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On Natural Coalgebra Decompositions of Tensor Algebras and Loop Suspensions
Paul Selick, University of Toronto, ON, Canada, and Jie Wu, National University of Singapore, Republic of Singapore
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Memoirs of the American Mathematical Society
2000; 109 pp; softcover
Volume: 148
ISBN-10: 0-8218-2110-5
ISBN-13: 978-0-8218-2110-7
List Price: US$49 Individual Members: US$29.40
Institutional Members: US\$39.20
Order Code: MEMO/148/701

Abstract. We consider functorial decompositions of $$\Omega\Sigma X$$ in the case where $$X$$ is a $$p$$-torsion suspension. By means of a geometric realization theorem, we show that the problem can be reduced to the one obtained by applying homology: that of finding natural coalgebra decompositions of tensor algebras. We solve the algebraic problem and give properties of the piece $$A^{\mathrm{min}}(V)$$ of the decomposition of $$T(V)$$ which contains $$V$$ itself, including verification of the Cohen conjecture that in characteristic $$p$$ the primitives of $$A^{\mathrm{min}}(V)$$ are concentrated in degrees of the form $$p^t$$. The results tie in with the representation theory of the symmetric group and in particular produce the maximum projective submodule of the important $$S_n$$-module $$\mathrm{Lie}(n)$$.

Graduate students and research mathematicians interested in topology and representation theory.

• Introduction
• Natural coalgebra transformations of tensor algebras
• Geometric realizations and the proof of Theorem 1.3
• Existence of minimal natural coalgebra retracts of tensor algebras
• Some lemmas on coalgebras
• Functorial version of the Poincaré-Birkhoff-Whitt theorem
• Projective $$\mathbf{k}(S_n)$$-submodules of Lie$$(n)$$
• The functor $$A^{\mathrm{min}}$$ over a field of characteristic $$p>0$$
• Proof of Theorems 1.1 and 1.6
• The functor $$L^\prime_n$$ and the associated $$\mathbf{k}(\Sigma_n)$$-module $$\mathrm{Lie}^\prime(n)$$
• Examples
• References