In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an infinitesimal transformation is a
limiting form of ''small''
transformation. For example one may talk about an
infinitesimal rotation In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\ ...
of a
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
, in three-dimensional space. This is conventionally represented by a 3×3
skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, i ...
''A''. It is not the matrix of an actual
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
in space; but for small real values of a parameter ε the transformation
:
is a small rotation, up to quantities of order ε
2.
History
A comprehensive theory of infinitesimal transformations was first given by
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius S ...
. This was at the heart of his work, on what are now called
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s and their accompanying
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s; and the identification of their role in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and especially the theory of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s. The properties of an abstract
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
are exactly those definitive of infinitesimal transformations, just as the axioms of
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
embody
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
. The term "Lie algebra" was introduced in 1934 by
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
, for what had until then been known as the ''algebra of infinitesimal transformations'' of a Lie group.
Examples
For example, in the case of infinitesimal rotations, the Lie algebra structure is that provided by the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
, once a skew-symmetric matrix has been identified with a 3-
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
. This amounts to choosing an axis vector for the rotations; the defining
Jacobi identity is a well-known property of cross products.
The earliest example of an infinitesimal transformation that may have been recognised as such was in
Euler's theorem on homogeneous functions
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''de ...
. Here it is stated that a function ''F'' of ''n'' variables ''x''
1, ..., ''x''
''n'' that is homogeneous of degree ''r'', satisfies
:
with
:
the
Theta operator In mathematics, the theta operator is a differential operator defined by
: \theta = z .
This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in ''z'':
:\theta (z^k) = k z^k,\quad k=0,1,2,\dots
In ...
. That is, from the property
:
it is possible to differentiate with respect to λ and then set λ equal to 1. This then becomes a
necessary condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth o ...
on a
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
''F'' to have the homogeneity property; it is also sufficient (by using
Schwartz distributions one can reduce the
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
considerations here). This setting is typical, in that there is a
one-parameter group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
:\varphi : \mathbb \rightarrow G
from the real line \mathbb (as an additive group) to some other topological group G.
If \varphi is ...
of
scalings operating; and the information is coded in an infinitesimal transformation that is a
first-order differential operator.
Operator version of Taylor's theorem
The operator equation
:
where
:
is an
operator version of
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is th ...
— and is therefore only valid under ''caveats'' about ''f'' being an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
. Concentrating on the operator part, it shows that ''D'' is an infinitesimal transformation, generating translations of the real line via the
exponential
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...
. In Lie's theory, this is generalised a long way. Any
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
Lie group can be built up by means of its
infinitesimal generators (a basis for the Lie algebra of the group); with explicit if not always useful information given in the
Baker–Campbell–Hausdorff formula.
References
*{{Springer, id=L/l058370, title=Lie algebra
*
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius S ...
(1893
Vorlesungen über Continuierliche Gruppen English translation by D.H. Delphenich, §8, link from Neo-classical Physics.
Lie groups
Transformation (function)