If H is a connected, graded Hopf algebra, then Takeuchi’s formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellationfree formulas. We apply this idea to the Hopf algebras of polynomials, graphs, and noncommutative symmetric functions.

Explores the problem of minimizing loss of significance when a concept set is considered too complex to be communicated comprehensibly to a particular audience. In such a case the set must be transformed, possibly by collapsing some distinctions, so as to reduce the number of conceptual elements in the set by a factor of 2, 3 or more. If collapsed in an orderly manner, maintaining integrative d...

We define the orthogonal sum of quadratic pairs and we show that there is no Witt cancellation theorem for this operation in characteristic 2. 1. Introduction. Quadratic pairs on central simple algebras were defined in [5]. They play the same role for quadratic forms as involutions for symmetric or skew-symmetric bilinear forms. In particular, they can be used to define twisted orthogonal group...

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