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A Leibniz variety with almost polynomial growth. (English) Zbl 1179.17002
Summary: Let \(F\) be a field of characteristic zero. In this paper we study the variety of Leibniz algebras \(\tilde{\mathcal V}_1\) defined by the identity \(y_1(y_2y_3)(y_4y_5)\equiv 0\). We give a complete description of the space of multilinear identities in the language of Young diagrams through the representation theory of the symmetric group. As an outcome we show that the variety \(\tilde{\mathcal V}_1\) has almost polynomial growth, i.e., the sequence of codimensions of \(\tilde{\mathcal V}_1\) cannot be bounded by any polynomial function but any proper subvariety of \(\tilde{\mathcal V}_1\) as polynomial growth.

MSC:
17A32 Leibniz algebras
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
20C30 Representations of finite symmetric groups
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