Direct Energy, 2018a

14 LIE ANALYSIS 321
Group property name
Summary of property
Identity X 1 · X id = X 1
Inverse
X 1 · X1 −1 = X id
Associativity (X 1 · X 2 ) · X 3 = X 1 · (X 2 · X 3 )
Closure
X 1 · X 2 is an element of the group
Table 14.1: Group properties.
are multiplied rst must be equal.
(X 1 · X 2 ) · X 3 = X 1 · (X 2 · X 3 ) (14.37)
The closure property says that when two elements of the group are multiplied
together, the result is another element of the group. Table 14.1
summarizes these properties where X 1 , X 2 , and X 3 are elements of the
group, and X id is the identity element which is also a member of the group.
However, groups may have more or less than three elements.
In general, the order in which group elements are multiplied matters.
X 1 · X 2 ≠ X 2 · X 1 . (14.38)
The quantity X 1 · X 2 · X −1
1 · X −1
2 is sometimes called the commutator,
and it is denoted [X 1 ,X 2 ] . Due to the closure property, the result of the
commutator is guaranteed to be another element of the group [14, p. 21,32]
[164, p. 39,50].
[X 1 ,X 2 ]=X 1 · X 2 · X −1
1 · X −1
2 (14.39)
Continuous symmetries of equations are described by innitesimal generators
that form a Lie group. The elements of the group are the innitesimal
generators scaled by a constant [164, p. 52]. The group multiplication
operation is regular multiplication also possibly scaled by a constant. Accordingto
this denition, U = ∂ t , U =2∂ t and U = −10.2∂ t are all the
same element of the group because the constant does not aect the element.
If we nd a few innitesimal generators of a group, we may be able to use
Eq. 14.39 to nd more generators. A complete set of innitesimal generators
describe all possible continuous (regular geometrical) symmetries of
the equation. All continuous (regular geometrical) symmetry operations of
the equation can be described as linear combinations of the innitesimal
generators.