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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
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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 \(\mathrm{Lie}(n)\).

Readership

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

Table of Contents

  • 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
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