Matches in DBpedia 2016-04 for { ?s ?p "In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave like matrices whose elements commute. In particular there is natural definition of the determinant for them and most linear algebra theorems like Cramer's rule, Cayley–Hamilton theorem, etc. hold true for them. Any matrix with commuting elements is a Manin matrix. These matrices have applications in representation theory in particular to Capelli's identity, Yangian and quantum integrable systems.Manin matrices are particular examples of Manin's general constructionof \"non-commutative symmetries\" which can be applied to any algebra.From this point of view they are \"non-commutative endomorphisms\"of polynomial algebra C[x1, ...xn].Taking (q)-(super)-commuting variables one will get (q)-(super)-analogsof Manin matrices, which are closely related to quantum groups. Maninworks were influenced by the quantum group theory.He discovered that quantized algebra of functions Funq(GL)can be definedby the requirement that T and Tt are simultaneouslyq-Manin matrices.In that sense it should be stressed that (q)-Manin matrices are definedonly by half of the relations of related quantum group Funq(GL), and these relations are enough for many linear algebra theorems."@en }
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- Manin_matrix abstract "In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave like matrices whose elements commute. In particular there is natural definition of the determinant for them and most linear algebra theorems like Cramer's rule, Cayley–Hamilton theorem, etc. hold true for them. Any matrix with commuting elements is a Manin matrix. These matrices have applications in representation theory in particular to Capelli's identity, Yangian and quantum integrable systems.Manin matrices are particular examples of Manin's general constructionof \"non-commutative symmetries\" which can be applied to any algebra.From this point of view they are \"non-commutative endomorphisms\"of polynomial algebra C[x1, ...xn].Taking (q)-(super)-commuting variables one will get (q)-(super)-analogsof Manin matrices, which are closely related to quantum groups. Maninworks were influenced by the quantum group theory.He discovered that quantized algebra of functions Funq(GL)can be definedby the requirement that T and Tt are simultaneouslyq-Manin matrices.In that sense it should be stressed that (q)-Manin matrices are definedonly by half of the relations of related quantum group Funq(GL), and these relations are enough for many linear algebra theorems.".