Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 145 / 2 / Solution

For a separated filtered group defineThe associated graded Lie algebra of a filtered group isEach quotient is abelian because $[G_{\lambda },G_{\lambda }]\subseteq G_{2\lambda }\subseteq G_{\lambda +}$. If $x\in G_{\lambda }$ and $y\in G_{\mu }$, setThe standard commutator identities $[xy,z]=[x,z{]}^y[y,z]$ and $[x,yz]=[x,z][x,y{]}^z$ prove well-defined bilinearity, while the Hall-Witt identity gives the Jacobi identity. If $\omega ([x,y])=\omega (x)+\omega (y)$, this bracket has nonzero initial form, so the graded Lie algebra is nonabelian.

For a p-valuation, defineThe p-power axiom and the Hall-Petrescu formula make this well defined and turn each homogeneous component into part of a graded ${\mathbb{F}}_p[t]$-module. The leading-term congruence $[x^p,y]\equiv [x,y{]}^p$ modulo terms of valuation greater than $\omega (x)+\omega (y)+1$ makes the bracket ${\mathbb{F}}_p[t]$-bilinear.

For the given upper-triangular group, write an element of degree $n$ to leading order asLet $X_n,Y_n$ denote the classes with $(\alpha ,\beta )=(1,0),(0,1)$. Matrix commutators giveand pth powers give $t{X}_n=X_{n+1}$ and $t{Y}_n=Y_{n+1}$. Thus