View source for Noether's theorems/Mathematics/Lie groups and Lie algebras

{{DISPLAYTITLE:Lie groups and Lie algebras}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Groups of continuous motions|Groups of continuous motions]].''

[[File:NT-F-19.png|thumb|right|360px|alt=A curved closed region representing the group, with a marked point labelled the identity. A straight line touching the region at that point is labelled the algebra, the set of smallest steps. A caption describes closure under the commutator as the structure the theorems use.|A Lie group and its algebra as the tangent at the identity]]

Noether's theorems are built on Lie's theory of continuous groups, and she cites
him at three separate points. This page carries only what the theorems use.

'''Notation.''' Throughout this track, <math>\psi</math>, <math>\mu</math> and
<math>\varepsilon</math> are reserved for Noether's meanings — the Lagrange
expressions, the count of dependent variables, and the group parameters. Symbols
imported from other traditions are renamed where they would collide, and
[[Noether's theorems/Notation|the notation page]] tabulates the substitutions.

== Generators ==

A one-parameter family of transformations of the variables
<math>x^{1},\dots,x^{m}</math>, reducing to the identity at parameter zero, acts
on functions. Its '''generator''' is the first-order operator

<math>X=\sum_{i}\xi^{i}(x)\frac{\partial}{\partial x^{i}}\qquad\qquad\mathrm{(1)}</math>

whose coefficients <math>\xi^{i}</math> are the components of the infinitesimal
displacement. This is
[[Noether's theorems/First theorem/One-parameter groups and their generators|the
object of the groups page]], written as an operator rather than as a
displacement.

For a <math>\varrho</math>-parameter group there are <math>\varrho</math>
generators <math>X_{1},\dots,X_{\varrho}</math>.

== The commutator, derived ==

Compose two infinitesimal transformations in each order and subtract. The claim
is that what remains is again a first-order operator, which is not obvious,
because each product separately is second order.

Apply <math>X_{a}</math> after <math>X_{b}</math> to a function <math>f</math>:

<math>X_{a}X_{b}f=\sum_{i,j}\xi_{a}^{i}\frac{\partial\xi_{b}^{j}}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}+\sum_{i,j}\xi_{a}^{i}\xi_{b}^{j}\frac{\partial^{2}f}{\partial x^{i}\partial x^{j}}\qquad\qquad\mathrm{(2)}</math>

Now interchange <math>a</math> and <math>b</math> and subtract. The second term of
(2) carries <math>\partial^{2}f/\partial x^{i}\partial x^{j}</math>, which is
symmetric in <math>i</math> and <math>j</math>, multiplied by
<math>\xi_{a}^{i}\xi_{b}^{j}</math>. Interchanging <math>a</math> and
<math>b</math> and relabelling the summation indices leaves that term unchanged,
so it cancels. What survives is

<math>[X_{a},X_{b}]f=\sum_{j}\left(\sum_{i}\left(\xi_{a}^{i}\frac{\partial\xi_{b}^{j}}{\partial x^{i}}-\xi_{b}^{i}\frac{\partial\xi_{a}^{j}}{\partial x^{i}}\right)\right)\frac{\partial f}{\partial x^{j}}\qquad\qquad\mathrm{(3)}</math>

first order, as claimed. The cancellation is the whole content: '''the second-order
part of a commutator vanishes because mixed partial derivatives commute.'''

== Closure and structure constants ==

If the transformations form a group, the commutator of two generators must again
be a generator of the same group, so

<math>[X_{a},X_{b}]=\sum_{d}c_{ab}^{d}X_{d}\qquad\qquad\mathrm{(4)}</math>

with constants <math>c_{ab}^{d}</math>. From the definition of the commutator,

<math>[X_{a},X_{b}]=-[X_{b},X_{a}]\qquad\text{and}\qquad[[X_{a},X_{b}],X_{d}]+[[X_{b},X_{d}],X_{a}]+[[X_{d},X_{a}],X_{b}]=0\qquad\qquad\mathrm{(5)}</math>

the second following by expanding all six products and cancelling in pairs. The
generators with the commutator form the '''Lie algebra''' of the group.

== Where Noether uses it ==

At exactly one place, and it is load-bearing.

In her third section, proving
[[Noether's theorems/First theorem/The converse of the first theorem|the converse]],
she has recovered a family of transformations and must show they form a group. Her
argument is that if they did not, the commutator of two of them — she calls it the
Lie bracket process — would produce an infinitesimal transformation that is not a
linear combination of the <math>\varrho</math> she started with. Since the
integral admits that transformation too, there would be more than
<math>\varrho</math> independent divergence relations, contrary to assumption.

That argument is (4) used as a test rather than as a fact: closure is what has to
be checked, and failure of closure is what would contradict the count.

She also relies on Lie's theorem that a family of infinitesimal transformations of
the infinite kind belongs to a genuine group of finite ones, citing it by number
and page.

== Not derived here ==

'''UNVERIFIED.''' That every finite-dimensional Lie algebra is the algebra of some
Lie group, and the classification of the simple Lie algebras, are stated in the
literature and are not derived on this page. Neither theorem needs either result.
The reader who wants them should go to a text on Lie theory; this page exists only
to make Noether's three citations legible.

''Provenance: (1) to (5) are standard and are derived above rather than cited. The
account of where Noether uses the bracket is read from her third and fourth
sections in the public-domain German original. Lie's works were not reached; her
citations to him are reported as hers. See
[[Noether's theorems/Sources|Sources]].''