# On Natural Coalgebra Decompositions of Tensor Algebras and Loop Suspensions (Memoirs of the American Mathematical Society)

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### Description

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 $\mathrm {Lie} (n)$.

### 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 Poincare-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.

### Product Details

• ISBN13: 9780821821107
• Format: Paperback
• Number Of Pages: 109
• ID: 9780821821107
• ISBN10: 0821821105

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