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Given any superassociative algebra, a Lie superalgebra with a Lie superbracket $[a,b]=ab- (-1)^{|a||b|} ba$ is constructed and satisfies

  • $[a,b]=-(-1)^{|a||b|}[b,a]$
  • $[a,[b,c]] = [[a,b],c] + (−1)^{|a||b}[b,[a,c]].$

Let we have a parity reversion functor which changes the parity of the components of superspace. We consider the odd Lie bracket of parity $\epsilon$ such that $|[a,b]|=|a|+|b|+ \epsilon $ (mod 2), then how can it be shown that the odd case is reduced to the even?

Nil
  • 1,336
  • I'm not sure I understand what you're asking. The parity reversion functor isn't monoidal, so it doesn't send superalgebras to superalgebras or Lie superalgebras to Lie superalgebras. – Qiaochu Yuan Aug 18 '20 at 06:20

0 Answers0