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The product operation in B, and similarly in A, induces a graded Lie algebra (GLA) structure (here called the cup structure) on H*(B, B) and H*(B, A) (cf., e.g., Gerstenhaber [2], Nijenhuis and Richardson [6]), and i* is known to be a homomorphism of these structures. The cup structure on H*(B, B) is abelian; cf. [2]. I t is also known that H*(B, B) has another GLA structure (here called the comp structure) with respect to the reduced grading (elements of H(B, B) have reduced degree w — 1; cf. [2], [7]). The following theorem supplements this information.

A Lie product for the cohomology of subalgebras with coefficients in the quotient