The functor L of Quillen

4 Pages·2011·0.118 MB·English

by  Marco Manetti

THE FUNCTOR L OF QUILLEN
DEFORMATIONTHEORY2011-12;M.M.
Let R be a commutative ring, by a nonassociative (= not necessarily associative) graded R-
algebra we mean a graded R-module M = ⊕Mi endowed with a R-bilinear map Mi ×Mj →
Mi+j.
The nonassociative algebra M is called unitary if there exist a “unity” 1 ∈ M0 such that
1m=m1=m for every m∈M.
A left ideal (resp.: right ideal) of M is a graded submodule I ⊂M such that MI ⊂I (resp.:
IM ⊂I). A graded submodule is called an ideal if it is both a left and right ideal.
AhomomorphismofR-modulesd: M →M iscalledaderivationofdegreekifd(Mi)⊂Mi+k
and satisfies the graded Leibniz rule d(ab)=d(a)b+(−1)kaad(b).
If M is as associative graded algebra we denote by M the associated graded Lie algebra,
L
with bracket equal to the graded commutator [a,b]=ab−(−1)abba.
It is easy to see that if f: M →M is a derivation, then also f: M →M is a derivation.
L L
Notation: For a graded Lie algebra H we denote [a,b,c]=[a,[b,c]] and more generally
[a ,...,a ]=[a ,[a ,...,a ]]=[a ,[a ,[a ,...,[a ,a ]...].
1 n 1 2 n 1 2 3 n−1 n
Notice that the descending central series H[n] may be defined as
H[n] =Span{[a ,...,a ]}, a ,...,a ∈H.
1 n 1 n
Jacoby identity becomes
[[a,b],c]=[a,b,c]−(−1)ab[b,a,c].
1. Free graded Lie algebras
Let V be a graded vector space over K, we denote by
(cid:77) (cid:79)n (cid:77) (cid:79)n
T(V)= V, T(V)= V.
n≥0 n≥1
The tensor product induce on T(V) a structure of unital associative graded algebra and T(V)
is an ideal of T(V). The algebra T(V) is called tensor algebra generated by V and T(V) is
called the reduced tensor algebra generated by V.
Lemma 1.1. Let V be a K-vector space and ı: V → T(V) the natural inclusion. For every
graded associative K-algebra R and every linear map f ∈ Hom0(V,R) there exists a unique
K
homomorphism of K-algebras φ: T(V)→R such that f =φı.
Proof. Clear. (cid:3)
Lemma 1.2. Let V be a K-vector space and ı: V → T(V) the natural inclusion. For every
f ∈Homk(V,T(V)) there exists a unique derivation φ: T(V)→T(V) such that f =φı.
K
Proof. Leibniz rule forces to define φ as
n
(cid:88)
φ(v ⊗···⊗v )= (−1)k(v1+···+vi−1) v ⊗···⊗f(v )⊗···⊗v .
1 n 1 i n
i=1
(cid:3)
1
2 DEFORMATIONTHEORY2011-12;M.M.
Definition 1.3. Let V be a graded K-vector space; the free Lie algebra generated by V is
the smallest graded Lie subalgebra L(V)⊂T(V) which contains V.
L
Equivalently L(V) is the intersection of all the Lie subalgebras of T(V) containing V.
L
For every integer n > 0 we denote by L(V) ⊂ L(V)∩(cid:78)nV the linear subspace generated
n
by all the elements
[v ,v ,...,v ], n>0, v ,...,v ∈V.
1 2 n 1 n
By definition L(V) = V, L(V) = [V,L(V) ] and therefore ⊕ L(V) ⊂ L(V). On the
1 n n−1 n>0 n
other hand the Jacobi identity [[x,y],z]=[x,[y,z]]−[y,[x,z]] implies that
[L(V) ,L(V) ]⊂[V,[L(V) ,L(V) ]]+[L(V) ,[V,L(V) ]]
n m n−1 m n−1 m
and therefore, by induction on n, [L(V) ,L(V) ]⊂L(V) .
n m n+m
This implies that the direct sum ⊕ L(V) is a graded Lie subalgebra of L(V); therefore
n>0 n
⊕ L(V) =L(V) and for every n
n>0 n
(cid:79)n (cid:77)
L(V) =L(V)∩ V, L(V)[n] = L(V) .
n i
i≥n
TheconstructionV (cid:55)→L(V)isafunctorfromthecategoryofgradedvectorspacestothecat-
egory of graded Lie algebras, since every morphism of vector spaces V →W induce a morphism
of algebras T(V)→T(W) which restricts to a morphism of Lie algebras L(V)→L(W).
Theorem 1.4 (Dynkyn-Sprecht-Wever). Assume that V is a vector space and H a graded Lie
algebra with bracket [,]. Let σ ∈ Hom0(V,H) be a linear map and define, for every n ≥ 2, the
1
maps
(cid:79)n
σ : V →H, σ (v ⊗···⊗v )=[σ (v ),σ (v ⊗···⊗v )]=[σ (v ),σ (v ),...,σ (v )].
n n 1 n 1 1 n−1 2 n 1 1 1 2 1 n
Then the linear map
∞
σ = (cid:88) σn: L(V)→H, σ(v ⊗···⊗v )= 1[σ (v ),σ (v ),...,σ (v )],
n 1 n n 1 1 1 2 1 n
n=1
is the unique homomorphism of graded Lie algebras extending σ .
1
Proof. The adjoint representation
θ: V →Hom∗(H,H), θ(v)x=[σ (v),x],
1
extends to a morphism of graded associative algebras θ: T(V)→Hom∗(H,H) by the composi-
tion rule
θ(v ⊗···⊗v )x=θ(v )θ(v )···θ(v )x.
1 s 1 2 s
By definition
σ (v ⊗···⊗v )=θ(v ⊗···⊗v )σ (v )
n 1 n 1 n−1 1 n
and more generally, for every v ,...,v ,w ,...,w ∈V we have
1 n 1 m
σ (v ⊗···⊗v ⊗w ⊗...⊗w )=θ(v ⊗···⊗v )σ (w ⊗···⊗w ).
n+m 1 n 1 m 1 n m 1 m
Since every element of L(V) is a linear combination of homogeneous elements it is sufficient to
prove, by induction on n≥1, the following properties
A : If m≤n, x∈L(V) and y ∈L(V) then σ([x,y])=[σ(x),σ(y)].
n m n
B : If m≤n, y ∈L(V) and h∈H then θ(y)h=[σ(y),h].
n m
The initial step n=1 is straightforward, assume therefore n≥2.
[A +B ⇒ B ] We have to consider only the case m = n. The element y is a linear
n−1 n−1 n
combination of elements of the form [a,b], a∈V, b∈L(V) and, using B we get
n−1 n−1
θ(y)h=[σ(a),θ(b)h]−(−1)abθ(b)[σ(a),h]=[σ(a),[σ(b),h]]−(−1)ab[σ(b),[σ(a),h]].
THE FUNCTOR L OF QUILLEN 3
Using A we get therefore
n−1
θ(y)h=[[σ(a),σ(b)],h]=[σ(y),h].
[B ⇒A ]
n n
σ ([x,y])=θ(x)σ (y)−(−1)xyθ(y)σ (x)=[σ(x),σ (y)]−(−1)xy[σ(y),σ (x)]
n+m n m n m
=n[σ(x),σ(y)]−m(−1)xy[σ(y),σ(x)]=(n+m)[σ(x),σ(y)].
Since L(V) is generated by V as a Lie algebra, the unicity of σ follows. (cid:3)
Corollary 1.5. For every vector space V the linear map
1
π: T(V)→L(V), π(v ⊗···⊗v )= [v ,v ,...,v ]
1 n n 1 2 n
is a projection. In particular for every n>0
L(V) ={x∈V⊗n |π(x)=x}.
n
Proof. The identity on L(V) is the unique Lie homomorphism extending the natural inclusion
V →L(V). (cid:3)
Lemma 1.6. Every f ∈Homk(V,L(V)) extends to a unique derivation L(V)→L(V).
Proof. The composition f: V → L(V) (cid:44)→ T(V) extends to a derivation F: T(V) → T(V).
Leibniz rule gives the unicity and F(L(V))⊂L(V). (cid:3)
2. The functor L of Quillen
Let s be a formal symbol of degree +1. For every graded vector space V denote
sV ={sv |v ∈V}, and s∈Hom1(V,sV), s(v)=sv.
Lemma 2.1. For every x∈V ⊗V we have
(cid:18) (cid:19)
x+tw(x)
π((s⊗s)x)=(s⊗s) .
2
In particular (s⊗s)x∈L(sV) if and only if x=tw(x).
2
Proof. By linearity may assume x=u⊗v. Then tw(x)=(−1)uvv⊗u and
2π((s⊗s)x)=2π((−1)usu⊗sv)=(−1)usu⊗sv−(−1)u+(u+1)(v+1)sv⊗su=
=(−1)usu⊗sv+(−1)v+uvsv⊗su=(s⊗s)(x+tw(x)).
(cid:3)
Let (C,∆,δ) be a differential graded cocommutative coalgebra. Denote by:
(1) d ∈Der1(L(sC),L(sC)) the derivation induced by the map
1
d : sC →L(sC), d (sv)=−sδ(v).
1 1
(2) d ∈Der1(L(sC),L(sC)) the derivation induced by the map
2
d : sC →L(sC), d (sv)=−(s⊗s)∆(v)
2 2
(this makes sense since tw◦∆=∆).
Theorem 2.2. In the above notation d2 = d2 = [d ,d ] = 0 and then (L(sC),[,],d +d ) is a
1 2 1 2 1 2
DGLA.
4 DEFORMATIONTHEORY2011-12;M.M.
Proof. d2(sv)=sδ2(v)=0.
1
d d (sv)=d (−sδ(v))=(s⊗s)∆(δ(v))=(s⊗s)(δ⊗Id+Id⊗δ)∆(v)=
2 1 2
=(−sδ⊗s+s⊗sδ)∆(v)=(d ⊗Id+Id⊗d )(s⊗s)∆(v)=d d (sv).
1 1 1 2
Remainstoprovethat[d ,d ]=2d2 =0.Givenx∈C⊗C wehavethestraighforwardidentities
2 2 2
(d ⊗Id)(s⊗s)(x)=−(s⊗s⊗s)(∆⊗Id)(x),
2
(Id⊗d )(s⊗s)(x)=(s⊗s⊗s)(Id⊗∆)(x),
2
and then for v ∈C we have
d2(sv)=−(d ⊗Id+Id⊗d )(s⊗s)∆(v)=−(s⊗s⊗s)(∆⊗Id−Id⊗∆)∆(v)=0.
2 2 2
(cid:3)
Definition 2.3. For every differential graded cocommutative coalgebra C denote L(C) the
differential graded Lie algebra (L(sC),[,],d +d ).
1 2
It is quite obvious that L is a functor.
Proposition 2.4 (Quillen). The functor L preserves quasiisomorphisms.
Proof. Omitted. (cid:3)
3. Twisting morphisms
Let (C,∆,δ) be a differential graded cocommutative coalgebra and (H,[,],∂) a DGLA. Then
the space Hom∗(C,H) has a natural structure of DGLA with differential
d(f)=∂f −(−1)ffδ
and bracket [f,g] equal to the composition
∆ f⊗g [,]
C−→C⊗C −−−→H ⊗H−→H.
Exercise Verify that this is a DGLA.
Definition3.1. Atwistingmorphismisamapα∈Hom1(C,H)satisfyingtheMaurer-Cartan
equation.
Thecompositionwiths: C →sC giveanisomorphismHom1(C,H)=Hom0(sC,H)andthen
everyelementα ∈Hom1(C,H)givesamorphimsofgradedLiealgebrasα: L(C)=L(sC)→H.
1
Lemma 3.2. α ∈ Hom1(C,H) is a twisting morphism if and only if α: L(C) → H is a
1
morphism of DGLA:
MC(Hom∗(C,H))=Hom (L(C),H).
DGLA
Proof. α is a morphims of DGLA if and only if
dα=α(d +d ).
1 2
Beingtheabovemapstwoα-derivations,byLeibnizruleitissufficienttoprovethattheycoincide
on sC, i.e. that for every v ∈C
1
dα (sv)=α d (sv)+ α (d (sv))
1 1 1 2 2 2
We have dα (sv)=dα (v), α d (sv)=−α (δ(v)) and then
1 1 1 1 1
dα (sv)−α d (sv)=(dα +α δ)v =(dα )v.
1 1 1 1 1 1
Similarly
α (d (sv))=−α ((s⊗s)∆(v))=−[α ,α ](v).
2 2 2 1 1
(cid:3)