Quantum Groups and Deformed Special Relativity

Abstract

The structure and properties of possible q‐Minkowski spaces are reviewed and the corresponding non‐commutative differential calculi are developed in detail and compared with already existing proposals. This is done by stressing the covariance properties of these algebras with respect to the corresponding q‐deformed Lorentz groups as described by appropriate reflection equations. This allow us to give an unified treatment for different q‐Minkowski algebras. Some isomorphisms among the space‐time and derivative algebras are demonstrated, and their representations are described briefly. Finally, some, physical consequences and open problems are discussed.

The Dirac operator and gamma matrices for quantum Minkowski spaces

“…One should mention that formally taking k −→ 0, next c −→ 0 (in cases 2)3)4)) or t −→ 1 (in case 1)), we can deform all the objects related to the cases 1)-4) with s = 1 (and any allowed H, T ) to the classical case [ 1), s = t = 1, H = T = 0]. Therefore k, c and 1 − t can serve as small deformation parameters.…”

Twisted $h$ -spacetimes and invariant equations

“…where R (3) = PR (2) P is given in (2.8), which is preserved under the h-Lorentz coaction; they are explicitly given in [14]. The commutation relations among the entries of K and Y may be expressed by an inhomogeneous reflection equation [16,17] from the right and using the commutation relations in (2.6). Eq.…”

“…In Sec. 2 we recall the properties of the hdeformed Minkowski spacetime [12,14] following the R-matrix formalism [16,17]. The different deformed Lorentz groups and corresponding Minkowski spaces as well as their properties are directly related to a set of four R-matrices R (j) (j=1,2,3,4), solutions to the Yang-Baxter equation (YBE) and FRT-relations.…”

“…The properties of the resulting R-matrix (16×16) are derived as consequences of the corresponding properties of its components, triangularity and twisting character, and the twisting matrices are constructed. Using the exchange R-matrix and a general approach to the construction of deformed Dirac operators [17], the problem of finding the h-Dirac matrices is solved in Sec. 6.…”

Dirac operator, bicovariant differential calculus and gauge theory on -Minkowski space

“…e m = −ie P 0 /κ P m , satisfy following commutation relations with elements of M κ [e µ , x ν ] = i κ (g 0µ e ν − g µν e 0 − g µν e 4 ),[e 4 , x µ ] = − i κ e µ ,(1 11). and the additional relation ✷ κ ≡ e µ e ν = g µν e µ e ν = κ 2 + (e 4 ) 2 .(1.12)Eqs.…”