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**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.

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.

Reviewer: Alice Fialowski (Budapest)

### MSC:

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 |

### Citations:

Zbl 0656.17014
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\textit{A. Jadczyk} and \textit{D. Kastler}, Ann. Phys. 179, 169--200 (1987; Zbl 0637.17013)

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### References:

[1] | Kastler, D.; Stora, R., Lie-Cartan pairs, J. geom. phys., 2, No. 3, 1-31, (1985) · Zbl 0593.17009 |

[2] | {\scR. Jadczyk and D. Kastler}, Graded Lie-Cartan Pairs I, Rep. Math. Phys., in press. · Zbl 0656.17014 |

[3] | de Witt, B., Supermanifolds, () |

[4] | Batchelor, M., Graded manifolds and supermanifolds, () · Zbl 0539.53058 |

[5] | {\scV. Molotkov}, “Infinite-dimensional Z_{2}K-Supermanifolds, Trieste IC/84/183.” · Zbl 1362.58001 |

[6] | Rogers, A., Graded manifolds, supermanifolds and infinite-dimensional Grassmann algebras, Commun. math. phys., 105, 375-384, (1986) · Zbl 0602.58003 |

[7] | Connes, A., Non-commutative differential geometry, (), 41-144 · Zbl 0592.46056 |

[8] | {\scD. Kastler}, Introduction to A. Connes’ “Non-Commutative Differential Geometry,” Fields and Geometry Vol. XXII, Karpacz School of Theoretical Physics Lectures, (A. Jadczyk, Ed.), World Scientific, Singapore · Zbl 0867.58003 |

[9] | {\scD. Kastler}, \(Z\)/2 graded Cohomology within the differential envelope, an Introduction to A. Connes’ “Non-Commutative Differential Geometry” (Travaux en cours, Ed.), Scient. Hermann, Paris · Zbl 0662.55001 |

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