Let \(M_ n(k)\) be the algebra of \(n\times n\) matrices over a field \(k\). To any element \(R\in M_ n(k)^{\otimes 2}\) is associated a bialgebra \(A(R)\), which is generated as an algebra by \(n^ 2\) elements \(t_{ij}\), \(1\leq i,j\leq n\), with defining relations (*) \(R(T\otimes 1)(1\otimes T)= (1\otimes T)(T\otimes 1)R\), where \(T=(t_{ij})\). Under certain circumstances, one can find distinguished elements of \(A(R)\), called quantum determinants, such that the quotient of \(A(R)\) by the ideal they generate is a Hopf algebra \(H(R)\).
There is a natural action of \(H(R)\) on \(A(R)\), but, in general, this is not ‘covariant’, i.e. \(A(R)\) is not an \(H(R)\)-module algebra. To achieve covariance, the defining relations (*) must be modified – this leads to an algebra \(B(R)\) which the author calls a ‘braided group’. The general properties of \(B(R)\) are discussed, and several examples are given arising from certain standard \(R\)-matrices.