Graded Lie-Cartan pairs. II: The fermionic differential calculus. (English) Zbl 0637.17013

In a previous paper of the authors [Rep. Math. Phys. 25, No. 1, 1–51 (1988; Zbl 0656.17014)], the Lie-Cartan pairs, describing the algebraic features of the classical operators of differential geometry, are generalized to the case of “anticommuting variables”.
In this paper the authors present another variant of that formalism, which also gives another proof of the previous results. Let \(L\) be a Lie superalgebra and \(A\) a graded commutative algebra. In Section 2 the graded Lie-Cartan pairs \((L,A)\) and their \(E\)-connections are defined, where \(E\) is a graded unital \(A\)-module. The classical operators attached to a given \(E\)-connection are introduced in Section 3; their property of preserving graded alternation, their “locality properties” for a local connection, and the classical identities which they fulfil are deduced.
In Section 4 the particular case \(E=A\) is studied. It turns out that the sets \(\Lambda^*(L,A)\) of graded alternate forms are graded commutative bigraded differential algebras and the classical operators are graded derivatives of those algebras. In Section 5, for general \(E\)-valued case, the module-derivation properties of the \(\Lambda^*(L,A)\)-valued classical operators are reduced to the derivation properties of the \(\Lambda^*(L,A)\)-valued ones. With the help of these, the identities between classical operators are proved.


17B70 Graded Lie (super)algebras
58A50 Supermanifolds and graded manifolds
53C80 Applications of global differential geometry to the sciences
53C05 Connections (general theory)
17A70 Superalgebras


Zbl 0656.17014
Full Text: DOI


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