Document Zbl 1386.55021 - zbMATH Open

Let \(p\) be an odd prime and \(\mathcal{Q}(p)\) be the universal Steenrod algebra (see [M. Brunetti et al., Manuscr. Math. 118, No. 3, 271–282 (2005; Zbl 1092.55013), A. Ciampella and L. A. Lomonaco, Commun. Algebra 32, No. 7, 2589–2607 (2004; Zbl 1061.55014)]). In [Bol. Soc. Mat. Mex., III. Ser. 23, No. 1, 487–500 (2017; Zbl 1376.55014)], the authors proved that no length-preserving strict monomorphisms turn out to exist in \(\mathcal{Q}(p)\). This implies that, unlike the \(p=2\) case, \(\mathcal{Q}(p)\) does not have a fractal structure that preserves the length of monomials, but it did not exclude the existence of fractal structures for proper subalgebras of \(\mathcal{Q}(p)\).
The main result of this paper is that there exist two subalgebras \(\mathcal{Q}^{\epsilon}\), \(\epsilon \in \{0, \; 1\}\), and a chain \(\{\mathcal{Q}^{\epsilon}_{i}, \; i \in \mathbb{N}\}\) of nested subalgebras of \(\mathcal{Q}(p)\) such that: \(\mathcal{Q}^{\epsilon}_{i} \cong \mathcal{Q}^{\epsilon},\; i \geq 0\) as length-graded algebras.

MSC:

55S10	Steenrod algebra
13A50	Actions of groups on commutative rings; invariant theory

References:

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