Completed group algebra ($R[[G]]$)

For a profinite group $G$ and a finite or profinite coefficient ring $R$, the completed group algebra iswhere $U$ ranges over the open normal subgroups. It records convergent noncommutative power series in topological generators of $G$.

An Iwasawa algebra is a completed group algebra such as ${\mathbb{Z}}_p[[G]]$ or ${\mathbb{F}}_p[[G]]$ for a compact p-adic analytic group $G$.

A filtration on a group $G$ is a function $\omega :G\to \mathbb{R}\cup \{\infty \}$ satisfyingIt is separated when $\omega (x)=\infty$ only for $x=1$.

For a prime number $p$, a p-valuation is a separated filtration on a group such that every $x\ne 1$ satisfiesA group equipped with one is called a p-valued group.

An ordered basis $(g_1,\dots ,g_d)$ of a complete finite-rank p-valued group gives each element a unique convergent expression $g_1^{{\lambda }_1}\cdots g_d^{{\lambda }_d}$ with ${\lambda }_i\in {\mathbb{Z}}_p$, and its valuation is the minimum of $\omega (g_i)+v_p({\lambda }_i)$ over the nonzero coordinates.

A p-valued group is p-saturated when it is complete and every $x$ withhas a pth root in the group.

For $G_{\lambda }=\{g:\omega (g)\ge \lambda \}$ and $G_{\lambda +}=\{g:\omega (g)>\lambda \}$, the associated graded groupis a graded Lie algebra with bracket induced by the group commutator. For a p-valuation, $t{\mathrm{gr}}_{\lambda }(g)={\mathrm{gr}}_{\lambda +1}(g^p)$ makes it a graded ${\mathbb{F}}_p[t]$-Lie algebra.

For a p-valued group $(G,\omega )$, give $p^r(g_1-1)\cdots (g_s-1)$ weight $r+{\sum }_i\omega (g_i)$. The spans of elements of weight at least $\lambda$ form a multiplicative filtration on ${\mathbb{Z}}_p[G]$ and its completions.

The initial form of $g-1$ depends only on the initial form of $g$ in $\mathrm{gr}G$. The resulting Lie map extends to a surjective graded algebra homomorphismFor a complete finite-rank p-valued group, ordered-basis expansions show that it is an isomorphism.

If $G$ is a complete finite-rank p-valued group with center $Z(G)$, thenThe key facts are that the finite-conjugacy center of a p-valued group equals its center and that compatibility through the finite group-algebra quotients eliminates coefficients on infinite conjugacy classes.

Inside the completed rational Iwasawa algebra, the convergent seriessends a finite-rank p-saturated group to a ${\mathbb{Z}}_p$-Lie algebra. The Baker--Campbell--Hausdorff formula and its commutator expansion prove closure under addition and the commutator bracket.

If $(g_1,\dots ,g_d)$ is an ordered basis of a finite-rank p-saturated group, thenwhere a primitive element $x$ satisfies $\Delta (x)=x\otimes 1+1\otimes x$ for the completed Hopf algebra coproduct.