Is a Superalgebra with Only Three Basis Possible?

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SUMMARY

A superalgebra with only three basis elements, specifically c, a, and b, is not feasible due to inconsistencies with the Jacobi identity. In this configuration, c is defined as even, while a and b are odd, leading to the relations [c,a]=a, [c,b]=-b, and [a,b]=c. The failure to satisfy the Jacobi identity confirms that such a superalgebra cannot exist.

PREREQUISITES
  • Understanding of superalgebras and their properties
  • Familiarity with the Jacobi identity in algebra
  • Knowledge of even and odd elements in algebraic structures
  • Basic concepts of Lie brackets and their applications
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  • Study the properties of superalgebras in depth
  • Research the Jacobi identity and its implications in algebra
  • Explore examples of superalgebras with more than three basis elements
  • Investigate the role of even and odd elements in algebraic structures
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The discussion is beneficial for mathematicians, theoretical physicists, and algebra students interested in the structure and properties of superalgebras.

johnstrass
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Why isn't there a superalgebra having only three basis, c, a,b where c is even and a,b are odd and [c,a]=a, [c,b]=-b, [a,b]=c?
 
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I am so stupid. Too simple. It is inconsistent with the Jacobi id.
 

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