拟洗牌代数上的Hoffman-Ihara算子和反同构

Hoffman-Ihara Operators and Anti-isomorphisms on Quasi-shuffle Algebras

Hoffman-Ihara算子是拟洗牌代数上由形式幂级数所确定的一类线性映射.本文证明了对于任意非零元μ_(1)和μ_(2),由形式幂级数f(t)所确定的Hoffman-Ihara算子,是从由字母表上任意满足结合律的乘积所定义的权为μ_(1)的拟洗牌代数到权为μ_(2)的拟洗牌代数的(反)同构当且仅当f(t)=μ_(1)/μ_(2)t(f(t)=(-μ_(1)t)/(μ_(2)+μ_(1)μ_(2)t)).

Hoffman-Ihara operators are linear maps on quasi-shuffle algebras defined by formal power series.It is proved that for any nonzero elementsμ_(1)andμ_(2),the Hoffman-Ihara operator defined by a formal power series f(t)is an isomorphism(respectively,anti-isomorphism)from a quasi-shuffle algebra with weightμ_(1)to a quasi-shuffle algebra with weightμ_(2)induced by any given associative product on any alphabet if and only if f(t)=μ_(1)/μ_(2)t(respectively,f(t)=(-μ_(1)t)/(μ_(2)+μ_(1)μ_(2)t)).