The deformation of Lie algebra ?

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SUMMARY

The deformation of a Lie algebra involves manipulating its structure to derive new commutation relations, specifically transitioning from relation (19) to (20) as outlined in the discussion. The structure of a Lie algebra is defined as a (1,2) tensor Ckij, which is antisymmetric under indices i and j, and adheres to the Jacobi identity. Deformations can yield 1-parameter analytic families of solutions, some of which may be equivalent to the original algebra through linear transformations, while others represent distinct structures.

PREREQUISITES
  • Understanding of Lie algebra structures and properties
  • Familiarity with tensor calculus concepts
  • Knowledge of the Jacobi identity in algebra
  • Experience with linear transformations in algebraic contexts
NEXT STEPS
  • Research the process of deriving commutation relations in Lie algebras
  • Study the implications of the Jacobi identity on Lie algebra deformations
  • Explore 1-parameter analytic families of solutions in algebra
  • Investigate linear transformations and their effects on algebraic structures
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Mathematicians, theoretical physicists, and students of algebra interested in advanced topics related to Lie algebras and their deformations.

Esmaeil
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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)?
 

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What’s the question? From the perspective of tensor calculus, the structure of a Lie algebra is (1,2) tensor Ckij 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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