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By Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards

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It is easy to verify that G2 preserves Conf C n;m ; easy compuC tations also show that G2 acts freely on Conf n;m precisely when 2n C m 2. In this C case, we may take the quotient space Conf C n;m =G2 , which will be denoted by Cn;m : in fact, we will refer to it, rather than to Conf C n;m , as to the configuration space of n points in HC and m points in R. It is also a real manifold of dimension 2n C m 2. 2. We will not be too much concerned about orientations of configuration C spaces; anyway, it is still useful to point out that Cn;m is an orientable manifold.

These weights are defined as integrals over suitable configuration spaces of points in the upper half-plane. We therefore introduce these configuration spaces, and also their compactifications à la Fulton–MacPherson, which ensure that the integral weights truly exist. Furthermore, the algebraic identities illustrated in Chapters 7 and 8 follow from factorization properties of these integrals, which in turn rely on Stokes’ Theorem: thus, we need to discuss the boundary stratification of the compactified configuration spaces.

1/jajjbj ba 28 4 Superspaces and Hochschild cohomology is a graded Lie algebra. A/ of super derivations of A: a degree k graded linear map d W A ! a/b C . A/ of (degree non-preserving) linear maps A ! A (with product the composition). 8. Let g be a graded Lie algebra. 1. A graded g-module is a graded vector space V with a degree 0 graded linear map g ˝ V ! y v/ . x v/ D Œx; y v: In other words it is a morphism g ! V / of graded Lie algebras. 2. A/. In this case A is called a g-module algebra. 2 Hochschild cohomology strikes back Hochschild cohomology of a graded algebra.

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