2018/02/14 by Gerstenhaber, Murray
#FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1802.04919
Universal Deformation Formulas (UDFs) for the deformation of associative algebras play a key role in deformation quantization. Here we present examples for certain classes of infinitesimals. A basic representable 2-cocycle F of an associative algebra \mathcal A is one for which there exist commuting derivations D,…, Dn of \mathcal A such that F = ∑ijaijDi \smile Dj, where the aij are central elements of \mathcal A. When \mathcal A is defined over the rationals, there is a natural definition of the exponential of such a cocycle. With this exp ℏ F defines a formal one-parameter family of deformations of \mathcal A, where ℏ is a deformation parameter. The rational quantization of smooth functions on a smooth manifold using a bivector field as an infinitesimal deformation is a special case.