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.
^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
^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.
^Cite error: The named reference properties was invoked but never defined (see the help page).
Cayley–Hamilton theorem, etc. hold true for them. Any matrix with commuting elements is a Maninmatrix. These matrices have applications in representation...
Yuri Ivanovich Manin (Russian: Ю́рий Ива́нович Ма́нин; 16 February 1937 – 7 January 2023) was a Russian mathematician, known for work in algebraic geometry...
square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the...
linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Construction of Instantons." The ADHM construction uses...
worked in mathematical physics. In collaboration with his advisor Yuri Manin, he constructed the moduli space of Yang–Mills instantons, a result that...
Princeton Companion to Mathematics, Princeton University Press, p. 222 Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory...
of Manin, the obstructions to the Hasse principle holding for cubic forms can be tied into the theory of the Brauer group; this is the Brauer–Manin obstruction...
special classes of varieties, but not in general. Manin used the Brauer group of X to define the Brauer–Manin obstruction, which can be applied in many cases...
elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is...
the Malcev algebra Yuri Manin, author of the Gauss–Manin connection in algebraic geometry, Manin-Mumford conjecture and Manin obstruction in diophantine...
the form of the so-called cohomological field theories by Kontsevich and Manin), which was explored and studied systematically by B. Dubrovin and Y. Zhang...
Archived from the original on 17 February 2023. Retrieved 17 July 2023. Manin, Giuseppina (2 September 2009). "Dialetto o doppiaggio? Tornatore inaugura...
singularities, one can consider the isomonodromy equations as nonhomogeneous Gauss–Manin connections. This leads to alternative descriptions of the isomonodromy...
exponential increase in overhead when simulating quantum dynamics, prompting Yuri Manin and Richard Feynman to independently suggest that hardware based on quantum...
term was coined by John Preskill in 2012, but the concept dates to Yuri Manin's 1980 and Richard Feynman's 1981 proposals of quantum computing. Conceptually...
^{k_{1}+k_{2}}(V)\right).} It can be verified (as is done by Kostrikin and Manin) that the resulting product is in fact commutative and associative. In some...
function Mumford, David (2008). Abelian varieties. C. P. Ramanujam, I︠U︡. I. Manin. Published for the Tata Institute of Fundamental Research. ISBN 978-8185931869...
American Mathematical Society, pp. 41–97, ISBN 978-0-8218-1198-6, MR 1701597 Manin, Yuri Ivanovich (1997), Gauge Field Theory and Complex Geometry (2nd ed...
called the conductor-discriminant formula. Cohen, Diaz y Diaz & Olivier 2002 Manin, Yu. I.; Panchishkin, A. A. (2007), Introduction to Modern Number Theory...
the Malcev algebra Yuri Manin, author of the Gauss–Manin connection in algebraic geometry, Manin-Mumford conjecture and Manin obstruction in diophantine...
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