On star-varieties with almost polynomial growth (Q2747690)

All algebras considered in the paper are over a field \(F\) of characteristic 0, and with involution \(*\). Let \(c_n(V,*)\) denote the \(n\)-th \(*\)-codimension of the variety \(V\). The authors give a complete description of the varieties \(V\) with involution such that the sequence \(c_n(V,*)\) is polynomially bounded. Namely they prove that this happens if and only if the algebras \(G_2\) and \(M\) do not belong to \(V\). Here \(G_2=F\oplus F\) with the exchange involution while \(\dim M=4\) and a linear basis of \(M\) consists of the elements \(\{a,b,c,c^*\}\). The multiplication in \(M\) is the induced by: \(a=a^*=a^2\), \(b=b^*=b^2\), \(ac=cb=c\), \(c^*a=bc^*=c^*\); all other products being 0. (It turns out that \(M\) is related to the algebra \(UT_2\) of \(2\times 2\) upper triangular matrices; its role in the PI theory with involution is similar to that of \(UT_2\), while \(G_2\) is an analogue of the Grassmann algebra.)NEWLINENEWLINENEWLINEFinally, the authors relate the \(*\)-identities to the ordinary ones. Furthermore, they show that the \(*\)-varieties generated by \(G_2\) and by \(M\) are the only two \(*\)-varieties with almost polynomial growth, and that there is no \(*\)-variety with intermediate growth (between polynomial and exponential ones).