dear jim:  i have no idea what is meant by an "abstract definition" of a graded hall basis.  but i think i know how to produce one in two ways, depending on what you mean.  both ways involve some choices.  

1)  you could mean a basis for the graded free lie algebra LV on a graded vector space V.  that is probably what you mean but it is not what hilton meant and used!  he used the classical case.  in some sense, he wanted a geometric basis for the wedge X \vee Y.  you get this in the classical ungraded way.  see 2 below.

i shall assume that 2 is a unit in the ground ring .  it is simpler then.  the reference for all this is my book which also tells you how to proceed when 2 is not a unit.

if you mean the true graded case above, some preliminaries should be recalled where the graded case is different from the ungraded case.

in the graded case, recall that, 

a)  if x has odd degree, Lx = <x,[x,x]} has a basis of two elements and , 

b)  if x has even degree, Lx = <x> has a basis of one element.

in either case, we start by picking off one element x as a basis element and see what we have left.  we do this as follows.

consider the exact sequence

0 \to K \to LV \to <x> \to 0 where

I) x is one of the basis elements of V, write V = <x> \oplus W and (it can be shown that) 

II) K is a free graded lie algebra with generating graded vector space as follows

a)  if x has even degree, the generating vector space for K is

Z = W \oplus [x,W] \oplus [x,[x,W]] \oplus ......, that is, K = LZ where Z is the direct sum

\bigoplus_{k \geq 0} ad^k(x)W

b)  if x has odd degree, the generating vector space for K is

Z = <[x,x]> \oplus W \oplus [x,W], that is, K = LZ

note:  you can see this in the trivial one dimensional case LV = Lx where W is 0.

now iterate this, picking off a generator of Z in the same way.

the result is the (usually infinite) hall basis for LV

for example, if LV = L<x,y> is generated by two elements with x even and y odd, you get

x, y, [x,y],[x[x,y]], ....

then [y,y], [y,[x,y]],[y,[x,[x,y]]], ...

continue picking off one at a time to grow the list.

in the classical ungraded case (that is, everything is concentrated in even degree) hall gave a more specific method of choice for the order of picking off one generator at a time.

2)  but you might not mean the graded case.  you might mean the usual ungraded hall basis. after all this is what hilton used and milnor also.  the hilton-milnor theorem is based on the following variation.  suppose L = LV where V is a graded vector space.  write V = W \oplus U as a direct sum.

consider the exact sequence

0 \to K \to L \to LW \to 0

where K is the free graded lie algebra with generating basis Z =

U \oplus [W,U] \oplus [W,[W,U]] \oplus ..... = \bigoplus_{k \geq 0} ad^k (U) (W), that is, K = LZ

hilton's choice of a hall basis is the result of iterating this, picking off one summand at a time, starting with U, then [W,U], then [W,[W.U]], etc

if U is nonzero, this is produces a countable collection of vector spaces.

note that this is really just the ungraded case of 1, that is, where everything is concentrated in even degrees.  it is therefore the classical case.

closing remarks:  if 2 is not a unit in the ground ring and x has odd degree, then [x,x] has to be replaced by the square x^2.  (note that [x,x] = xx + xx = 2x^2 when x has odd degree.)

as i said before, the reference is my new book, "algebraic methods in unstable homotopy theory", cambridge university press.

i hope this answers your question.

best wishes,

joe neisendorfer

On Sat, Feb 5, 2011 at 9:51 AM, jim stasheff <jds-vZPEP/eVim7G6jMu9gPlYQ@public.gmane.org> wrote:

A graded Hall basis is implicit in Hilton's On the homotopy groups of the union of spheres

Does anyone have a reference to an abstract defn of graded Hall basis for a graded Lie algebra?

Google let me down.

jim

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