Abstract
Let p be an odd prime. Investigating the existence of a fractal structure for the universal Steenrod algebra \(\mathcal Q(p)\) and the Lambda algebra leads to determine the group of their length-preserving automorhisms. Contrarily to the \(p=2\) case, no length-preserving strict monomorphism turns out to exist, and this makes reasonable to conjecture that the algebras above do not contain proper subalgebras isomorphic to themselves.
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Acknowledgments
In 2003, the first author gave a talk at the conference ATM-03 ‘Algebraic Topology in Malaga’ on the embedding of the \(E_2\)-page of the mod 2 Adams Spectral sequence into \( {{\mathrm{Ext}}}_{Q_1(2)} (\mathbb {F}_2, \mathbb {F}_2)\), where \(Q_1(2)\) denotes the mod 2 universal Steenrod algebra with an exotic augmentation (see [4, 8]). On that occasion Sam Gitler encouraged the speaker to explore the existence in \(\mathcal Q(p)\) of sequences of smaller subalgebras in order to circumvent problems arising from its ‘bigness’. Such suggestion led along the years to many insights concerning the universal Steenrod algebra, among which its koszulity (see [2, 6, 9]), and more recently the existence of a fractal structure for \(p=2\) (see [3, 11]). Some results in this paper are also inherent to that very hint. All the authors did know Sam Gitler personally, and were fascinated by his humanity and willingness in sharing knowledge.
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This research has been carried out as part of “Programma STAR”, financially supported by UniNA and Compagnia di San Paolo.
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Brunetti, M., Ciampella, A. & Lomonaco, L.A. Length-preserving monomorphisms for some algebras of operations. Bol. Soc. Mat. Mex. 23, 487–500 (2017). https://doi.org/10.1007/s40590-016-0089-7
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DOI: https://doi.org/10.1007/s40590-016-0089-7