These matrices have applications in representation theory in particular to Capelli's identity, Yangian and quantum integrable systems.
From this point of view they are "non-commutative endomorphisms" of polynomial algebra C[x1, ...xn].
He discovered that quantized algebra of functions Funq(GL) can be defined by the requirement that T and Tt are simultaneously q-Manin matrices.
Matrices with generic noncommutative elements do not admit a natural construction of the determinant with values in a ground ring and basic theorems of the linear algebra fail to hold true.
On the other hand, if one considers certain specific classes of matrices with non-commutative elements, then there are examples where one can define the determinant and prove linear algebra theorems which are very similar to their commutative analogs.
Manin matrices is a general and natural class of matrices with not-necessarily commutative elements which admit natural definition of the determinant and generalizations of the linear algebra theorems.
A rectangular matrix M is called a Manin matrix if for any 2×2 submatrix, consisting of rows i and k, and columns j and l: the following commutation relations hold Below are presented some examples of the appearance of the Manin property in various very simple and natural questions concerning 2×2 matrices.
The general idea is the following: consider well-known facts of linear algebra and look how to relax the commutativity assumption for matrix elements such that the results will be preserved to be true.
Where detcolumn of 2×2 matrix is defined as ad − cb, i.e. elements from first column (a,c) stands first in the products.
Manin's ideology one can associate to any algebra certain bialgebra of its "non-commutative symmetries (i.e. endomorphisms)".
Manin matrices considered here are examples of this general construction applied to polynomial algebras C[x1, ...xn].
Many concepts of geometry can be respelled in the language of algebras and vice versa.
Also maps from a space to itself can be composed (they form a semigroup), hence a dual object Fun(G) is a bialgebra.
Then it is coassociative and is compatible with coaction on the polynomial algebra defined in the previous proposition.
The following gives an example of a Manin matrix which is important for Capelli identities: One can replace X, D by any matrices whose elements satisfy the relation: Xij Dkl - Dkl Xij = δikδkl, same about z and its derivative.
Calculating the determinant of this matrix in two ways: direct and via Schur complement formula essentially gives Capelli's identity and its generalization (see section 4.3.1,[4] based on[5]).
The determinant of a Manin matrix can be defined by the standard formula, with the prescription that elements from the first columns comes first in the product.
In particular, the determinant can be defined in the standard way using permutations and it satisfies a Cramer's rule.
[3] MacMahon Master theorem holds true for Manin matrices and actually for their generalizations (super), (q), etc.
The only difference with commutative case is that one should pay attention that all determinants are calculated as column-determinants and also adjugate matrix stands on the right, while commutative inverse to the determinant of M stands on the left, i.e. due to non-commutativity the order is important.
This proposition is somewhat non-trivial, it implies the result by Enriquez-Rubtsov and Babelon-Talon in the theory of quantum integrable systems (see section 4.2.1[4]).
MacMahon Master theorem [6] The Capelli identity from 19th century gives one of the first examples of determinants for matrices with non-commuting elements.
Introduce a formal variable z which commute with Eij, respectively d/dz is operator of differentiation in z.
On the right hand side of this equality one recognizes the Capelli determinant (or more precisely the Capelli characteristic polynomial), while on the left hand side one has a Manin matrix with its natural determinant.
Also it gives an easy way to prove that this expression belongs to the center of the universal enveloping algebra U(gln), which is far from being trivial.
Indeed, it's enough to check invariance with respect to action of the group GLn by conjugation.
For the sake of quantum integrable systems it is important to construct commutative subalgebras in Yangian.
It is well known that in the classical limit expressions Tr(Tk(z)) generate Poisson commutative subalgebra.
The correct quantization of these expressions has been first proposed by the use of Newton identities for Manin matrices: Proposition.
But around and some after this date Manin matrices appeared in several not quite related areas:[6] obtained certain noncommutative generalization of the MacMahon master identity, which was used in knot theory; applications to quantum integrable systems, Lie algebras has been found in;[4] generalizations of the Capelli identity involving Manin matrices appeared in.