# The Exterior Lie Algebras

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I define an exterior Lie algebra by the following axioms:

1)

$$\alpha \wedge (\beta \wedge \gamma)=(\alpha \wedge \beta)\wedge \gamma$$

2)

$$\alpha \wedge \beta = (-1)^{deg(\alpha)deg(\beta)}(\beta \wedge \alpha)$$

3)

$$[\alpha,\beta]=-[\beta,\alpha]$$

4)

$$[\alpha,[\beta,\gamma]]=[[\alpha,\beta],\gamma]+[\beta,[\alpha,\gamma]]$$

5)

$$[\alpha\wedge \beta,\gamma]=[\alpha,\beta]\wedge \gamma+(-1)^{deg(\alpha)deg(\gamma)}(\alpha \wedge [\beta,\gamma])$$

i)

$$deg(\alpha \wedge \beta)=deg(\alpha)+deg(\beta)$$

ii)

$$deg([\alpha,\beta])=deg(\alpha)+deg(\beta)$$

Is such an algebra a supersymmetric algebra?

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