized hypergeometric series, 18 generalized Laguerre polynomial, 22 generalized Vandermonde determinant, 125 Genocchi number, 17

The history of q-calculus and a new method

Theq-differential calculus for theq-Minkowski space is developed. The algebra of theq-derivatives with theq-Lorentz generators is found giving theq-deformation of the Poincare algebra. The reality structure of theq-Poincare algebra is given. The reality structure of theq-differentials is also found. The real Laplacian is constructed. Finally the comultiplication, counit and antipode for theq-Poincare algebra are obtained making it a Hopf algebra.

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q-Deformed Poincaré algebra

We investigate the structure of the tensor product representation of the quantum groupSLq(2,C) by using the 2-dimensional quantum plane as a building block. Two types of 4-dimensional spaces are constructed applying the methods used in twistor theory. We show that the 4-dimensional real representation ofSLq(2,C) generates a consistent non-commutative algebra, and thus it provides a quantum deformation of Minkowski space. The transformation of this 4-dimensional space gives the quantum Lorentz groupSOq(3, 1).

Tensor representation of the quantum group SL q (2,C) and quantum Minkowski space

Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalization of symmetry groups for certain integrable systems, and on the other as part of a generalization of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of space–time, in particular, amounts to a postulated new force or physical effect called cogravity.

Quantum groups and noncommutative geometry

The q‐Poincare group of M. Schlieker et al. [Z. Phys. C 53, 79 (1992)] is shown to have the structure of a semidirect product and coproduct B× SOq(1,3) where B is a braided‐quantum group structure on the q‐Minkowski space of four‐momentum with braided‐coproduct Δ_p=p⊗1+1⊗p. Here the necessary B is not a usual kind of quantum group, but one with braid statistics. Similar braided vectors and covectors V(R’), V*(R’) exist for a general R‐matrix. The abstract structure of the q‐Lorentz group is also studied.

Braided momentum in the q-poincare group

We examine the properties of the quantum Lorentz group SOq(3, 1) using the matrix given in Ref. 14. We show that this matrix together with the q-deformed metric C provide a representation of a BWM algebra. Using the projection operators which decompose the matrix into irreducible components, we give the general definition of the corresponding quantum space, i.e. the q-deformed Minkowski space and the q-deformed Clifford algebra. We also construct the q analog of Dirac matrices and show that they form a matrix representation of the q-deformed Clifford algebra.

A quantum lorentz group

We construct a differential calculus on theN-dimensional non-commutative Euclidean space, i.e., the space on which the quantum groupSO q (N) is acting. The differential calculus is required to be manifestly covariant underSO q (N) transformations. Using this calculus, we consider the Schrodinger equation corresponding to the harmonic oscillator in the limit ofq→1. The solution of it is given byq-deformed functions.

SO-q(N) covariant differential calculus on quantum space and quantum deformation of Schrodinger equation

An outline of a quantum group covariant spinor calculus is presented. The use of the calculus is demonstrated, for example, by setting up the commutation relations of coordinates of differentq-planes.

M. Schlieker

Papers

Tensor representation of the quantum group SL q (2,C) and quantum Minkowski space

A quantum lorentz group

SO-q(N) covariant differential calculus on quantum space and quantum deformation of Schrodinger equation

Spinor calculus for quantum groups