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Table of Contents
Introduction
Chapter 1 - Capelli's Identity
Chapter 2 - Binet–Cauchy Identity, Brahmagupta–Fibonacci Identity and
Green's Identities
Chapter 3 - Difference of Two Squares, Euler's Identity and Jacobi Triple
Product
Chapter 4 - Differentiation Rules
Chapter 5 - Abel's Identity and Morrie's Law
Chapter 6 - Lagrange's Identity (Boundary Value Problem) and Liouville's
Formula
Chapter 7 - Newton's Identities
Chapter 8 - Lagrange's Identity and Polarization Identity
Chapter 9 - Pascal's Rule, Polynomial Identity Ring and q-Vandermonde
Identity
Chapter 10 - Pythagorean Trigonometric Identity

Chapter 11 - Squared Triangular Number, Tangent half-angle Formula and
Vandermonde's Identity
Chapter 12 - Vector Calculus Identities

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Introduction

In mathematics, the term identity has several different important meanings:
•

An identity is a relation which is tautologically true. This is usually taken to
mean something that is true by definition, either directly by the definition, or as a
consequence of it. For example, algebraically, this occurs if an equation is
satisfied for all values of the involved variables. Definitions are often indicated by
the 'triple bar' symbol ≡, such as A2 ≡ x·x. The symbol ≡ can also be used with
other meanings, but these can usually be interpreted in some way as a definition,
or something which is otherwise tautologically true (for example, a congruence
relation).

•

In algebra, an identity or identity element of a set S with a binary operation · is
an element e that, when combined with any element x of S, produces that same x.
That is, e·x = x·e = x for all x in S. An example of this is the identity matrix.

•


The identity function from a set S to itself, often denoted id or idS, is the function
which maps every element to itself. In other words, id(x) = x for all x in S. This
function serves as the identity element in the set of all functions from S to itself
with respect to function composition.

Examples
Identity relation
A common example of the first meaning is the trigonometric identity

which is true for all complex values of θ (since the complex numbers
sin and cos), as opposed to

are the domain of

which is true only for some values of θ, not all. For example, the latter equation is true
when
false when
.

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Identity element
The concepts of "additive identity" and "multiplicative identity" are central to the Peano
axioms. The number 0 is the "additive identity" for integers, real numbers, and complex
numbers. For the real numbers, for all
and
Similarly, The number 1 is the "multiplicative identity" for integers, real numbers, and
complex numbers. For the real numbers, for all

and

Identity function
A common example of an identity function is the identity permutation, which sends each
element of the set
to itself or
to itself in natural order.

Comparison
These meanings are not mutually exclusive; for instance, the identity permutation is the
identity element in the group of permutations of
under composition.

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Chapter-1

Capelli's Identity

In mathematics, Capelli's identity, named after Alfredo Capelli (1887), 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.

Statement
Suppose that xij for i,j = 1,...,n are commuting variables. Write Eij for the polarization
operator


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

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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:
E = XDt,

where E,X,D are matrices with elements Eij, xij,
respectively. If all elements in these
matrices would be commutative then clearly det(E) = det(X)det(Dt). 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: (n − i)δij on the left hand
side. For generic noncommutative matrices formulas like
det(AB) = det(A)det(B)
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 alraeady 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 [a,b] denotes the commutator ab − ba. These are the same commutation relations
which are satisfied by the matrices eij which have zeros everywhere except the position
(i,j), where 1 stands. (eij 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.

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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 Eij restricted to the space of first-order polynomials is exactly the
same as the action of matrix units eij 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 Eij 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:

equals to (k, 0, ... ,0), indeed:

for i < j. Its highest weight

.

Such representation is sometimes called bosonic representation of

. Similar formulas

define the so-called fermionic representation, here ψi 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 representations 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 xi,pj. The matrix Ec is of rank one and hence its determinant is equal to

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zero. Elements of matrix E are defined by the similar formulas, however, its elements do
not commute. The Capelli identity shows that the commutative identity: det(Ec) = 0 can
be preserved for the small price of correcting matrix E by (n − i)δij.
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

and its center

An interesting property of the Capelli determinant is that it commutes with all operators
Eij, that is the commutator [Eij,det(E + (n − i)δij)] = 0 is equal to zero. It can be generalized:
Consider any elements Eij in any ring, such that they satisfy the commutation relation
[Eij,Ekl] = δjkEil − δilEkj, (so they can be differential operators above, matrix units eij or any
other elements) define elements Ck as follows:

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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 + (k
− i)δij. 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 algebra


can defined as an algebra generated by
Eij
subject to the relations
[Eij,Ekl] = δjkEil − δilEkj
alone. The proposition above shows that elements Ckbelong to the center 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 (E11 + E22) commute with Eij. (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 Eij. Let us do it for E12:

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[E12,E11E22 − E21E12 + E22]
= [E12,E11]E22 + E12[E11,E22] − [E12,E21]E12 − E21[E12,E12] + [E12,E22]
= − E12E22 + E11E12 − (E11 − E22)E12 − 0 + E12
= − E12E22 + E22E12 + E12 = − E12 + E12 = 0.
We see that the naive determinant E11E22 − E21E12 will not commute with E12 and the
Capelli's correction + E22 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: E = XDt,
where E is
matrix with elements Eij; X is
matrix with elements xij; D 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 minors 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 mm=n we return to the identity above.
Let us also mention that similar to the commutative case, 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
MKL 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

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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 + (n − i)δij 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 Howe 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 Eij. Such
operators preserve the degree of polynomials. Let us look at the polynomials of degree 1:
Eijxkl = xilδjk, 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 xil,
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 Eij 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
operators Ekl.
The Lie group
acts on the vector space
can show that the corresponding action of Lie algebra

commute with differential
in a natural way. One
is given by the

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differential operators
these operators.

and

respectively. This explains the commutativity of

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

GLn and GLm can be given as follows:

The summands are indexed by the Young diagrams D, and representations ρD are
mutually non-isomorphic. And diagram D determine D' and vice versa.
•

In particular the representation of the big group
free, that is each irreducible representation occurs only one time.

is multiplicity

One easily observe the strong similarity to Schur–Weyl duality.

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.
Howe, B. Kostant , Fields medalist A. Okounkov , A. Sokal , 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.

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 algebras and their super , (q) , 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
It has been believed for quite a long time that the identity is intimately related with semisimple Lie algebras. Surprisingly a new purely algebraic generalization of the identity

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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.

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

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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.
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.

The Mukhin–Tarasov–Varchenko identity and the Gaudin model

Statement
Consider matrices X and D as in Capelli's identity, i.e. with elements xij and
position (ij).

at

Let z be another formal variable (commuting with x). Let A and B be some matrices

which elements are complex numbers.

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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 mechanics).

The Gaudin quantum integrable system and Talalaev's theorem
The matrix

is a Lax matrix 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 permanents, 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:

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"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.".

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Chapter-2

Binet–Cauchy Identity, Brahmagupta–
Fibonacci Identity and Green's Identities

Binet–Cauchy identity
In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and
Augustin-Louis Cauchy, states that

for every choice of real or complex numbers (or more generally, elements of a
commutative ring). Setting ai = ci and bi = di, it gives the Lagrange's identity, which is a
stronger version of the Cauchy–Schwarz inequality for the Euclidean space .

The Binet–Cauchy identity and exterior algebra
When n = 3 the first and second terms on the right hand side become the squared
magnitudes of dot and cross products respectively; in n dimensions these become the
magnitudes of the dot and wedge products. We may write it

where a, b, c, and d are vectors. It may also be written as a formula giving the dot
product of two wedge products, as

In the special case of unit vectors a=c and b=d, the formula yields

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When both vectors are unit vectors, we obtain the usual relation
1 = cos2(φ) + sin2(φ)
where φ is the angle between the vectors.

Proof
Expanding the last term,

where the second and fourth terms are the same and artificially added to complete the
sums as follows:

This completes the proof after factoring out the terms indexed by i.

Generalization
A general form, also known as the Cauchy–Binet formula, states the following: Suppose
A is an m×n matrix and B is an n×m matrix. If S is a subset of {1, ..., n} with m elements,
we write AS for the m×m matrix whose columns are those columns of A that have indices

from S. Similarly, we write BS for the m×m matrix whose rows are those rows of B that
have indices from S. Then the determinant of the matrix product of A and B satisfies the
identity

where the sum extends over all possible subsets S of {1, ..., n} with m elements.
We get the original identity as special case by setting

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Brahmagupta–Fibonacci identity
In algebra, Brahmagupta's identity, also called Fibonacci's identity, implies that the
product of two sums of two squares is itself a sum of two squares. In other words, the set
of all sums of two squares is closed under multiplication. Specifically:

For example,

The identity is a special case (n = 2) of Lagrange's identity, and is first found in
Diophantus. Brahmagupta proved and used a more general identity, equivalent to

showing that the set of all numbers of the form x2 + ny2 is closed under multiplication.
Both (1) and (2) can be verified by expanding each side of the equation. Also, (2) can be
obtained from (1), or (1) from (2), by changing b to −b.
This identity holds in both the ring of integers and the ring of rational numbers, and more
generally in any commutative ring.
In the integer case this identity finds applications in number theory for example when
used in conjunction with one of Fermat's theorems it proves that the product of a square
and any number of primes of the form 4n + 1 is also a sum of two squares.

History
The identity is first found in Diophantus's Arithmetica (III, 19). The identity was
rediscovered by Brahmagupta (598–668), an Indian mathematician and astronomer, who
generalized it. His Brahmasphutasiddhanta was translated from Sanskrit into Arabic by
Mohammad al-Fazari, and was subsequently translated into Latin in 1126. The identity
later appeared in Fibonacci's Book of Squares in 1225.

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Related identities
Euler's four-square identity is an analogous identity involving four squares instead of two
that is related to quaternions. There is a similar eight-square identity derived from the
Cayley numbers which has connections to Bott periodicity.

Relation to complex numbers
If a, b, c, and d are real numbers, this identity is equivalent to the multiplication property
for absolute values of complex numbers namely that:

since

by squaring both sides

and by the definition of absolute value,

Interpretation via norms
In the case that the variables a, b, c, and d are rational numbers, the identity may be
interpreted as the statement that the norm in the field Q(i) is multiplicative. That is, we
have

and also

Therefore the identity is saying that

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Application to Pell's equation
It its original context, Brahmagupta applied his discovery to the solution of Pell's
equation, namely x2 − Ny2 = 1. Using the identity in the more general form

he was able to "compose" triples (x1, y1, k1) and (x2, y2, k2) that were solutions of
x2 − Ny2 = k, to generate the new triple

Not only did this give a way to generate infinitely many solutions to x2 − Ny2 = 1 starting
with one solution, but also, by dividing such a composition by k1k2, integer or "nearly
integer" solutions could often be obtained. The general method for solving the Pell
equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was also
based on this identity.

Green's identities
In mathematics, Green's identities are a set of three identities in vector calculus. They
are named after the mathematician George Green, who discovered Green's theorem.

Green's first identity
This identity is derived from the divergence theorem applied to the vector field
: Let φ and ψ be scalar functions defined on some region U in R3, and
suppose that φ is twice continuously differentiable, and ψ is once continuously

differentiable. Then

where Δ is the Laplace operator,
is the boundary of region U and n is the outward
pointing unit normal of surface element dS.

Green's second identity
If φ and ψ are both twice continuously differentiable on U in R3, and ε is once
continuously differentiable:

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For the special case of ε = 1 all across U in R3 then:

In the equation above ∂φ / ∂n is the directional derivative of φ in the direction of the
outward pointing normal n to the surface element dS:

Green's third identity
Green's third identity derives from the second identity by choosing
fundamental solution of the Laplace equation. This means that:

For example in

, where G is a

, the fundamental solution has the form:

Green's third identity states that if ψ is a function that is twice continuously differentiable

on U, then

A further simplification arises if ψ is itself a harmonic function, i.e. a solution to the
Laplace equation. Then

and the identity simplifies to:

On manifolds
Green's identities hold on a Riemannian manifold, In this setting, the first two are

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where u and v are smooth real-valued functions on M, dV is the volume form compatible
with the metric,
is the induced volume form on the boundary of M, N is oriented unit
vector field normal to the boundary, and

is the Laplacian.

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Chapter-3

Difference of Two Squares, Euler's Identity
and Jacobi Triple Product

Difference of two squares
In mathematics, the difference of two squares, or the difference of perfect squares, is
when a number is squared, or multiplied by itself, and is then subtracted from another
squared number. It refers to the identity

from elementary algebra.

Proof
The proof is straightforward, starting from the RHS: apply the distributive law to get a
sum of four terms, and set as an application of the commutative law. The resulting
identity is one of the most commonly used in all of mathematics.

The proof just given indicates the scope of the identity in abstract algebra: it will hold in
any commutative ring R.
Also, conversely, if this identity holds in a ring R for all pairs of elements a and b of the
ring, then R is commutative. To see this, we apply the distributive law to the right-hand
side of the original equation and get

and if this is equal to a2 − b2, then we have

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