Noether's theorems/Mathematics/Lie groups and Lie algebras

Plain-language counterpart: Groups of continuous motions.

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, $\psi$, $\mu$ and $\epsilon$ 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 the notation page tabulates the substitutions.

Generators

A one-parameter family of transformations of the variables $x^1,\dots ,x^m$, reducing to the identity at parameter zero, acts on functions. Its generator is the first-order operator

$X=\sum _i{\xi }^i(x)\frac{\partial }{\partial x^i}\mathrm{(}\mathrm{1}\mathrm{)}$

whose coefficients ${\xi }^i$ are the components of the infinitesimal displacement. This is the object of the groups page, written as an operator rather than as a displacement.

For a $\varrho$-parameter group there are $\varrho$ generators $X_1,\dots ,X_{\varrho }$.

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 $X_a$ after $X_b$ to a function $f$:

$X_a{X}_{b}f=\sum _{i,j}{\xi }_a^i\frac{\partial {\xi }_b^j}{\partial x^i}\frac{\partial f}{\partial x^j}+\sum _{i,j}{\xi }_a^i{\xi }_b^j\frac{{\partial }^{2}f}{\partial x^i\partial x^j}\mathrm{(}\mathrm{2}\mathrm{)}$

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

$[X_a,X_b]f=\sum _j\left(\sum _i({\xi }_a^i\frac{\partial {\xi }_b^j}{\partial x^i}-{\xi }_b^i\frac{\partial {\xi }_a^j}{\partial x^i})\right)\frac{\partial f}{\partial x^j}\mathrm{(}\mathrm{3}\mathrm{)}$

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

$[X_a,X_b]=\sum _d{c}_{ab}^d{X}_d\mathrm{(}\mathrm{4}\mathrm{)}$

with constants $c_{ab}^d$. From the definition of the commutator,

$[X_a,X_b]=-[X_b,X_a]\text{and}[[X_a,X_b],X_d]+[[X_b,X_d],X_a]+[[X_d,X_a],X_b]=0\mathrm{(}\mathrm{5}\mathrm{)}$

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 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 $\varrho$ she started with. Since the integral admits that transformation too, there would be more than $\varrho$ 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 Sources.