Capelli's identity
Encyclopedia
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 
. It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω process
.
The Capelli identity states that the following differential operators, expressed as determinants, are equal:
Both sides are differential operators. The determinant on the left has non-commuting entries, and is expanded with all terms preserving their "left to right" order. Such a determinant is often called a column-determinant, since it can be obtained by the column expansion of the determinant starting from the first column. It can be formally written as
where in the product first come the elements from the first column, then from the second and so on. The determinant on the far right is Cayley's omega process, and the one on the left is the Capelli determinant.
The operators Eij can be written in a matrix form:
where
are matrices with elements Eij, xij, 
respectively. If all elements in these matrices would be commutative then clearly 
. The Capelli identity shows that despite noncommutativity there exists a "quantization" of the formula above. The only price for the noncommutivity is a small correction: 
on the left hand side. For generic noncommutative matrices formulas like
do not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices. That is why the Capelli identity still holds some mystery, despite many proofs offered for it. A very short proof does not seem to exist. Direct verification of the statement can be given as an exercise for n' = 2, but is already long for n = 3.
with the only difference that summation index a ranges from 1 to m. One can easily see that such operators satisfy the commutation relations:
Here
denotes the commutator
. These are the same commutation relations which are satisfied by the matrices 
which have zeros everywhere except the position (i,j), where 1 stands. (
are sometimes called matrix units). Hence we conclude that the correspondence 
defines a representation of the Lie algebra 
in the vector space of polynomials of xij.
In particular, for the polynomials of the first degree it is seen that:
Hence the action of
restricted to the space of first-order polynomials is exactly the same as the action of matrix units 
on vectors in 
. So, from the representation theory point of view, the subspace of polynomials of first degree is a subrepresentation
of the Lie algebra
, which we identified with the standard representation in 
. Going further, it is seen that the differential operators 
preserve the degree of the polynomials, and hence the polynomials of each fixed degree form a subrepresentation
of the Lie algebra
. One can see further that the space of homogeneous polynomials of degree k can be identified with the symmetric tensor power 
of the standard representation 
.
One can also easily identify the highest weight structure of these representations. The monomial
is a highest weight vector, indeed: 
for i < j. Its highest weight equals to (k, 0, ... ,0), indeed: 
.
Such representation is sometimes called bosonic representation of
. Similar formulas 
define the so-called fermionic representation, here 
are anti-commuting variables. Again polynomials of k-th degree form an irreducible subrepresentation which is isomorphic to 
i.e. anti-symmetric tensor power of 
. Highest weight of such representation is (0, ..., 0, 1, 0, ..., 0). These representations for k = 1, ..., n are fundamental representation
s of
.
the motivation for this equality is the following: consider
for some commuting variables 
. The matrix 
is of rank one and hence its determinant is equal to zero. Elements of matrix 
are defined by the similar formulas, however, its elements do not commute. The Capelli identity shows that the commutative identity: 
can be preserved for the small price of correcting matrix 
by 
.
Let us also mention that similar identity can be given for the characteristic polynomial:
where
. The commutative counterpart of this is a simple fact that for rank = 1 matrices the characteristic polynomial contains only the first and the second coefficients.
Let us consider an example for n = 2.
Using
we see that this is equal to:
. It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω processCayley's Ω process
In mathematics, Cayley's Ω process, introduced by , is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action....
.
Statement
Suppose that xij for i,j = 1,...,n are commuting variables. Write Eij for the polarization operatorThe Capelli identity states that the following differential operators, expressed as determinants, are equal:
Both sides are differential operators. The determinant on the left has non-commuting entries, and is expanded with all terms preserving their "left to right" order. Such a determinant is often called a column-determinant, since it can be obtained by the column expansion of the determinant starting from the first column. It can be formally written as
where in the product first come the elements from the first column, then from the second and so on. The determinant on the far right is Cayley's omega process, and the one on the left is the Capelli determinant.
The operators Eij can be written in a matrix form:
where
are matrices with elements Eij, xij, respectively. If all elements in these matrices would be commutative then clearly . The Capelli identity shows that despite noncommutativity there exists a "quantization" of the formula above. The only price for the noncommutivity is a small correction: on the left hand side. For generic noncommutative matrices formulas likedo not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices. That is why the Capelli identity still holds some mystery, despite many proofs offered for it. A very short proof does not seem to exist. Direct verification of the statement can be given as an exercise for n' = 2, but is already long for n = 3.
Relations with representation theory
Consider the following slightly more general context. Suppose that n and m are two integers and xij for i = 1,...,n,j = 1,...,m, be commuting variables. Redefine Eij by almost the same formula:with the only difference that summation index a ranges from 1 to m. One can easily see that such operators satisfy the commutation relations:
Here
denotes the commutatorCommutator
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:...
. These are the same commutation relations which are satisfied by the matrices which have zeros everywhere except the position (i,j), where 1 stands. ( are sometimes called matrix units). Hence we conclude that the correspondence defines a representation of the Lie algebra in the vector space of polynomials of xij.Case m = 1 and representation Sk Cn
It is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi:In particular, for the polynomials of the first degree it is seen that:
Hence the action of
restricted to the space of first-order polynomials is exactly the same as the action of matrix units on vectors in . So, from the representation theory point of view, the subspace of polynomials of first degree is a subrepresentationRepresentation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of the Lie algebra
, which we identified with the standard representation in . Going further, it is seen that the differential operators preserve the degree of the polynomials, and hence the polynomials of each fixed degree form a subrepresentationRepresentation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of the Lie algebra
. One can see further that the space of homogeneous polynomials of degree k can be identified with the symmetric tensor power of the standard representation .One can also easily identify the highest weight structure of these representations. The monomial
is a highest weight vector, indeed: for i < j. Its highest weight equals to (k, 0, ... ,0), indeed: .Such representation is sometimes called bosonic representation of
. Similar formulas define the so-called fermionic representation, here are anti-commuting variables. Again polynomials of k-th degree form an irreducible subrepresentation which is isomorphic to i.e. anti-symmetric tensor power of . Highest weight of such representation is (0, ..., 0, 1, 0, ..., 0). These representations for k = 1, ..., n are fundamental representationFundamental representation
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group...
s of
.Capelli identity for m = 1
Let us return to the Capelli identity. One can prove the following:the motivation for this equality is the following: consider
for some commuting variables . The matrix is of rank one and hence its determinant is equal to zero. Elements of matrix are defined by the similar formulas, however, its elements do not commute. The Capelli identity shows that the commutative identity: can be preserved for the small price of correcting matrix by .Let us also mention that similar identity can be given for the characteristic polynomial:
where
. The commutative counterpart of this is a simple fact that for rank = 1 matrices the characteristic polynomial contains only the first and the second coefficients.Let us consider an example for n = 2.
Using
we see that this is equal to:
-
The universal enveloping algebra
An interesting property of the Capelli determinant is that it commutes with all operators Eij, that is the commutator
and its center CommutatorIn 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:...
is equal to zero. It can be generalized:
Consider any elements Eij in any ring, such that they satisfy the commutation relation
, (so they can be differential operators above, matrix units eij or any other elements) define elements Ck as follows:
where
then:
- elements Ck commute with all elements Eij
- elements Ck can be given by the formulas similar to the commutative case:
i.e. they are sums of principal minors of the matrix E, modulo the Capelli correction
. In particular element C0 is the Capelli determinant considered above.
These statements are interrelated with the Capelli identity, as will be discussed below, and similarly to it the direct few lines short proof does not seem to exist, despite the simplicity of the formulation.
The universal enveloping algebraUniversal enveloping algebraIn mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
can defined as an algebra generated by
- Eij
subject to the relations
alone. The proposition above shows that elements Ckbelong to the centerCenter (algebra)The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum, meaning "center". More specifically:...
of
. It can be shown that they actually are free generators of the center of 
. They are sometimes called Capelli generators. The Capelli identities for them will be discussed below.
Consider an example for n = 2.
It is immediate to check that element
commute with 
. (It corresponds to an obvious fact that the identity matrix commute with all other matrices). More instructive is to check commutativity of the second element with 
. Let us do it for 
:
We see that the naive determinant
will not commute with 
and the Capelli's correction 
is essential to ensure the centrality.
General m and dual pairs
Let us return to the general case:
for arbitrary n and m. Definition of operators Eij can be written in a matrix form:
, where 
is 
matrix with elements 
; 
is 
matrix with elements 
; 
is 
matrix with elements 
.
Capelli–Cauchy–Binet identities
For general m matrix E is given as product of the two rectangular matrices: X and transpose to D. If all elements of these matrices would commute then one knows that the determinant of E can be expressed by the so-called Cauchy–Binet formula via minorMajor and minorIn Western music, the adjectives major and minor can describe a musical composition, movement, section, scale, key, chord, or interval.Major and minor are frequently referred to in the titles of classical compositions, especially in reference to the key of a piece.-Intervals and chords:With regard...
s of X and D. An analogue of this formula also exists for matrix E again for the same mild price of the correction
:
,
In particular (similar to the commutative case): if m
Let us also mention that similar to the commutative case (see Cauchy–Binet for minors), one can express not only the determinant of E, but also its minors via minors of X and D:
,
Here K = (k1 < k2 < ... < ks), L = (l1 < l2 < ... < ls), are arbitrary multi-indexes; as usually
denotes a submatrix of M formed by the elements M kalb. Pay attention that the Capelli correction now contains s, not n as in previous formula. Note that for s=1, the correction (s − i) disappears and we get just the definition of E as a product of X and transpose to D. Let us also mention that for generic K,L corresponding minors do not commute with all elements Eij, so the Capelli identity exists not only for central elements.
As a corollary of this formula and the one for the characteristic polynomial in the previous section let us mention the following:
where
. This formula is similar to the commutative case, modula 
at the left hand side and t[n] instead of tn at the right hand side.
Relation to dual pairs
Modern interest in these identities has been much stimulated by Roger HoweRoger Evans HoweRoger Evans Howe is the William R. Kenan Jr. Professor of Mathematics at Yale University. He is well known for his contributions to representation theory, and in particular for the notion of a reductive dual pair, sometimes known as a Howe pair, and the Howe correspondence.He attended Ithaca High...
who considered them in his theory of reductive dual pairs (also known as Howe duality). To make the first contact with these ideas, let us look more precisely on operators
. Such operators preserve the degree of polynomials. Let us look at the polynomials of degree 1: 
, we see that index l is preserved. One can see that from the representation theory point of view polynomials of the first degree can be identified with direct sum of the representations 
, here l-th subspace (l=1...m) is spanned by 
, i = 1, ..., n. Let us give another look on this vector space:
Such point of view gives the first hint of symmetry between m and n. To deepen this idea let us consider:
These operators are given by the same formulas as
modula renumeration 
, hence by the same arguments we can deduce that 
form a representation of the Lie algebra 
in the vector space of polynomials of xij. Before going further we can mention the following property: differential operators 
commute with differential operators 
.
The Lie group
acts on the vector space 
in a natural way. One can show that the corresponding action of Lie algebra 
is given by the differential operators 
and 
respectively. This explains the commutativity of these operators.
The following deeper properties actually hold true:
- The only differential operators which commute with
are polynomials in 
, and vice versa.
- Decomposition of the vector space of polynomials into a direct sum of tensor products of irreducible representations of
and 
can be given as follows:
The summands are indexed by the Young diagrams D, and representations
are mutually non-isomorphic. And diagram 
determine 
and vice versa.
- In particular the representation of the big group
is multiplicity free, that is each irreducible representation occurs only one time.
One easily observe the strong similarity to Schur–Weyl dualitySchur–Weyl dualitySchur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finite-dimensional representations of the general linear and symmetric groups...
.
Generalizations
Much work have been done on the identity and its generalizations. Approximately two dozens of mathematicians and physicists contributed to the subject, to name a few: R. HoweRoger Evans HoweRoger Evans Howe is the William R. Kenan Jr. Professor of Mathematics at Yale University. He is well known for his contributions to representation theory, and in particular for the notion of a reductive dual pair, sometimes known as a Howe pair, and the Howe correspondence.He attended Ithaca High...
, B. KostantBertram Kostant-Early life and education:Kostant grew up in New York City, where he graduated from the celebrated Stuyvesant High School in 1945. He went on to obtain an undergraduate degree in mathematics from Purdue University in 1950. He earned his Ph.D...
Fields medalistFields MedalThe Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
A. OkounkovAndrei OkounkovAndrei Yuryevich Okounkov is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions. He is currently a professor at Columbia University....
A. SokalAlan SokalAlan David Sokal is a professor of mathematics at University College London and professor of physics at New York University. He works in statistical mechanics and combinatorics. To the general public he is best known for his criticism of postmodernism, resulting in the Sokal affair in...
, D. Zeilberger.
It seems historically the first generalizations were obtained by Herbert Westren Turnbull in 1948, who found the generalization for the case of symmetric matrices (see for modern treatments).
The other generalizations can be divided into several patterns. Most of them are based on the Lie algebra point of view. Such generalizations consist of changing Lie algebra
to simple Lie algebrasSimple Lie groupIn group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
and their superLie superalgebraIn mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
(q)Quantum groupIn mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
, and current versions. As well as identity can be generalized for different reductive dual pairs. And finally one can consider not only the determinant of the matrix E, but its permanent, trace of its powers and immanants. Let us mention few more papers; still the list of references is incomplete. It has been believed for quite a long time that the identity is intimately related with semi-simple Lie algebras. Surprisingly a new purely algebraic generalization of the identity have been found in 2008 by S. Caracciolo, A. Sportiello, A. D. Sokal which has nothing to do with any Lie algebras.
Turnbull's identity for symmetric matrices
Consider symmetric matrices
Herbert Westren Turnbull in 1948 discovered the following identity:
Combinatorial proof can be found in the paper, another proof and amusing generalizations in the paper, see also discussion below.
The Howe–Umeda–Kostant–Sahi identity for antisymmetric matrices
Consider antisymmetric matrices
Then
The Caracciolo–Sportiello–Sokal identity for Manin matrices
Consider two matrices M and Y over some associative ring which satisfy the following condition
for some elements Qil. Or ”in words”: elements in j-th column of M commute with elements in k-th row of Y unless j = k, and in this case commutator of the elements Mik and Ykl depends only on i, l, but does not depend on k.
Assume that M is a Manin matrix (the simplest example is the matrix with commuting elements).
Then for the square matrix case
Here Q is a matrix with elements Qil, and diag(n − 1, n − 2, ..., 1, 0) means the diagonal matrix with the elements n − 1, n − 2, ..., 1, 0 on the diagonal.
See proposition 1.2' formula (1.15) page 4, our Y is transpose to their B.
Obviously the original Cappeli's identity the particular case of this identity. Moreover from this identity one can see that in the original Capelli's identity one can consider elements
for arbitrary functions fij and the identity still will be true.
Statement
Consider matrices X and D as in Capelli's identity, i.e. with elements
and 
at position (ij).
Let z be another formal variable (commuting with x). Let A and B be some matrices which elements are complex numbers.
-
Here the first determinant is understood (as always) as column-determinant of a matrix with non-commutative entries. The determinant on the right is calculated as if all the elements commute, and putting all x and z on the left, while derivations on the right. (Such recipe is called a Wick ordering in the quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
).
The Gaudin quantum integrable system and Talalaev's theorem
The matrix
is a Lax matrixLax pairIn mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that describe the corresponding differential equations. They were introduced by Peter Lax to discuss solitons in continuous media...
for the Gaudin quantum integrable spin chain system. D. Talalaev solved the long-standing problem of the explicit solution for the full set of the quantum commuting conservation laws for the Gaudin model, discovering the following theorem.
Consider
Then for all i,j,z,w
i.e. Hi(z) are generating functions in z for the differential operators in x which all commute. So they provide quantum commuting conservation laws for the Gaudin model.
Permanents, immanants, traces – "higher Capelli identities"
The original Capelli identity is a statement about determinants. Later, analogous identities were found for permanentPermanentThe permanent of a square matrix in linear algebra, is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix...
s, immanants and traces.
Turnbull's identity for permanents of antisymmetric matrices
Consider the antisymmetric matrices X and D with elements xij and corresponding derivations, as in the case of the HUKS identity above.
Then
Let us cite : "...is stated without proof at the end of Turnbull’s paper". The authors themselves follow Turnbull – at the very end of their
paper they write:
"Since the proof of this last identity is very similar to the proof of Turnbull’s symmetric analog (with a slight twist), we leave it as an instructive and pleasant exercise for the reader.".
