New Lie-algebraic and quadratic deformations of Minkowski space from twisted Poincaré symmetries

Abstract

We consider two new classes of twisted D=4 quantum Poincaré symmetries described as the dual pairs of noncocommutative Hopf algebras. Firstly we investigate a two-parameter class of twisted Poincaré algebras which provide the examples of Lie-algebraic noncommutativity of the translations. The corresponding associative star-products and new deformed Lie-algebraic Minkowski spaces are introduced. We discuss further the twist deformations of Poincaré symmetries generated by the twist with its carrier in Lorentz algebra. We describe corresponding deformed Poincaré group which provides the quadratic deformations of translation sector and define the quadratically deformed Minkowski space-time algebra.

Yang-Baxter deformations of Minkowski spacetime

“…It would be interesting to consider what happens to the Lax pairs and to study the associated algebras. In particular, deformed Poincaré algebras are studied in [90][91][92] in terms of classical r-matrices. It would be very interesting to clarify the correspondence between the list in [90][91][92] and our results.…”

“…In particular, deformed Poincaré algebras are studied in [90][91][92] in terms of classical r-matrices. It would be very interesting to clarify the correspondence between the list in [90][91][92] and our results. 13 It would be most important to generalize our argument to 10D Minkowski spacetime in order to extend our argument to a consistent string theory.…”

Twists of quantum groups and noncommutative field theory

“…There are many possible commutation relations (CR) for x µ , x ν with the right hand side linear or quadratic in x µ (see [12,13]). However, those written above follow from a special limit of string theory [14] and have attracted substantial interest.…”

“…In case that (12) is true, one can extend the Hopf algebra by adding the square root of the element u that was introduced in (9). It is straightforward that the composition of the coboundary twist with the element ∆(u − 1 2 )(u 1 2 ⊗u 1 2 ) and successive twist with the element (u − 1 2 ⊗u − 1 2 )F −1 (u 1 2 ⊗u 1 2 ) obeys (13). This double transformation is carried out by means of the 2-cocycle ∆(u − 1 2 )F −1 (u 1 2 ⊗u 1 2 ), and the required property (13) readily follows from (12) and the identity (u⊗u)τ (γ ⊗γ)(F −1 ) = F ∆(u) fulfilled for any solution to the twist equation [16] (the element u is exactly the same as the one taking part in the definition of the twisted antipode (9)).…”

$ N = \frac{1}{2} $ deformations of chiral superspaces from new quantum Poincaré and Euclidean superalgebras

“…The basic example of a twisted Poincaré deformation is provided by the canonical (Moyal-Weyl) twist [9]- [12] which preserves the constant values of the commutator of noncommutative Minkowski coordinates (for examples of other Poincaré twists providing more general covariant noncommutative space-times see e.g. [17]). We add that twisting of Poincaré symmetries was used for obtaining the quantum covariant formulation of noncommutative field theories (see e.g.…”