Overexponential codimension growth of identities with involution (Q6597857)

Let \(A\) be a (not necessarily associative) algebra over a field \(F\) of characteristic 0. One of the main quantitative characterizations of the polynomial identities of \(A\) is the codimension sequence \(c_n(A)=\dim(P_n/(P_n\cap \text{Id}(A)))\), \(n=1,2,\ldots\). Here \(P_n\) is the space of multilinear polynomials of degree \(n\) in the free nonassociative algebra and \(\text{Id}(A)\) is the ideal of the polynomial identities of \(A\). For an associative PI-algebra \(A\) the asymptotic behavior of the codimension sequence is well understood and has a lot of nice properties. In particular, the sequence is exponentially bounded, i.e. \(c_n(A)\leq a^n\) for some \(a>1\). Many of these properties are shared when \(A\) is a Lie algebra although some new phenomena appear. When \(A\) is nonassociative the behavior of the codimensions may be very exotic. In particular, starting from any binary infinite periodic or Sturmian word \(w\) with slop \(\pi(w)=\alpha\in(0,1)\) in [\textit{A. Giambruno} and \textit{M. Zaicev}, Adv. Appl. Math. 95, 53--64 (2018; Zbl 1430.17003)] it was constructed an algebra \(A(w)\) such that \(c_n(A)\) grows as \((n!)^{\alpha}\). The algebra \(A(w)\) can be equipped with a \({\mathbb Z}_2\)-grading and the sequence of graded codimensions \(c_n^{\text{gr}}(A(w))\) again grows as \((n!)^{\alpha}\). \N\NThe main result of the paper under review shows that the algebra \(A(w)\) gives rise to a commutative algebra \(A(w)^+\) which can be equipped not only with a \({\mathbb Z}_2\)-grading but also with an involution \(\ast\) such that both the graded codimensions \(c_n^{\text{gr}}(A(w)^+)\) and the codimensions \(c_n^{\ast}(A(w)^+)\) for algebras with involution grow as \((n!)^{\alpha}\).