Manin matrix

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 which takes values in the abelianization K^*/[K^*, K^*] of the multiplicative group K^* of ground ring K; and theory of quasideterminants. 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; super-matrices and Berezinian.

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.

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 [M_ij,M_kl] = [M_kj,M_il]. Here [a,b] denotes (ab − ba) the commutator of a and b.[3]

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:

${\displaystyle {\begin{pmatrix}\cdots &\cdots &\cdots &\cdots &\cdots \\\cdots &M_{ij}&\cdots &M_{il}&\cdots \\\cdots &\cdots &\cdots &\cdots &\cdots \\\cdots &M_{kj}&\cdots &M_{kl}&\cdots \\\cdots &\cdots &\cdots &\cdots &\cdots \end{pmatrix}}={\begin{pmatrix}\cdots &\cdots &\cdots &\cdots &\cdots \\\cdots &a&\cdots &b&\cdots \\\cdots &\cdots &\cdots &\cdots &\cdots \\\cdots &c&\cdots &d&\cdots \\\cdots &\cdots &\cdots &\cdots &\cdots \end{pmatrix}}}$

the following commutation relations hold

${\displaystyle ac=ca,~~~bd=db,~~~{\text{(entries in the same column commute) }}}$

${\displaystyle ad-da=cb-bc,~~~{\text{(cross commutation relation)}}.}$

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.[3] The proofs of all observations is direct 1 line check.

Consider a 2×2 matrix ${\displaystyle M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}.}$

Observation 1. Coaction on a plane.
Consider the polynomial ring C[x₁, x₂], and assume that the matrix elements a, b, c, d commute with x₁, x₂. Define y₁, y₂ by

${\displaystyle {\begin{pmatrix}y_{1}\\y_{2}\end{pmatrix}}={\begin{pmatrix}a&b\\c&d\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}.}$

Then y₁, y₂ commute among themselves if and only if M is a Manin matrix.

Proof:

${\displaystyle [y_{1},y_{2}]=[ax_{1}+bx_{2},cx_{1}+dx_{2}]=[a,c]x_{1}^{2}+[b,d]x_{2}^{2}+([a,d]+[b,c])x_{1}x_{2}.}$

Requiring this to be zero, we get Manin's relations.

Observation 2. Coaction on a super-plane.
Consider the Grassmann algebra C[ψ₁, ψ₂], and assume that the matrix elements a, b, c, d commute with ψ₁, ψ₂. Define φ₁, φ₂ by

${\displaystyle {\begin{pmatrix}\phi _{1},~\phi _{2}\end{pmatrix}}={\begin{pmatrix}\psi _{1},~\psi _{2}\end{pmatrix}}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}.}$

Then φ₁, φ₂ 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. The inverse matrix is given by the standard formula

${\displaystyle M^{-1}={\frac {1}{ad-cb}}{\begin{pmatrix}d&-b\\-c&a\end{pmatrix}}}$

if and only if M is a Manin matrix.

Proof:

${\displaystyle {\begin{pmatrix}d&-b\\-c&a\end{pmatrix}}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{pmatrix}da-bc&db-bd\\-ca+ac&-cb+ad\end{pmatrix}}={\text{if and only if }}M{\text{ is a Manin matrix}}={\begin{pmatrix}ad-cb&0\\0&ad-cb\end{pmatrix}}.}$

Observation 4. Cayley–Hamilton theorem. The equality

${\displaystyle M^{2}-(a+d)M+(ad-cb)1_{2\times 2}=0}$

holds if and only if M is a Manin matrix.

Observation 5. Multiplicativity of determinants.

det^column(MN) = det^column(M)det(N) holds true for all complex-valued matrices N if and only if M is a Manin matrix.

Where det^column of 2×2 matrix is defined as ad − cb, i.e. elements from first column (a,c) stands first in the products.

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. Manin matrices considered here are examples of this general construction applied to polynomial algebras C[x₁, ...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)${\displaystyle \otimes }$ 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.

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 End(A), such that there exists homomorphisms called coaction:

${\displaystyle coaction:~~End(A)\otimes A\leftarrow A,}$

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[x₁, ..., x_n] will give not the usual algebra of matrices Mat_n (more precisely algebra of function on it), but much bigger non-commutative algebra of Manin matrices (more precisely algebra generated by elements M_ij. More precisely the following simple propositions hold true.

Proposition. Consider polynomial algebra Pol = C[x₁, ..., x_n] and matrix M with elements in some algebra EndPol. The elements ${\displaystyle y_{i}=\sum _{k}M_{ik}\otimes x_{k}\in EndPol\otimes Pol}$ commute among themselves if and only if M is a Manin matrix.

Corollary. The map ${\displaystyle x_{i}\mapsto y_{i}=\sum _{k}M_{ik}\otimes x_{k}}$ is homomorphism from Pol to EndPol ${\displaystyle \otimes }$ Pol. It defines coaction.

Indeed, to ensure that the map is homomorphism the only thing we need to check is that y_i commute among themselves.

Proposition. Define the comultiplication map by the formula ${\displaystyle \Delta (M_{ij})=\sum _{l}M_{il}\otimes M_{lj}}$. 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.

• Any matrix with commuting elements is a Manin matrix.
• Any matrix whose elements from different rows commute among themselves (such matrices sometimes called Cartier-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 M^t are Manin matrices, then all elements of M commute with each other.
• No-go facts: M^k 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 M_ij in general (in that respect Manin matrices differs from quantum groups); det(e^M) ≠ e^Tr(M); log(det(M)) ≠ Tr(log(M)).
• Consider polynomial algebra C[x_ij] and denote by ${\displaystyle \partial _{ij}}$ the operators of differentiation with respect to

x_ij, form matrices X, D with the corresponding elements. Also consider variable z and corresponding differential operator ${\displaystyle \partial _{z}}$. The following gives an example of a Manin matrix which is important for Capelli identities:

${\displaystyle {\begin{pmatrix}zId&D^{t}\\X&\partial _{z}Id\end{pmatrix}}.}$

One can replace X, D by any matrices whose elements satisfy the relation: X_ij D_kl - D_kl X_ij = δ_ikδ_kl, 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.

Many linear algebra statements hold for Manin matrices even when R is not commutative. 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. analogs.

Proposition. Cramer's rule (See[2] or section 4.1.[3]) The inverse to a Manin matrix M can be defined by the standard formula: ${\displaystyle M^{-1}={\frac {1}{{\det }^{col}(M)}}M^{adj},}$ where M^adj is adjugate matrix 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.[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[4]).

Proposition. Cayley–Hamilton theorem (See section 7.1.[3])

${\displaystyle det^{column}(t-M)|_{t=M}^{right~substitute}=0,~~i.e.~~\sum _{i=0...n}(-1)^{i}\sigma _{i}M^{n-i}=0.}$

Where σ_i are coefficients of the characteristic polynomial ${\displaystyle det^{column}(t-M)=\sum _{i=0...n}(-1)^{i}\sigma _{i}t^{n-i}}$.

Proposition. Newton identities (See section 7.2.1.[3]) ${\displaystyle \forall k\geq 0:-(-1)^{k}k\sigma _{k}=\sum _{i=0...k-1}\sigma _{i}Tr(M^{k-i})}$

Where σ_i are coefficients of the characteristic polynomial ${\displaystyle det^{column}(t-M)=\sum _{i=0...n}(-1)^{i}\sigma _{i}t^{n-i}}$, and by convention σ_i=0, for i>n, where n is size of matrix M.

Proposition. Determinant via Schur complement (See section 5.2.[3]) Assume block matrix below is a Manin matrix and two-sided inverses M⁻¹, A⁻¹, D⁻¹ exist, then

${\displaystyle det^{column}{\begin{pmatrix}A&B\\C&d\\\end{pmatrix}}=det^{column}(A)det^{column}(D-CA^{-1}B)=det^{column}(D)det^{column}(A-BD^{-1}C).}$

Moreover, Schur complements ${\displaystyle (D-CA^{-1}B),(A-BD^{-1}C)}$ are Manin matrices.

Proposition. MacMahon Master theorem

[6]

The Capelli identity 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 gl_n and serves as a prototype for more complicated applications to loop Lie algebra for gl_n, Yangian and integrable systems.

Take E_ij be matrices with 1 at position (i,j) and zeros everywhere else. Form a matrix E with elements E_ij at position (i,j). It is matrix with elements in ring of matrices Mat_n. 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 E_ij, respectively d/dz is operator of differentiation in z. The only thing which will be used that commutator of these operators is equal to 1.

Observation. The matrix ${\displaystyle d/dzId-E/z}$ is a Manin matrix.

Here Id is identity matrix.

2 × 2 example: ${\displaystyle M={\begin{pmatrix}d/dz-E_{11}/z&-E_{12}/z\\-E_{21}/z&d/z-E_{22}/z\end{pmatrix}}.}$

It is instructive to check the column commutativity requirement: ${\displaystyle [d/dz-E_{11}/z,-E_{21}/z]=[d/dz,-E_{21}/z]+[-E_{11}/z,-E_{21}/z]=E_{21}/z^{2}-E_{21}/z^{2}=0}$.

Observation. The matrix ${\displaystyle exp(-d/dz)(Id+E/z)}$ is a Manin matrix.

The only fact required from E_ij for these observations is that they satisfy commutation relations [E_ij, E_kl]= δ_jkE_il - δ_liE_kj. So observations holds true if E_ij are generators of the universal enveloping algebra of Lie algebra gl_n, or its images in any representation. For example, one can take

${\displaystyle E_{ij}=x_{i}{\frac {\partial }{\partial x_{j}}};~~~~~E_{ij}=\sum _{a=1}^{n}x_{ia}{\frac {\partial }{\partial x_{ja}}};~~~~E_{ij}=\psi _{i}{\frac {\partial }{\partial \psi _{j}}}.}$

Here ψ are Grassmann variables.

Observation. ${\displaystyle z^{n-1}det^{col}(d/dz-E/z)=det^{col}(zd/dz-E-diag(n-1,n-2,...,1,0))}$

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. 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 U(gl_n), which is far from being trivial. Indeed, it's enough to check invariance with respect to action of the group GL_n by conjugation. ${\displaystyle det^{col}(d/dz-gEg^{-1}/z)=det^{col}(g(d/dz-E/z)g^{-1})=det(g)det^{col}(d/dz-E/z)det(g^{-1})=det^{col}(d/dz-E/z)}$. So the only property used here is that ${\displaystyle det(gM)=det(Mg)=det(M)det(g)}$ which is true for any Manin matrix M and any matrix g with central (e.g. scalar) elements.

Observation. Let T(z) be a generating matrix of the Yangian for gl_n. 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(T^k(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.)

Manin proposed general construction of "non-commutative symmetries" in,[1] the particular case which is called Manin matrices is discussed in,[2] where some basic properties were outlined. The main motivation of these works was to give another look on quantum groups. Quantum matrices Fun_q(GL_n) can be defined as such matrices that T and simultaneously T^t are q-Manin matrices (i.e. are non-commutative symmetries of q-commuting polynomials x_i x_j = q x_j x_i. 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:[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.[7] Directions proposed in these papers has been further developed.