The skew field of rational functions on \(\text{GL}_ q(n,K)\)
For every field \(K\), the ring of regular functions on the quantum matrix algebra \(M_q(n,K)\) is defined by generators and relations. This ring is a \(K\)-algebra and is generated, as a \(K\)-algebra, by the elements \(\{a_{ij}\mid 1\leq i,j\leq n\}\) and \(K[q,q^{-1}]\). The variable \(q\) commutes with \(a_{ij}\), and the matrix elements are related by the conditions \(a_{ij}a_{ik}=q^{-1}a_{ik}a_{ij}\) for \(j<k\), \(a_{ki}a_{mi}=q^{-1}a_{mi}a_{ki}\) for \(k<m\), \(a_{ij}a_{km}=a_{km}a_{ij}\) for \(i<k\), \(j>m\), and \(a_{ij}a_{km}-a_{km}a_{ij}=(q^{-1}-q)a_{im}a_{kj}\) for \(i<k\), \(j<m\). We denote the ring of regular functions on the quantum matrix algebra \(M_q(n,K)\) by \(K[M_q(n,K)]\) or \({\mathfrak F}_q\). The ring \(K[\text{GL}_q(n)]\) of regular functions on the quantum group \(\text{GL}_q(n,K)\) is defined as a localization of the ring \({\mathfrak F}_q\) by the multiplicative subset generated by \(\text{det}_q\). The ring \(K[\text{GL}_q(n)]\) is a Noetherian integral domain. A Noetherian integral domain has a skew field of fractions. The skew fields of fractions of the rings \(K[\text{GL}_q(n)]\) and \({\mathfrak F}_q\) coincide. Definition. The skew field of fractions \(\text{Fract }K[\text{GL}_q(n)]\) will be called the skew field of rational functions on the quantum group \(\text{GL}_q(n,K)\) and denoted by \(K(\text{GL}_q(n))\) or \(F_q\). The main purpose of this article is to describe the skew field of fractions \(F_q\) in simpler terms. To this end, we consider algebras of twisted polynomials and their skew fields of fractions.
scientific article; zbMATH DE number 795625