Noether's Theorem states that for every symmetry there exists a corresponding conservation law.pic.twitter.com/n9Gpaqeknt
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Yes, that is also well known in supersymmetry, introducing EL the algebra closes on-shell if we do not use the EL eqns the algebra closes off-shell provided auxiliary fields are introduced. The Noether procedure is also used to construct SUSY/SUGRA Lagrangians.
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