Noncommutative Physics on Lie Algebras, (ℤ2) n Lattices and Clifford Algebras

Abstract

Abstract. We survey noncommutative spacetimes with coordinates being enveloping algebras of Lie algebras. We also explain how to do differential geometry on noncommutative spaces that are obtained from commutative ones via a Moyal-product type cocycle twist, such as the noncommutative torus, θ-spaces and Clifford algebras. The latter are noncommutative deformations of the finite lattice (Z 2 ) n and we compute their noncommutative de Rham cohomology and moduli of solutions of Maxwell's equations. We exactly quantize noncommutative U (1)-Yang-Mills theory on Z 2 × Z 2 in a path integral approach.

Braided cyclic cocycles and nonassociative geometry

“…Next it is known that the cotwisted comodule algebra Ω(A F ) := Ω(A) F is a DC over the algebra A F in the category of H F -Com, (see [12]). We recall that differential forms in Ω(A F ) are as the same as differential forms in Ω(A) with same coaction of H on them and the differential operator, d, remain unchanged but the product of forms has been cotwisted to…”

“…Next, as in [12] we chose the cocycle F (u i 1 v i 2 , u j 1 v j 2 ) = e ıθi 2 j 1 to gauge transform the category of Z × Z-graded vector spaces. Then the algebra C F G after cotwisting of the product has the relations v.u = e ıθ u.v, which we call algebraic noncommutative torus.…”

“…Recent applications include [10][11] [12] in the associative case and [13], in the nonassociative case.…”

Moduli of quantum Riemannian geometries on ⩽4 points

“…We extend this to inner calculi with the assumption θ * = θ so that * anticommutes with d (other conventions are also possible). For models based on groups and conjugacy classes with elements of order 2 this is naturally implemented by e * a = e a (more generally, e a −1 ), as in [6,8,7]. For the models based on Z 2 × Z 2 connectivity in Section 4 we take e * a = e a .…”

“…There has been a lot of interest over the years [1,2,3,4,5,6,7,8] in the specific application of noncommutative geometry [9] to the commutative algebra of functions on a finite set Σ (usually a finite group) in which the differential forms do not commute with functions. This provides a systematic way of handling geometry on finite lattices which, at the level of cohomology, electromagnetism and YangMills theory has already proven interesting and computable.…”