This is a follow up question to Invariants of a PDE by Lie Symmetries, as I tried to follow the reasoning from the book Applications of Lie Groups to Differential Equations (Peter J. Olver, Example 3.13 on page 204-207).
As one can show the heat equation $u_t=u_{xx}$ has the following symmetries (infinitesimal transformations): $$ v_1 = \partial_x \quad v_2 = \partial_t \quad v_3 = u\partial_u \quad v_4 = x\partial_x +2t\partial_t$$ $$ v_5 = 2t\partial_x-xu\partial_u \quad v_6 = 4tx\partial_x+4t^2\partial_t-(x^2+2t)u\partial_u$$ $$ v_{\alpha} = \alpha(x,t)\partial_u$$
Where $\alpha(x,t)$ is a solution of the heat equation.
In example 3.13 the author wants to derive the optimal system of sub-algebras of the heat equation.
Getting the commutator table and adjoint table is not a problem. But from there on everything is not clear to me.
The author starts with the general vector $$v=a_1v_1+a_2v_2+a_3v_3+a_4v_4+a_5v_5+a_6v_6$$
and directly concludes that $\eta(v)=a_4^2-4a_2a_6$ is an invariant of the full adjoint action: $\eta(Ad g(v))=\eta(v), v \in g, g \in G$
My first question: How am I supposed to find this invariant?
EDIT: I found out that this has to do with the killing form of the lie algebra.
Then it is stated that:
$$\tilde{v}=\sum_{i=1}^6\tilde{a}_iv_i=Ad(\exp(\alpha v_6))\circ Ad(\exp(\beta v_2))v.$$
My second question: Where does this come from?
He then continues to simplify and finally gets to the set of optimal subalgebras of the heat equation
$$My last question: Can someone explain what he does?
Thank you alot for reading my question :).
The invariant of full adjoint action or formally called killing form is actually solution of set of linear partial differential equations of first order. I can suggest you well written article on this topic where author has described procedure for construction of such killing form. Please see article, this article is also available on arxiv.org.
The set of partial differential equations I am talking about are given by equation (13) on pp.053504-5 and their solution is killing form you are asking for. If you fully understood this paper you can definitely master the construction of optimal system.