On the multiplicative complexity of Lie algebras (Q1173543)

A lower bound for the tensor rank of simple Lie algebras \(L\) of type \(A_{\ell},B_{\ell},C_{\ell}\), or \(D_{\ell}\) is derived by proving the inequality \(\text{rk}(L)\geq\dim L-1+f\), where \(f=2\) if \(L=A_ 1,B_ 2\); \(f=2\ell-2\) if \(L=A_{\ell}\), \(\ell\geq2\); \(f=2\ell-1\) if \(L=B_{\ell}\), \(\ell\geq 3\); \(f=2\ell-4\) if \(L=C_{\ell}\), \(\ell\geq 3\), \(L=D_{\ell}\), \(\ell\geq 2\). An analogous result for associative algebras was given by \textit{A. Alder} and \textit{V. Strassen} [Theor. Comput. Sci. 15, 201-211 (1981; Zbl 0464.68045)]. For direct sums of simple Lie algebras \(L_ i\), the inequality \[ \text{rk}(L_ 1\oplus\dots\oplus L_ m)\geq\sum_{i=1}^ m\dim L_ i+\sum_{i=1}^ mf_ i-1 \] is proved. \{In the translation paper the `\(\geq\)' sign has been omitted.\}.