How can we deform a given Lie algebra? In particular, in the attachment file how can we arrive at the commutation relations (20) by starting from the commutation relation (19)?
What’s the question? From the perspective of tensor calculus, the structure of a Lie algebra is (1,2) tensor C^{k}_{ij} antisymmetric under i, j and satisfying a quadratic equation known as the Jacobi identity. The equation has very special form, so it not only has other solutions, but, as authors claim, even 1-parametric analytic families of solutions (it’s this that is usually called a deformation). Some solutions may be equivalent (up to linear transformations) to the original algebra, whereas others are not.
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