Manin matrix

In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88,[1][2][3] 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 construction of "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)-analogs of Manin matrices, which are closely related to quantum groups. Manin works were influenced by the quantum group theory. He discovered that quantized algebra of functions Funq(GL) can be defined by the requirement that T and Tt are simultaneously q-Manin matrices. In that sense it should be stressed that (q)-Manin matrices are defined only by half of the relations of related quantum group Funq(GL), and these relations are enough for many linear algebra theorems.

  1. ^ Manin, Yuri (1987), "Some remarks on Koszul algebras and quantum groups", Annales de l'Institut Fourier, 37 (4): 191–205, doi:10.5802/aif.1117, Zbl 0625.58040
  2. ^ Manin, Y. (1988). "Quantum Groups and Non Commutative Geometry". Université de Montréal, Centre de Recherches Mathématiques: 91 pages. ISBN 978-2-921120-00-5. Zbl 0724.17006.
  3. ^ Cite error: The named reference properties was invoked but never defined (see the help page).