In mathematics, Manin matrices, named after
Yuri Manin
Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logi ...
who introduced them around 1987–88,
are a class of
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with elements in a not-necessarily
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, which in a certain sense behave like matrices whose elements commute. In particular there is natural definition of the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
for them and most
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
theorems like
Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...
,
Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies it ...
, etc. hold true for them. Any matrix with commuting elements is a Manin matrix. These matrices have applications in
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
in particular to
Capelli's identity
In mathematics, Capelli's identity, named after , is an analogue of the formula det(''AB'') = det(''A'') det(''B''), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra \mathfrak_n ...
,
Yangian In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians first appeared in physics in the work of Ludvig Faddeev and his school in the late 1970s and early 1980s concerning the quantum inverse ...
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''
1, ...''x''n">'x''1, ...''x''n
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
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras) ...
theory.
He discovered that quantized algebra of functions ''Fun
q(GL)'' can be defined by the requirement that ''T'' and ''T
t'' 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 ''Fun
q(GL)'', and these relations are enough for many linear algebra theorems.
Definition
Context
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. There are several modifications of the determinant theory:
Dieudonné determinant In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by .
If ''K'' is a division ring, then the Dieudonné determinant is a homomorphism ...
which takes values in the
abelianization
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
''K''
*/
*, ''K''*">'K''*, ''K''*of the multiplicative group ''K''
* of ground ring ''K''; and theory of
quasideterminant In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows:
:
\left, \begin
a_ & a_ \\
a_ & a_ \end
\_ = a_ - ...
s. But the analogy between these determinants and commutative determinants is not complete. 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. Examples include: quantum groups and q-determinant; Capelli matrix and
Capelli determinant
In mathematics, Capelli's identity, named after , is an analogue of the formula det(''AB'') = det(''A'') det(''B''), for certain matrices with noncommuting entries, related to the Lie algebra representation, representation theory of ...
; super-matrices and
Berezinian In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when considerin ...
.
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.
Formal definition
An ''n'' by ''m'' matrix ''M'' with entries ''M
ij'' over a ring ''R'' (not necessarily commutative) is a Manin matrix if all elements in a given column commute and if for all ''i'',''j'',''k'',''l'' it holds that
''ij'',''M''''kl''">'M''''ij'',''M''''kl''=
''kj'',''M''''il''">'M''''kj'',''M''''il'' Here
'a'',''b''denotes (''ab'' − ''ba'') the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
of ''a'' and ''b''.
The definition can be better seen from the following formulas.
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
:
:
Ubiquity of 2 × 2 Manin matrices
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. The answer is: if and only if ''M'' is a Manin matrix.
[ The proofs of all observations is direct 1 line check.
Consider a 2×2 matrix
Observation 1. Coaction on a plane. ]
Consider the polynomial ring ''C'' 1, ''x''2">'x''1, ''x''2 and assume that the matrix elements ''a'', ''b'', ''c'', ''d'' commute with ''x''1, ''x''2.
Define ''y''1, ''y''2 by
:
Then ''y''1, ''y''2 commute among themselves if and only if ''M'' is a Manin matrix.
Proof:
:
Requiring this to be zero, we get Manin's relations.
Observation 2. Coaction on a super-plane.
Consider the Grassmann algebra ''C'' 1, ''ψ''2">'ψ''1, ''ψ''2 and assume that the matrix elements ''a'', ''b'', ''c'', ''d'' commute with ''ψ''1, ''ψ''2.
Define ''φ''1, ''φ''2 by
:
Then ''φ''1, ''φ''2 are Grassmann variables (i.e. anticommute among themselves and ''φ''i2=0) if and only if ''M'' is a Manin matrix.
Observations 1,2 holds true for general ''n'' × ''m'' Manin matrices.
They demonstrate original Manin's approach as described below (one should thought of usual matrices as homomorphisms of polynomial rings, while Manin matrices are more general "non-commutative homomorphisms").
Pay attention that polynomial algebra generators are presented as column vectors, while Grassmann algebra as row-vectors, the same can be generalized to arbitrary pair of Koszul dual algebras and associated general Manin matrices.
Observation 3. Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...
.
The inverse matrix is given by the standard formula
:
if and only if ''M'' is a Manin matrix.
Proof:
:
Observation 4. Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies it ...
.
The equality
:
holds if and only if ''M'' is a Manin matrix.
Observation 5. Multiplicativity of determinants.
detcolumn(''MN'') = detcolumn(''M'')det(''N'') holds true for all complex-valued matrices N if and only if ''M'' is a Manin matrix.
Where detcolumn of 2×2 matrix is defined as ''ad'' − ''cb'', i.e. elements from first column (''a'',''c'') stands first in the products.
Conceptual definition. Concept of "non-commutative symmetries"
According to Yu. Manin's ideology one can associate to any algebra certain bialgebra of its "non-commutative symmetries (i.e. endomorphisms)". More generally to a pair of algebras ''A'', ''B'' one can associate its algebra of "non-commutative homomorphisms" between ''A'' and ''B''.
These ideas are naturally related with ideas of non-commutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some ge ...
.
Manin matrices considered here are examples of this general construction applied to polynomial algebras ''C'' 1, ...''x''n">'x''1, ...''x''n
The realm of geometry concerns of spaces, while the realm of algebra respectively with algebras, the bridge between the two realms is association to each space an algebra of functions on it, which is commutative algebra.
Many concepts of geometry can be respelled in the language of algebras and vice versa.
The idea of symmetry ''G'' of space ''V'' can be seen as action of ''G'' on ''V'', i.e. existence of a map ''G× V -> V''.
This idea can be translated in the algebraic language as existence of homomorphism ''Fun(G)'' ''Fun(V) <- Fun(V)'' (as usually maps between functions and spaces go in opposite directions).
Also maps from a space to itself can be composed (they form a semigroup), hence a dual object ''Fun(G)'' is a bialgebra
In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. ...
.
Finally one can take these two properties as basics and give purely algebraic definition of "symmetry" which can be applied to an arbitrary algebra (non-necessarily commutative):
Definition. Algebra of non-commutative symmetries (endomorphisms) of some algebra ''A'' is a bialgebra
In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. ...
''End(A)'', such that there exists homomorphisms called ''coaction'':
which is compatible with a comultiplication in a natural way.
Finally ''End(A)'' is required to satisfy only the relations which come from the above, no other relations, i.e. it is universal coacting bialgebra for ''A''.
Coaction should be thought as dual to action ''G× V -> V'', that is why it is called coaction. Compatibility of the comultiplication map with the coaction map, is dual to ''g (h v) = (gh) v''. One can easyly write this compatibility.
Somewhat surprising fact is that this construction applied to the polynomial algebra ''C'' 1, ..., ''x''n">'x''1, ..., ''x''nwill give not the usual algebra of matrices ''Matn'' (more precisely algebra of function on it), but much bigger non-commutative algebra of Manin matrices (more precisely algebra generated by elements ''Mij''.
More precisely the following simple propositions hold true.
Proposition. Consider polynomial algebra ''Pol = C'' 1, ..., ''x''n">'x''1, ..., ''x''nand matrix ''M'' with elements in some algebra ''EndPol''.
The elements commute among themselves if and only if ''M'' is a Manin matrix.
Corollary. The map is homomorphism from ''Pol'' to ''EndPol'' ''Pol''. It defines coaction.
Indeed, to ensure that the map is homomorphism the only thing we need to check is that ''yi'' commute among themselves.
Proposition. Define the comultiplication map by the formula .
Then it is coassociative and is compatible with coaction on the polynomial algebra defined in the previous proposition.
The two propositions above imply that the algebra generated by elements of a Manin matrix is a bialgebra coacting on the polynomial algebra. If one does not impose other relations ones get algebra of non-commutative endomorphisms of the polynomial algebra.
Properties
Elementary examples and properties
* Any matrix with commuting elements is a Manin matrix.
* Any matrix whose elements from different rows commute among themselves (such matrices sometimes called Cartier Cartier may refer to:
People
* Cartier (surname), a surname (including a list of people with the name)
* Cartier Martin (born 1984), American basketball player
Places
* Cartier Island, an island north-west of Australia that is part of Australia' ...
- Foata matrices) is a Manin matrix.
* Any submatrix of a Manin matrix is a Manin matrix.
* One can interchange rows and columns in a Manin matrix the result will also be a Manin matrix. One can add row or column multiplied by the central element to another row or column and results will be Manin matrix again. I.e. one can make elementary transformations with restriction that multiplier is central.
* Consider two Manin matrices ''M'',''N'' such that their all elements commute, then the sum ''M+N'' and the product ''MN'' will also be Manin matrices.
* If matrix ''M'' and simultaneously transpose matrix Mt are Manin matrices, then all elements of ''M'' commute with each other.
* No-go facts: ''Mk'' is not a Manin matrix in general (except ''k''=-1 discussed below); neither det(''M''), nor Tr(''M'') are central in the algebra generated by ''Mij'' in general (in that respect Manin matrices differs from quantum groups); det(''eM'') ≠ ''e''Tr(''M''); log(det(''M'')) ≠ Tr(log(''M'')).
* Consider polynomial algebra ''C'' ij''">'xij''and denote by the operators of differentiation with respect to
''xij'', form matrices ''X, D'' with the corresponding elements.
Also consider variable ''z'' and corresponding differential operator . 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
In mathematics, Capelli's identity, named after , is an analogue of the formula det(''AB'') = det(''A'') det(''B''), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra \mathfrak_n ...
and its generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
(see section 4.3.1, based on).
Determinant = column-determinant
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.
Linear algebra theorems
Many linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
statements hold for Manin matrices even when R is not commutative. In particular, the determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
can be defined in the standard way using permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s and it satisfies a Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...
.[ ]MacMahon Master theorem
In mathematics, MacMahon's master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph ''Combinatory analysis'' (1916). It is often used to derive binomial iden ...
holds true for Manin matrices and actually for their generalizations (super), (q), etc. analogs.
Proposition. Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...
(See or section 4.1.)
The inverse to a Manin matrix ''M'' can be defined by the standard formula:
where Madj is adjugate matrix
In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a differe ...
given by the standard formula - its (i,j)-th element is the column-determinant of the (n − 1) × (n − 1) matrix that results from deleting row ''j'' and column ''i'' of M and multiplication by (-1)i+j.
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.
Proposition. Inverse is also Manin. (See section 4.3.)
Assume a two-sided inverse to a Manin matrix ''M'' exists, then it will also be a Manin matrix.
Moreover, ''det(M−1) = (det(M))−1''.
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).
Proposition. Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies it ...
(See section 7.1.)
:
Where ''σi'' are coefficients of the characteristic polynomial
.
Proposition. Newton identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomi ...
(See section 7.2.1.)
Where ''σi'' are coefficients of the characteristic polynomial
,
and by convention ''σi''=0, for ''i>n'', where ''n'' is size of matrix ''M''.
Proposition. Determinant via Schur complement In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows.
Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' ...
(See section 5.2.)
Assume block matrix below is a Manin matrix and two-sided inverses M−1, A−1, D−1 exist, then
:
Moreover, Schur complements
are Manin matrices.
Proposition. MacMahon Master theorem
In mathematics, MacMahon's master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph ''Combinatory analysis'' (1916). It is often used to derive binomial iden ...
[
]
Examples and applications
Capelli matrix as Manin matrix, and center of U(gln)
The Capelli identity
In mathematics, Capelli's identity, named after , is an analogue of the formula det(''AB'') = det(''A'') det(''B''), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra \mathfrak_n ...
from 19th century gives one of the first examples of determinants for matrices with non-commuting elements. Manin matrices give a new look on this classical subject. This example is related to Lie algebra ''gln'' and serves as a prototype for more complicated applications to loop Lie algebra for ''gln'', Yangian and integrable systems.
Take ''Eij'' be matrices with 1 at position (''i,j'') and zeros everywhere else.
Form a matrix ''E'' with elements ''Eij'' at position (''i,j''). It is matrix with elements in ring of matrices ''Matn''. It is not Manin matrix however there are modifications which transform it to Manin matrix as described below.
Introduce a formal variable ''z'' which commute with ''Eij'', respectively ''d/dz'' is operator of differentiation in ''z''. The only thing which will be used that commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
of these operators is equal to 1.
Observation. The matrix is a Manin matrix.
Here ''Id'' is identity matrix.
2 × 2 example:
It is instructive to check the column commutativity requirement:
.
Observation. The matrix is a Manin matrix.
The only fact required from ''Eij'' for these observations is that they satisfy commutation relations ij'', ''Ekl''">'Eij'', ''Ekl'' δjk''Eil'' - δli''Ekj''. So observations holds true if ''Eij'' are generators of the universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representati ...
of Lie algebra gln, or its images in any representation.
For example, one can take
:
Here ψ are Grassmann variables.
Observation.
On the right hand side of this equality one recognizes the Capelli determinant
In mathematics, Capelli's identity, named after , is an analogue of the formula det(''AB'') = det(''A'') det(''B''), for certain matrices with noncommuting entries, related to the Lie algebra representation, representation theory of ...
(or more precisely the Capelli characteristic polynomial), while on the left hand side one has a Manin matrix with its natural determinant.
So Manin matrices gives new look on Capelli's determinant. Moreover, Capelli identity and its generalization can be derived by techniques of Manin matrices.
Also it gives an easy way to prove that this expression belongs to the center of the universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representati ...
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. . So the only property used here is that which is true for any Manin matrix ''M'' and any matrix ''g'' with central (e.g. scalar) elements.
Loop algebra for gln, Langlands correspondence and Manin matrix
Yangian type matrices as Manin matrices
Observation.
Let ''T(z)'' be a generating matrix of the Yangian In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians first appeared in physics in the work of Ludvig Faddeev and his school in the late 1970s and early 1980s concerning the quantum inverse ...
for ''gln''.
Then the matrix ''exp(-d/dz) T(z)'' is a Manin matrix.
The quantum determinant for Yangian can be defined as ''exp (n d/dz)''det''column(exp(-d/dz) T(z))''. Pay attention that ''exp(-d/dz)'' can be cancelled, so the expression does not depend on it. So the determinant in Yangian theory has natural interpretation via Manin matrices.
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. Coefficients of ''Tr(T(z+k-1)T(z+k-2)...T(z))'' for all ''k'' commute among themselves. They generate commutative subalgebra in Yangian. The same subalgebra as coefficients of the characteristic polynomial det''column(1-exp(-d/dz) T(z))'' .
(The subalgebra sometimes called Bethe subalgebra, since Bethe ansatz is a method to find its joint eigpairs.)
Further questions
History
Manin proposed general construction of "non-commutative symmetries" in,
the particular case which is called Manin matrices is discussed in, where some basic properties were outlined. The main motivation of these works was to give another look on quantum groups. Quantum matrices ''Funq''(''GLn'') can be defined as such matrices that ''T'' and simultaneously ''Tt'' are q-Manin matrices (i.e. are non-commutative symmetries of q-commuting polynomials ''xi xj'' = ''q xj xi''.
After original Manin's works there were only a few papers on Manin matrices until 2003.
But around and some after this date Manin matrices appeared in several not quite related areas: 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; generalizations of the Capelli identity involving Manin matrices appeared in.
Directions proposed in these papers has been further developed.
References
*
*
*
*
*
*
* {{cite journal , ref=Ko07B , last1=Konvalinka , first1=Matjaž , year=2007 , title=Non-commutative Sylvester's determinantal identity , url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v14i1r42 , journal=Electron. J. Combin. , volume=14 , issue=1 , doi=10.37236/960 , at=#R42 , issn=1077-8926 , arxiv=math/0703213, bibcode=2007math......3213K , s2cid=544799
Matrix theory
Matrices