We formulate the algebraic version of recoupling theory suitable for commutation quantization over any gradation. This gives a generalization of graded Lie algebra. Underlying this is the new notion of an -algebra defined in this paper. -algebra is a generalization of algebra that goes beyond nonassociativity. We construct the universal enveloping -algebra of recoupling Lie algebras and prove a generalized Poincaré–Birkhoff–Witt theorem. As an example we consider the algebras over an arbitrary recoupling of graded Heisenberg Lie algebra. Finally we uncover the usual coalgebra structure of a universal envelope and substantiate its Hopf structure.
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