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.

  • 1
    $\begingroup$ Thank you mskalsi, you seem to be a master of Lie algebra :D $\endgroup$ – MrYouMath May 9 '16 at 19:27
  • $\begingroup$ Thanks for complement, but I am struggling with Lie groups and algebra everyday. I believe if one have knowledge of functional analysis, differential geometry and manifolds then Lie group and Lie algebra can be better understood. $\endgroup$ – IgotiT May 10 '16 at 3:55
  • $\begingroup$ I have read through the paper and it is very easy to understand. Still the calculations are very time consuming. Do you know by any chance a way how to apply these calculations in Maple or Mathematica? A tutorial would be nice or even a file. I did't the calculations by hand but I iam not quiet sure if I have done everything correctly. $\endgroup$ – MrYouMath May 12 '16 at 12:09
  • $\begingroup$ @MrYouMath Do you know how they select representative elements ?(See cases after equation (32)). How the killing form help to select those representative elements ? $\endgroup$ – IgotiT May 24 '16 at 4:31

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