Lengths of cyclic algebras and commutative subalgebras of quaternion matrices
Given an associative unital algebra \(A\) of finite dimension over a field \(F\) and a finite set of generators \(S\) of \(A\) over \(F\), there exists a natural number \(k\) for which \(A=\sum_{i=0}^k \sum_{b_1,\dots,b_i \in S} F b_1 \cdot \dots \cdot b_i\). The minimal such \(k\) is the length of \(A\) with respect to \(S\). The general length of \(A\) is the supremum of its length with respect to \(S\) where \(S\) ranges over all finite sets of generators. For example, when \(A\) is a cyclic central simple algebra of degree \(n\), one can take \(S\) to be the set consisting of the two standard generators \(x\) and \(y\) (where \(F[x]/F\) is the cyclic extension with \(\sigma\) generating its Galois group, and \(yxy^{-1}=\sigma(x)\)), and then the length of \(A\) with respect to \(S\) is exactly \(2(n-1)\), which means its general length is at most \(2n-2\). The author uses this observation and other known results in the literature to show that the general length of \(A\) is exactly \(2(n-1)\) for \(n\leq 5\). The author also studies commutative subalgebras of the matrix algebra over \(F\), and extends some of the results to the matrix algebra over \(\mathbb{H}\), Hamilton's division ring of real quaternions.
scientific article; zbMATH DE number 7390153
scientific article