FREE GROUPS OF LIE TYPE

Y. Billig  and  A. Pianzola
Dept. of Mathematics & Statistics  Dept. of Mathematical Sciences
University of New Brunswick  University of Alberta
Fredericton, New Brunswick  Edmonton, Alberta
Canada E3B 5A3  Canada T6G 2G1

Abstract

Groups are constructed which afford rings of polynomial functions. Using invariant derivations of these rings along classical lines, yields Lie algebras which are completions of free Lie algebras.

Throughout this report I will be a finite set ^{†} and IK a field of characteristic 0.

For each i ∈ I we consider a multiplicative copy E_{i} := {E_{i}(λ), λ ∈ IK} of (IK, +).

Thus E_{i}(λ)E_{i}(μ) = E_{i}(λ + μ). Our group F is the free product F = ∗_{i∈I}E_{i}.

Let W be the free associative monoid (words) on I. Then for w w w ∈ W, w w w ≠ 1,

one has w w w = w w w_{1} ⋯ w w w_{w} with w > 0 and w w w_{1}, ⋯,w w w_{w} in I unique (so we use bold face

characters for words, and the corresponding unbold characters for their lengths). There is

also a unique expression, called reduced, of the form w w w = \overline{w w w}_{1}^{n_{1}} ⋯ \overline{w w w}n\overline{w} ; \overline{w} where \overline{w w w}_{n} ≠ \overline{w w w}_{n+1}.

We set \overline{w w w} := \overline{w w w}_{1} ⋯ \overline{w w w}\overline{w} and \overline{W} = \{\overline{w} : w ∈ W \} (reduced words).

Both authors acknowledge the generous support of NSERC Canada.

† This assumption is not essential. It is made in this report for ease of exposition, to rend some of the notation and concepts used more managable.

To each w w w ∈ W we attach a function π_{w w w} : IK^{w} → F by
π_{w w w} : (λ_{1}, …, λ_{w}) ↦ E_{w w w_{1}}(λ_{1}) ⋯ E_{w w w_{w}}(λ_{w}).

A function f : F → IK is polynomial if for all w w w ∈ W the function f_{w w w} := f ∘ π_{w w w} is polynomial. The ring of all such functions we denote by Pol (F).

Let A be the free associative IK-algebra on the set of symbols {x_{i}}_{i∈I}. To each w w w ∈ W we attach the element x^{w w w} := w ; ∏_{i=1}x_{w w w_{i}} ∈ A. Consider the completion \overline{A} of A and corresponding Magnus group

M := \{1 + ∑_{w w w∈W,w w w≠∅} c_{w w w}x^{w w w}, c_{w w w} ∈ IK\} ⊂ \overline{A}.

For each i ∈ I there is a group homomorphism ε_{i} : IK → M given by

ε_{i} : λ ↦ ∑_{n≥0} \frac{λ^{n}x^{n}}{n!} .

By universal nonsense these yield a (unique) group homomorphism ε : F → M.

For each a a a ∈ W define X^{(a a a)} ∈ \overline{A}^{∗} (dual space) by ⟨X^{(a a a)}, ∑_{bbb∈W}c_{bbb}x_{bbb}⟩ = δ_{a a a,bbb}c_{bbb}

(Kronecker δ). This yields a function

X^{a a a} := X^{(a a a)} ∘ ε : F → IK.

These functions are polynomial and allow us to describe Pol (F) : When can a formal expression ∑_{s s s∈W}c_{s s s}X^{s s s} be thought of as a function on F? The answer is that the set S := \{sss ∈ W : c_{s s s} ≠ 0\} must have the property that the set S_{L} := \{sss ∈ S : \overline{s} ≤ L\} be finite for all L ∈ IN. We call such sets summable.

Proposition 1. Every polynomial function on F can uniquely be written in the form

∑_{s s s∈W}c_{s s s}X^{s s s}; c_{s s s} ∈ IK, where the set S := {sss ∈ W : c_{s s s} ≠ 0} is summable. □

In view of this result the multiplicative structure of Pol (F) is completely determined by

Proposition 2. Let a a a, bbb ∈ W. Then

X^{a a a}X^{bbb} = ∑_{s s s∈a a a↭bbb} X^{s s s}

where ↭ is the shuffle product. □

We now begin to look at the derivations of Pol (F). Recall that Pol (F) has a natural right F -module structure via (x·f)(y) = f(yx^{−1}). Recall also that ∂ ∈ End Pol (F) (the IK algebra of IK-linear endomorphisms of Pol (F)) is right invariant if (x·∂)(f) = ∂(x · f) for all x ∈ F and f ∈ Pol (F).

Let ∂ ∈ End Pol (F) and for all sss ∈ W write ∂(X^{s s s}) = ∑_{w w w∈W}∂\begin{matrix}s & s & s \\ w & w & w \end{matrix}X^{w w w}. Assume that for all summable sets S we have

Cont 1: For all w w w ∈ W the set \{sss ∈ S : ∂\begin{matrix}s & s & s \\ w & w & w \end{matrix} ≠ 0\} is finite. Cont 2: The set \{w w w ∈ W : ∃sss ∈ S with ∂\begin{matrix}s & s & s \\ w & w & w \end{matrix} ≠ 0\} is summable.

Then if f = ∑ c_{s s s}X^{s s s} ∈ Pol (F) we have ∂(f) = ∑ c_{s s s}∂(X^{s s s}). Such endomorphisms we call continuous.

For w w w ∈ W define d(w w w) ∈ ℤ^{I} by d(w w w)(i) := Card {n : w w w_{n} = i}.

Proposition 3. Let ∂ ∈ End Pol (F). For ω ∈ ℤ^{I} and sss ∈ W write ∂_{ω}(X^{s s s}) = ∑_{d(w w w)=ω+d(s s s)}∂\begin{matrix}s & s & s \\ w & w & w \end{matrix}X^{w w w} (a finite sum). Assume ∂ is continuous. Then

(i) ∂_{ω} can uniquely be extended to a continuous operator on Pol (F). Moreover ∂ = ∑_{ω∈ℤ^{I}}∂_{ω} (this last sum with the obvious meaning).

(ii) If ∂ is a derivation then so is ∂_{ω}. (iii) If ∂ is right invariant then so is ∂_{ω}. Furthermore ∂_{ω} = 0 unless ω ∈ ℤ_{−}^{I},

(i.e. ω(i) ≤ 0 for all i ∈ I). □

For each i ∈ I, and sss ∈ W with reduced expression sss = \overline{sss}1^{n_{1}} ⋯ \overline{sss}\overline{s}^{n\overline{s}} , define

∂_{i}(X^{s s s}) = \{ \begin{matrix}n_{1}X^{\overline{s s s}{(n_{1}−1) \atop 1}\overline{s s s}{n_{2} \atop 2} ··· \overline{s s s}n\overline{s} ; \overline{s}} & if i = \overline{sss}_{1} \\ 0 & otherwise. \end{matrix}

It is easy to see that ∂_{i} extends uniquely to a continuous operator on Pol (F) which is also a right invariant derivation of Pol (F).

For each w w w ∈ W let ∂^{w w w} := ∂_{w w w_{1}} ⋯ ∂_{w w w_{w}}. By acting on the X^{s s s}’s one sees that the ∂^{w w w}’s are linearly independent, and hence that the associative subalgebra A of End Pol (F) generated by the ∂_{i}’s is free. This together with Proposition 3 and Friedrichs’ theorem yields:

Theorem 1. Let Lie (F) be the Lie algebra of right invariant continuous derivations of Pol (F). Then

(i) The ∂_{i}’s generate a subalgebra L of Lie (F) which is free. (ii) If ∂ ∈ Lie (F) and we write ∂ = ∑_{ω∈ℤ_{−}^{I}}∂_{ω} then each ∂_{ω} belongs to L. (iii) Lie (F) is a completion of the free Lie algebra L. More precisely

Lie (F) = ( ∏_{ω∈ℤ_{−}^{I}} L_{ω}) ∩ ( ⨁_{r r r∈\overline{W}} ( ∏_{\begin{matrix}w w w∈W \\ \overline{w w w}=r r r \end{matrix}} IK∂^{w}) ). □

References

D. Peterson and V. Kac, Infinite flag varieties and conjugacy theorems, Proc. Nat’l. Acad. Sci. USA 80 (1983), 1778-1782.