RESOLUTIONS OF ASSOCIATIVE AND LIE ALGEBRAS
RON ADIN AND DAVID BLANC
Abstract. Certain canonical resolutions are described for free associati*
*ve and free
Lie algebras in the category of non-associative algebras. These resolutio*
*ns derive
in both cases from geometric objects, which in turn reflect the combinato*
*rics of
suitable collections of leaf-labeled trees.
1.Introduction
We here describe certain explicit canonical resolutions for free associative *
*and free
(graded) Lie algebras, in the category of non-associative algebras. Both resol*
*utions
are based on the combinatorics of suitable collections of leaf-labeled trees.
The Lie case was needed for the second author's description of higher homotopy
operations in rational homotopy theory, in [B2 ]: it turns out that in order to*
* describe
all such higher operations, one must resolve the differential graded Lie algebr*
*a L*
over Q (representing the rational homotopy type of a given space X) simpliciall*
*y,
by suitable free (differential) graded Lie algebras. The higher homotopy operat*
*ions
correspond to relations and syzygies for these free graded Lie algebras, though*
*t of
as non-associative algebras over Q. Since we must replace all the Lie algebras*
* by
the corresponding free differential algebras in a functorial manner (to preserv*
*e the
simplicial structure of the original resolution of L*), we need canonical res*
*olutions
of free Lie algebras in the category of non-associative algebras, as described *
*in this
paper. The construction is closely related to "strongly homotopy Lie algebras" *
*(see
x3.21 below).
Our main interest is indeed in the Lie case. The associative case, which is b*
*ased on
work of Stasheff in [Sts], is included mainly as a preliminary illustration of *
*the ideas
involved, and to fix notation.
As might be expected, the resolutions, being canonical, are far from minimal:*
* this
is reflected in the fact that the resolution for the free Lie algebra L = L
has generators in all dimensions g, while if L is considered as a non-associa*
*tive
(skew-commutative) algebra, its homology vanishes above dimension 1, by Theorem
4.6 below (so that the generators for a minimal resolution are restricted to d*
*imen-
sions 1). Nevertheless, such canonical resolutions are often needed for funct*
*orial
constructions (as noted above), and we hope the combinatorics involved may be of
independent interest.
___________
Date: February 22, 1998.
1991 Mathematics Subject Classification. Primary 18G10; Secondary 05C05, 16S*
*10, 17B01,
17A50, 18G50.
Key words and phrases. resolutions, homology, Lie algebras, associative algeb*
*ras, non-associative
algebras, Jacobi identity, leaf-labeled trees, associahedron.
First author supported in part by the Israel Science Foundation, administered*
* by the Israel
Academy of Sciences and Humanities, and by an Internal Research Grant from Bar-*
*Ilan University.
1
2 RON ADIN AND DAVID BLANC
1.1. Notation and conventions. A graded object over any category C is a sequence
of objects X* = (X0; X1; : :): from C; we write |x| = n if x 2 Xn.
All vector spaces and algebras will be over a field k of characteristic 0 (th*
*ough the
application we have in mind is to the case of k = Q, the rationals). The (gra*
*ded)
vector space with the (graded) set X as its basis is denoted by V, and the v*
*ector
space dual of V is V ?:= Homk(V; k).
We denote by Alg the category of not-necessarily-associative algebras over k*
*, by
Alga Alg the full subcategory of associative algebras, and by Algc Alg the
full subcategory of skew-commutative not-necessarily-associative algebras, sati*
*sfying
xy = -yx for all x; y. Lie denotes the category of Lie algebras.
Similarly, we denote by A the category of graded not-necessarily-associativ*
*e al-
gebras, which we shall call GNAs. An object A* 2 A is thus a graded vector s*
*pace
A* = 1n=0An, equipped with a bilinear graded product . : Ap Aq ! Ap+q for
each p; q 0.
We denote by Aa A the full subcategory of graded associative algebras (GAAs),
and by Ac A the full subcategory of graded-skew-commutative not-necessarily-
associative algebras, satisfying y . x = (-1)|x||y|+1x . y, which we call GCA*
*s. L Ac
denotes the subcategory of graded Lie algebras (GLAs); the product [ ; ] : LpLq*
* !
Lp+q in a GLA L* satisfies the (graded) Jacobi identity
(1.2) (-1)|x||z|[[x; y]; z] + (-1)|y||x|[[y; z]; x] + (-1)|z||y|[[z; x]; y*
*] = 0:
Note that we can embed Alg in A by thinking of A 2 Alg as a graded algebra
A* with A0 = A and Ai = {0} for i 1; similarly for Alga Aa, and so o*
*n.
Thus results stated for graded algebras of various sorts include the ungraded v*
*ersions
as a special case.
There are also differential versions of all the above categories of graded al*
*gebras.
In particular, a differential graded (not-necessarily-associative) skew-commuta*
*tive al-
gebra, called a DGCA, is a GCA (A*; .) 2 Ac, equipped with a differential (i*
*.e., a
map @ = @An: An ! An-1 for each n > 0 such that @2 = 0) which is a graded
derivation in the sense that if x 2 Ap, y 2 Aq then @(x.y) = @(x).y +(-1)px.@(y*
*).
The category of DGCAs is denoted by dAc. Similarly for differential graded no*
*t-
necessarily-associative algebras, or DGNAs.
1.3. Notation. For any GNA (A*; .) 2 A, let [x; y] denote 1_2(x.y+(-1)|x||y|+*
*1y.x).
We then have [y; x] = (-1)|x||y|+1[x; y], so (A*; [ ;)]is now a (non-associativ*
*e) graded
algebra with a graded-skew-commutative multiplication.
1.4. Definition. A differential bigraded (not-necessarily-associative) skew-com*
*mu-
tative algebra, or DBGCA, is a bigraded vector space A*;*= 1p=01s=0Ap;s, equipp*
*ed
with a bilinear graded product . : Ap;s Aq;t! Ap+q;s+tfor each p; q; s; t 0 *
* and
a differential @ = @Ap;s: Ap;s! Ap-1;s satisfying x . y = (-1)(p+s)(q+t)+1y .*
* x and
@(x . y) = @(x) . y + (-1)p+sx . @(y) for x 2 Ap;s and y 2 Aq;t. The categor*
*y of such
DBGCAs will be denoted by dbAc.
L Each DBGCA (A*;*; @A) has an associated DGCA (A*; @A), defined An =
A ^
p+q=n Ap;q (same @ ); some authors re-index A*;* so that Ap;s= Ap;p+s, *
*and
RESOLUTIONS OF ASSOCIATIVE AND LIE ALGEBRAS 3
then A* is obtained from A^*;*by ignoring the first (homological) grading. n =*
* p+s
is called the total degree in A*;*.
1.5. Organization. In section 2 we describe the simpler case of resolutions of *
*free
associative algebras, and in section 3 we describe resolutions of free Lie alge*
*bras. In
section 4 we explain the connection to the homology of non-associative algebras.
1.6. Acknowledgements. We would like to thank Jean-Louis Loday for pointing out
Theorem 4.6 to us, Alan Robinson for providing us with a preprint of [RW ], an*
*d Steve
Shnider and Richard Stanley for several useful conversations. We would also lik*
*e to
thank the referee for his comments.
2. Associative algebras
We begin with a description of our canonical resolution for a free associativ*
*e algebra
by free non-associative algebras. We do so mainly because the underlying combin*
*a-
torics, as well as the corresponding geometric objects, are more transparent in*
* this
case than for Lie algebras. For simplicity we deal here only with the non-grade*
*d case.
First, some definitions. Fix once and for all a finite set X = {x1; : :;:xg} *
*(which we
think of as a set of generators for a free algebra).
2.1. Trees. Recall that a rooted plane tree T (see [Stn]) consists of a (non-em*
*pty)
finite set of nodes, with one designated node called the root r(T ); each nod*
*e v has
a linearly-ordered set of kv other nodes, called its children; v is called th*
*eir parent.
If kv = 0, then v is called a leaf ; otherwise it is called an internal node *
*of T , and
the set of all internal nodes is denoted by int(T ). The set of all leaves o*
*f T has
the obvious natural linear order "from left to right". In this paper we requir*
*e that
kv 6= 1 for all nodes v, i.e., all internal nodes have at least two children.
Note that the smallest rooted plane tree has a single node which is both the *
*root
and a leaf; in all other trees the root is an internal node.
2.2. Definition. Let In = {1; 2; : :;:g}n. For I = (i1; : :;:in) 2 In, let*
* T[I]
denote the collection of all rootedSplane trees with nSleaves labeled xi1; xi2*
*; : :;:xin,
in that order. Write Tn for I2InT[I], and T := 1n=1Tn.
Defining the excess of an internal node v of T to be e(v) := kv - 2, the to*
*tal
excess of each tree determines a lower grading on T by
X
(2.3) T 2 Tnk , k = e(v):
v2int(T)
Thus Tn0 consists precisely of the binary trees, for which every internal no*
*de
has exactly two children; such trees correspond to complete parenthesizations o*
*n (the
labels of) the leaves, e.g.: (x1((x3x2)x1)). More generally, trees in Tnk(0 k*
* n-2)
correspond to partial parenthesizations with n-k -1 pairs of parentheses (inclu*
*ding
an external pair, when n 2) - e.g., (x1(x3x2)x1) 2 T41.
2.4. Associahedra. Consider the (n - 2)-dimensional associahedron Kn-2 of [St*
*s,
xx2,6], whose vertices are indexed by the possible "associations" (i.e., full *
*parenthe-
sizations) on n letters: it has a realization as a convex polytope in Rn-2, *
*and its
boundary @Kn-2 is thus homeomorphic to the (n - 3)-sphere Sn-3 (cf. [Z , p*
*. 18]).
4 RON ADIN AND DAVID BLANC
The dual polytope is simplicial, so that its boundary complex Pn is an (n - *
*3)-
dimensional simplicial complex, in which the top-dimensional faces correspond t*
*o the
vertices of Kn-2, i.e., to binary trees. In general, the k-simplices of Pn *
*are in one-
to-one correspondence with the trees in Tnn-3-k. Note that the indexing is the*
* reverse
of the one we described above: the binary trees now appear in the top dimension.
By choosing various sequences of labels I 2 In = Xn* to serve as the "letter*
*s", we
obtain isomorphic copies of Pn, which we denote by Pn[I], with the correspo*
*nding
rational simplicial chain complexes being C[I]* := C*(Pn[I]; k); similarly C*
*[I]* :=
Homk(C[I]*; k) are the simplicial cochain complexes.
2.5. Definition. We denote by Alg the free non-associative algebra generat*
*ed
by the set X. This is just the non-associativeLtensor algebra on the vector s*
*pace
V, so we may write Alg = 1i=0An(X), where An(X) = V (cf. x1.1*
*).
The multiplication in Alg is defined by concatenation: if T 2 Tp0and T *
*02 Tq0,
then T . T 02 Tp+q0 is obtained by adjoining a root r(T . T 0) as the common*
* parent
of r(T ) and r(T 0), in that order. The free associative algebra on X, den*
*oted by
Alga, and the free graded non-associative algebra on a graded set X*, den*
*oted
by Alg, are defined similarly.
2.6. Definition. Given an associative algebra B 2 Alga, we may think of it as*
* an
object in Alg. As such, it cannot be free (even if B = Alga, say), so we*
* can
try to resolve it: that is, construct a DGNA (E*; @E) 2 dA which is free as a*
* GNA,
together with an augmentation " : E0 ! B such that the augmented chain complex
E* ! B - called a (dA-)resolution - is acyclic. Of course, the same can be*
* done
for any GNA B (e.g., if B is a Lie algebra).
A bigraded dbA-resolution F*;* of a graded algebra A* 2 Aa is defined ana*
*lo-
gously.
2.7. Constructing the resolution. Since @Kn-2 ' Sn-3, we have "HiC[I]* = k
for i = n - 3 and "HiC[I]* = 0 for i 6= n - 3. Let us re-index each C* = *
*C[I]*
by setting ^Ci= Cn-4-i for -1 i n - 4, and ^Cn-3= k, so ^C*= ^C[I]* is
an acyclic augmented chain complex. Note that ^C-1 is the free vector space o*
*n all
full parenthesizations of i1; : :;:in. Thus if we set
M1 M
E* = C^[I]*;
n=1 I2In
we have a dA-resolution of E-1 ~=B = Alga. Moreover, E* has the structu*
*re
of a DGNA, with the product extended bilinearly from the concatenation of trees
defined in x2.5.
3. Lie Algebras
We can now deal with the analogous resolution of a free graded Lie algebra, t*
*hought
of as an object in Ac. Let L 2 L denote the free graded Lie algebra ge*
*nerated
by the graded set X* = {x1; : :;:xg}. Ideally, we would like a Lie analogue *
*of
the associahedron Kn-2 (cf. x2.4): i.e., a (combinatorial) topological space *
*which
encodes the combinatorics of the resolution of L. Apparently this does *
*not
exist, in general; however, there is a version of the dual simplicial complex *
*Pn -
RESOLUTIONS OF ASSOCIATIVE AND LIE ALGEBRAS 5
namely, Boardman's "space of fully-grown trees" (see [Bo , x6]). This can be th*
*ought
of as an n-dimensional generalization of the "Lie-hedron" of [MS ] - but only*
* for Lie
expressions without repetitions (see x3.6 below).
In this section we again use the notation of x2.2, but now we must pay greater
attention to the grading on X*, as well as to the resulting signs. This is be*
*cause in
the case of Lie algebras we must deal separately with expressions in which the *
*same
generator appears more than once.
3.1. Definition. Let I = (i1; : :;:in) be an n-tuple of distinct indices of e*
*lements
in X*, and T 2 T[I] a rooted plane tree with leaves labeled xi1; : :;:xin*
*, in
that order. For each node v 2 int(T ) the symmetric group kv permutes the
kv children of v, changing T into a combinatorially isomorphic tree T 02 T[I*
*0]
(where I0 is the permutation of I), and the actions of the symmetric groups at
different nodesQcommute; so we define the branch automorphism group of T to be
B-Aut (T ) := v2int(T)kv. (The elements ' 2 B-Aut (T ) are, strictly speak*
*ing,
not automorphisms of T , but only of the collection of all rooted plane *
*trees
combinatorially isomorphic to T .)
Equivalently, we may think of B-Aut (T ) as the subgroup of the symmetric g*
*roup
n consisting of all linear orderings of the leaf labels xi1; : :;:xin of T w*
*hich are
compatible with the tree structure of T .
If we identify a tree T with the corresponding partially parenthesized expres*
*sion ff
in the letters xi1; : :,:xin, then we may write B-Aut (ff) for B-Aut (T ), *
* and think
of the group as permuting letters or parenthesized sub-blocks of ff.
3.2. Definition. For T as above, define the degree |v| of any node v of T induc*
*tively
by setting the degree of a leaf labeled by x 2 X* to be |x| (as in x1.1), *
*and if
v 2 int(T ) has children u1; : :;:uk, let |v| := |u1| + . .+.|uk| + k - 2. *
* In particular,
the total degree of T , denoted by |T |, is defined to be the degree of its r*
*oot r(T ).
Thus T[I] is bigraded (with the homological degree defined by (2.3)) and we w*
*rite
T 2 T[I]k;s if |T | = k and T is in homological degree s. If all the generato*
*rs in X*
have degree 0, the two degrees are the same.
Note that the action of B-Aut (T ) respects the degrees of the nodes, so we*
* may
define the Koszul sign "(') of a branch automorphism ' 2 B-Aut (T ) to be
the product of Koszul signs signX*(oe), taken over all the constituent permut*
*ations
oe 2 kv. (The Koszul sign of a permutation acting on a graded set X* is defi*
*ned
by letting signX*((k; k + 1)) = (-1)pq+1, for an adjacent transposition (k; *
*k + 1)
which switches two elements (in our case: nodes) of degrees p, q respectively*
*.)
3.3. Remark. The Koszul sign we use actually differs by -1 from that usually us*
*ed
by algebraists, so as to conform to the topological usage needed for our applic*
*ation
in [B2 ].
3.4. Definition. For T as above, we define the complexity cx(v) of any node v
inductively by setting cx(r(T )) = 0, where r(T ) is the root of T , and if*
* v 2 int(T )
has k children, then cx(u) = cx(v) + k for each child u of v.
3.5. Definition. For each T 2 T[I] Tnk as above, let + Tnk denote
the collection of all trees T 0 obtainable from T under some ' 2 B-Aut (T ) *
*with
ffl(') = +1, and similarly define - (with ffl(') = -1). We think of a*
*s the
6 RON ADIN AND DAVID BLANC
equivalence class of the tree T , with respect to the relation of abstract com*
*binatorial
isomorphism, partitioned by sign into two subclasses.
Write ^In forStheScollection of (unordered) n-multisets of elements of X*, *
* and
set ^T[I^] := I2^I T2T[I]. We may think of ^T[I^] as the collection o*
*f all rooted
trees T^ with n leaves labeled xi1; xi2; : :;:xin, without a specified planar e*
*mbedding,
but with a sign determiningSwhich of the two classes of possible embeddings we *
*have
chosen. Set ^Tn:= ^I2^In^T[I^].
3.6. Definition. The space of trees is the n-dimensional simplicial complex who*
*se
k-simplices consist, in our notation, of the unsigned equivalence classes *
*= +[
- of rooted trees T 2 Tn+3n-k[I^], for some fixed set ^I2 ^In+3of n + *
*3 distinct
labels. It is denoted by Tn in [RW , x1], but to avoid over-use of the lette*
*r T we shall
denote it here by Mn. See also [HW ].
Thus, the k-simplices Mnk are in one-to-one correspondence with the isomorphi*
*sm
classes of leaf-labeled trees - without a specified planar embedding - having e*
*xactly
k + 2 internal vertices, and with leaves labeled xi1; : :;:xin+3, say, not n*
*ecessarily
in that order. As Robinson and Whitehouse show, Mn is homotopy equivalent to
a wedge of (n + 2)! n-spheres (cf. [RW , Thm. 1.5]). However, we cannot use*
* their
results as they stand, since we need to be careful with signs. So we make the f*
*ollowing
definitions:
3.7. Definition. For I = (i1; : :;:in) as above, let J* = J[I]* denote the (bi)*
*graded
vector space with J[I]k spanned by T^[I]*;n-3-k for -1 k n-2. (For simplicity
we suppress the "topological" grading due to the grading of X*, since it is not*
* relevant
at this stage.) We define a differential @ = @Jk: Jk ! Jk-1 as follows:
Represent any 2 ^T[I]*;n-3-k by a partially parenthesized expression ff*
* in the
letters xi1; : :,:xin; then @[T^] will be represented by the sum of all exp*
*ressions
obtained from ff by omitting a pair of parentheses (equivalently: by contractin*
*g one
internal edge of T , i.e., an edge connecting two internal vertices) - with a*
*ppropriate
signs. These signs are determined recursively by the following three rules:
(1) If ff = ((a1a2: :a:k)b1: :b:m), where each ai or bj is a partially p*
*aren-
thesized expression (possibly just a generator x 2 X*), then the summand
(a1a2: :a:kb1: :b:m) appears in the expansion of @[ff] with the sign (*
*-1)m+1 .
(2) If ff = (ab1: :b:m), where a (and each bj) is a partially parenthesize*
*d expres-
sion, then the sum comprising (@[a]b1: :b:m) appears in the expansion of*
* @[ff]
with the sign (-1)m+1 .
(3) If ff = (a1: :a:kbc1: :c:m), where each of ai, b and cj is a partially*
* parenthe-
sized expression, then
P k
@[ff] = (-1)( i=1|ai|)|b|+k@[(ba1: :a:kc1: :c:m)]:
We set @[x] := 0 for any generator x 2 X*.
3.8. Example. For any partially parenthesized expressions a; b; c; d we have
@[((ab)c)] = (abc) + ((@[a]b)c) + (-1)|a|((a@[b])c) + (-1)|a|+|b|((ab)@[*
*c])
RESOLUTIONS OF ASSOCIATIVE AND LIE ALGEBRAS 7
and
@[(((ab)c)d)] =((ab)cd) + ((abc)d) + (((@[a]b)c)d) + (-1)|a|(((a@[b])*
*c)d)
+ (-1)|a|+|b|(((ab)@[c])d) + (-1)|a|+|b|+|c|(((ab)c)@[d])
3.9. Note.Rules (1) and (2) say that if ff = ((. .(.(a)b11: :b:1m1) . .).bt1: *
*:b:tmt), where
the expression a := a1a2: :a:k is initial in ff (with only left parentheses p*
*receding
it), then the sequence ffa = ((. .(.a1a2: :a:kb11: :b:1m1) . .).bt1: :b:tmt),*
* obtained from
ff by omitting the outer parentheses around a, appears in the expansion of @*
*[ff]
with the sign (-1)cx(a)(Def. 3.4).
Rule (3) says that if one wishes to omit the outer parentheses around a whe*
*n it
is not initial in ff, and T is the rooted plane tree corresponding to ff, then*
* one must
move a to the left of all its siblings, and similarly for all its ancestors, b*
*y a suitable
branch automorphism ' 2 B-Aut (T ) - which introduces the sign "(') - and then
apply the previous rule. Note that this ' is not unique (unless we specify a p*
*referred
choice for the automorphism); but it is not hard to see that the various choic*
*es of
' differ by elements of the isotropy subgroup of B-Aut (T ) which leave the *
*node
corresponding to a fixed, and that the correspondence ff 7! ffa commutes wit*
*h the
action of this subgroup. So the sign of the resulting class in the exp*
*ansion of
@J[ff] - and thus @J itself - is well defined.
3.10. Lemma. @J is a differential on J*.
Proof.Given a partially parenthesized expression ff = ((. .(.a1)a2) : :a:t), *
* with each
ai= ai1ai2: :a:iki, we must verify that, for any two pairs of parentheses in *
*ff, omitting
them in the two possible orders yields opposite signs in the expansion of @[@*
*[ff]].
Using the recursive rules of x3.7, one must check the following cases:
(i) ff = ((a)(b)c), where a := a1: :a:k, b := b1: :b:` and c := c1: :c:m.*
* We
see that if |a| := |a1| + . .+.|ak| and so on, then the two orders of o*
*mitting
parentheses yield
b|+`-2)+k
((a)(b)c) 7!(-1)2+m (a(b)c) = (-1)(2+m)+|a|(| ((b)ac) 7!
b|+`)+k)+k+m+1 (|a|(|b|+`)+1)+|a||b|+k`
(-1)(m+|a|(| (bac) = (-1) (abc) =
(-1)(|a|+k)`+1(abc)
and
b|+`)+1 ((|a|+k)(|b|+`)+1)+2+m
((a)(b)c) =(-1)(|a|+k)(| ((b)(a)c) 7! (-1) (b(a)c) =
(-1)(|a|+k)`+`+1+m((a)bc) 7! (-1)(|a|+k)`(abc)
respectively, which indeed differ in sign.
The remaining cases, namely:
(ii) (((a)b)c) 7! (abc,
(iii) (a(b)c) 7! (@[a]bc),
(iv) ((ab)c) 7! (@[a]bc),
(v) (abc) 7! (@[a]@[b]c),
*
* __
are dealt with in a similar fashion. *
* |__|
8 RON ADIN AND DAVID BLANC
To show that J[I]* is acyclic except in the top dimension for each I, we mi*
*mic
the geometric proof of Robinson and Whitehouse. This requires another
3.11. Definition. Given any subset A = {i1; : :;:ik} of I with k 2, let *
*"J[A]*
denote the subcomplex of J[I]* spanned by all trees T in which xi1; : :;:xik al*
*l have
the same parent node (i.e., in the corresponding expression ff, the letters xi*
*j are
not separated by unbalanced pairs of parentheses). Since @J is defined by omi*
*tting
parentheses, this is clearly a subcomplex. Compare [RW , Def. 1.3], and [RW ,*
* Lemma
1.4] for the following
3.12. Lemma. For any A I with |A| 2, the complex "J[A]* is acyclic.
Proof.Write x := xi1: :x:ik. We define a contracting homotopy ffi = ffiAm: "*
*J[A]m !
"J[A]m+1 on basis elements T 2 ^T[I]*;n-3-m (or equivalently, on the corresp*
*onding
partially parenthesized expression ff), and extend linearly:
If T has a node whose leaves are precisely xi1; : :;:xik (i.e., if ff has (*
*x) as a
sub-expression), then ffi(ff) := 0; while if ff = ((. .(.xa1) . .).at) (wh*
*ere each
ai= ai1ai2: :a:iki), we set ffi(ff) := (-1)cx(x)((. .(.(x)a1) . .).at). By r*
*equiring that
ffi('(ff)) = (-1)"(')'(ffi(ff)) for any ' 2 B-Aut (ff) (as long as both side*
*s of the
equation make sense), we have defined ffi on all of "J[A]*.
Using the rules of x3.7 above, one may verify that ffi is indeed a contractin*
*g homo- __
topy for "J[A]* (i.e., @ O ffi + ffi O @ = id). *
* |__|
This implies the following variant of [RW , Thm. 1.5]:
3.13. Proposition. For any n distinct indices I = (i1; : :;:in) we have Hi(J[I*
*]*) =
0 for -1 i < n - 3, and Hn-3(J[I]*) ~=k(n-1)!.
S
Proof.Let C* := 1k<`, we must again re-index, as in x2.7, by setting G[I^]i := (J[I^]n-*
*3-i)?
(vector space dual) for 0 i n - 3, so that is (up to sign) the
re-indexed cochain complex for Mn. We then define
M1 M
(3.17) F*;*:= G[I^]*:
n=0 ^I2^In+3
(We have re-inserted the "topological" grading into our notation at this stage,*
* to call
attention to the fact that we have constructed a bigraded resolution.)
3.18. Remark. Note that G[I^]* once more reverses the indexing, so that for I*
* con-
sisting of distinct indices, at least, G[I^]k is spanned by all trees of lower *
*(homological)
degree k, as defined in (2.3). Similarly, @?(T ), which we defined by the vec*
*tor space
dual of @J, could be described directly as the signed sum of all trees obtained*
* from T
by adding internal edges - or equivalently, adding parentheses to the corresp*
*onding
partially parenthesized expression ff, with the signs again given by x3.7. Thi*
*s is in
fact more natural algebraically, as the following examples show:
3.19. Example. For any partially parenthesized expressions a, b, and c in *
*F*;*,
one has
@?[(abc)] =((ab)c) + (-1)|a||b|+|a||c|((bc)a) + (-1)|a||c|+|b||c|((c*
*a)b)
+ ((@?[a]b)c) + (-1)|a|((a@?[b])c) + (-1)|a|+|b|((ab)@?[c*
*])
(compare Example 3.8). In particular, for any three generators x; y; z 2 X* one*
* has
@?[(xyz)] = ((xy)z) + (-1)|x||y|+|x||z|((yz)x) + (-1)|x||z|+|y||z|((zx)*
*y);
which up to the action of B-Aut (T ) is the usual graded Jacobi identity of (*
*1.2).
Similarly, for x; y; z; w 2 X*
(3.20)
@F [(xyzw)] =- ((xy)zw) + (-1)|y||z|((xz)yw) + (-1)|y||w|+|z||w|+1((xw)yz) +
(-1)|x||y|+|x||z|+1((yz)xw) + (-1)|x||y|+|x||w|+|z||w|+1((yw)xz*
*) +
(-1)(|x|+|y|)(|z|+|w|)((zw)xy) + ((xyz)w) + (-1)|z||w|+1((xyw)z*
*) +
(-1)|y|(|z|+|w|)((xzw)y)(-1)|x|(|y|+|z|+|w|+1)((yzw)x);
which can be thought of as a "second order Jacobi identity".
F*;* has an augmentation " : F*;0! L, which takes any fully parenthesi*
*zed
expression in the elements xi to the corresponding iterated Lie bracket. In fac*
*t, with
the product structure extended linearly from concatentation of trees, as in x2.*
*5, F*;*
10 RON ADIN AND DAVID BLANC
is a DBGCA (see x1.4; or a DGCA, when X* is ungraded). The product is graded-
skew-commutative, and Rule (2) of x3.7 implies that @?[a.b] = @?[a].b+(-1)|a|a.*
*@?[b]
for any a; b 2 F*;*.
3.21. Remark. In fact, F*;* is not merely a free bigraded skew-commutative not-
necessarily-associative algebra, but also is the free strongly homotopy Lie alg*
*ebra on
the graded set X*. The analogous singly-graded objects, first introduced by St*
*asheff
and Schlessinger in [SS2 ] (see also [SS2 ]) play a role in deformation theory,*
* in rational
homotopy theory, and in mathematical physics. See also [GK , x1.3.9], and [LM *
* , 2.1],
where these are called L(1)-structures. Martin Markl has pointed out to us th*
*at
the resolution for free Lie algebras we define can also be obtained by the meth*
*ods of
[GK ] and [M ].
3.22. Theorem. F*;* is a resolution of L = L.
Proof.It is clear from the construction that H0(F*;*) ~= L, and that F*;* is*
* free
as a DBGCA, so it suffices to show that F*;* is acyclic in positive degrees. *
*Since
F*;* is defined as a direct sum of chain complexes (3.17), it is enough to cons*
*ider
each summand separately. Thus, for each ^I2 ^In+3(fixed for the remainder of t*
*he
proof), it suffices to show that J[I^]* is acyclic in degrees < n.
To do so, first consider the corresponding multiset I0 without repetitions. B*
*ecause
J[I0]* is acyclic by Proposition 3.13 above, it has (many possible) contractin*
*g chain
homotopies. We now proceed to make a specific choice of such a homotopy (depend*
*ent
on the original I):
Assume that I = I0=L-Aut (I^) for G = L-Aut (I^) n as above. Since @k+1
commutes with the action of G, the summand Im(@k+1) of J[I0]k is invariant under
this action. Thus, by Maschke's Theorem (see [CR , 10.8]), for each 0 < k n *
*- 3
we may choose a splitting
J[I0]k = Im(@k+1) Sk;
where Sk is also invariant under the action of G, and of course @k|Sk is an*
* isomor-
phism onto Im(@k) J[I0]k-1 (because Im(@k+1) = Ker(@k)).
We may thus define a linear map ffi0k: J[I0]k ! J[I0]k+1 by ffi0k(@k+1Ti) *
*= Ti, and
ffi0k|Sk 0; this is a contracting homotopy for J[I0]*. Moreover, it commutes*
* with
the action of G, so it induces a contracting homotopy ffi on J[I^]*, which *
*is_thus
acyclic. *
*|__|
4. Homology of DGLs
We may use the resolutions constructed above to calculate the homology of a f*
*ree
Lie or associative algebra, considered as a non-associative algebra. We first *
*recall
Quillen's definition of homology in model categories:
4.1. Definition. An object X in a category C is said to be abelian if it is an *
*abelian
group object - that is, if HomC(Y; X) has a natural abelian group structure*
* for
any Y 2 C. When C is Lie, Alg, Alga, L, or A, for example, this is equi*
*valent
to requiring that all products vanish in X.
The full subcategory of abelian objects in C is denoted by Cab C. It is equi*
*valent
to the category Vect of vector spaces if C = Lie, Alg, Alga, and so on, a*
*nd to
RESOLUTIONS OF ASSOCIATIVE AND LIE ALGEBRAS 11
the category V of graded vactor spaces if C = L or A; so we see that Cab is
an abelian category, in the cases of interest to us. We then have an abelianiz*
*ation
functor Ab : C ! Cab, along with a natural transformation : Id ! Ab having*
* the
appropriate universal property. In all the examples above, Ab(X) = X=I(X), wh*
*ere
I(X) is the ideal in X 2 C generated by all non-trivial products.
4.2. Homology of algebras. Let C be a category as above, which also has a model
category structure (see [Q2 , II, x1]). In [Q1 , II, x5] (or [Q3 , x2]), Quill*
*en defines the
homology of an object X 2 C to be the total left derived functor L(Ab) of *
*Ab,
applied to X (cf. [Q1 , I, x4]).
In more familiar terms, this means that we construct a resolution A ! X (i.*
*e.,
replace X by a weakly equivalent cofibrant object A 2 C), and then define the *
*i-th
homology group of X by HiX := Hi(Ab(A)), where Ab(A) is (equivalent to) a
chain complex in an abelian category, so its homology is defined as usual. One *
*must
verify, of course, that this definition is independent of the choice of the res*
*olution
A ! X.
If C itself does not have a closed model category structure, one often define*
*s the
homology of X 2 C by embedding C in some category which does have such a struc-
ture, which in most cases may be taken to be sC, the category of simplicial o*
*bjects
over C (see [Q1 , II, x4]). Thus, if : C ,! sC is the embedding of categori*
*es defined
by taking (C) to be the constant simplicial object equal to C in all dimensio*
*ns,
then Hi(C) := ssi(L(Ab O )C).
This is the approach usually taken for C = Lie, Alg, A, and so on: to de*
*fine
the homology of a graded Lie algebra L* 2 L, say, one chooses a free simplici*
*al
resolution Ao;*! L* and then calculates the homotopy groups of the simplicial
graded vector space Ab(Ao;*) 2 sV.
As for graded Lie algebras and skew-commutative algebras, one can define clos*
*ed
model category structures on sAc and dbAc (see [BS , x2], and [B1 , x4]), and b*
*ecause
we are working over a field of characteristic 0, we have the following analogue*
* of [Q2 ,
Props. 2.3 & 4.6, Thm. 4.4]
N
4.3. Proposition. There are adjoint functors sAN*dbA, which induce equivalen*
*ces
of the corresponding homotopy categories ho(sA) ho(dbAc). N* takes free DBG-
CAs to free simplicial GCAs.
*
* __
Proof.See [B2 , Props. 2.9, 7.2, 7.3]. *
* |__|
Thus we may use DGCAs (resp. DBGCAs) instead of simplicial commutative al-
gebras (resp. simplicial GCAs) as our free resolutions - as in x2.6 - and r*
*eplace
the homotopy groups by the homology groups of the corresponding (bigraded) chain
complex.
4.4. Remark. We gave the definition of homology in its simplicial version, which
applies to more general types of universal algebras, in order to emphasize that*
* our
methods do not apply to associative or Lie algebras over an arbitrary (commutat*
*ive)
ground ring k, because in that case one cannot resort to differential graded al*
*gebras
as resolutions. (The case of k = Z would have been of special interest.)
12 RON ADIN AND DAVID BLANC
4.5. Calculating the homology. In particular, we may use the resolutions E* !
Alga and F*;*! L defined above to calculate the homology of a free
associative or (graded) Lie algebra, considered as an object in Alg or A. Exp*
*licitly,
if Eo is the simplicial algebra corresponding to the DGNA E*, then Hn(Alga<*
*X>)
is defined to be the n-th homotopy group of the simplicial vector space Ab(Eo),
where the abelianization functor is applied in each simplicial dimension separa*
*tely;
and similarly for H*(L).
However, the definition of the correspondence between E* and Eo (cf. [B2 ,
Proof of Prop. 2.9]) implies that the indecomposables in the two cases are in b*
*ijective
correspondence, so that in fact we may calculate Hn(Alga) as the n-th homolo*
*gy
group of the differential vector space (i.e., chain complex) Ab(E*) := E*=I(E*)*
*. This
simply means that we must replace by 0 all trees in E* whose roots have only *
*two
children, and compute the homology of the resulting chain complex. Similarly f*
*or
F*;*.
4.6. Theorem. Hi(L) = 0 for i 2.
Proof.As before, let I = (i1; : :;:in) be some n-tuple of distinct indices of*
* elements
in the graded set X*, and let N* = N[I]* denote the subcomplex of J* = J[I*
*]*
spanned by all trees T with kr(T) 3. We will say that a subcomplex C* J* is
`-coconnected if Hi(C*) = 0 for i n - 3 - `.
(I) Given any subset A = {i1; : :;:ik} of I, let N"[A]* denote the subcom*
*plex
of N[I]* spanned by all trees T in which xi1; : :;:xik all have the same pa*
*rent
node (compare x3.11 above). We claim that "N[A]* is k-coconnected, for any A w*
*ith
k 2.
This is shown essentially as in the proof of Lemma 3.12. We define a (partia*
*l)
contracting homotopy ffi : "N[A]i! "N[A]i+1 for i < n - 3 - k as follows:
Write x := xi1; : :;:xik. If ff has (x) as a sub-expression, then ffi[f*
*f] = 0; if
ff = ((. .(.xa1) . .).at), we set ffi[ff] = (-1)cx(x)((. .(.(x)a1) . .).at) *
*(and require that
ffi['(ff)] = (-1)"(')'(ffi[ff]) for any ' 2 B-Aut (ff)). Any other basis el*
*ement of
"N[A]* is in the subcomplex xJ[I \ A]* - i.e., of the form ff = (xa ) for*
* some
a = a1. .a.tfor some t 1, where (a) 2 J[I \ A]* (again, up to the B-Aut *
*(ff)-
action). But xJ[I \ A]* is isomorphic to the complex J[I \ A]* shifted up *
*by k,
and this has a contracting homotopy ffi0 in degrees < n - k - 3 by Propositio*
*n 3.13;
set ffi[(xa )] := ffi0[(a)] if t = 1, and ffi[(xa )] := ffi0[(a)] + ((x)a)*
* if t 2.
(II) We now show that N* = N[I]* is 2-coconnected. If we denote the sequence
of chain complexes
(N"[(i1; i2)]*; N"[(i1; i3)]*; : :;:"N[(i1; in-1)]*; N"[(i2; i3)]*; : :;:"N*
*[(in-2; in-1)]*)
S t
by (Dt*)mt=1,S and set Ct*= i=1Di* for 1 t m, then the chain complex
Cm*= 1k<`