E-Book Content
CHAPTER 1
L∞ -algebras 1.1. Koszul sign and unshuffles Given a graded vector space V , the twist map extends naturally, for every n ≥ 0, to an Nn action of the symmetric group Σn on the graded vector space V: tw : V ⊗n × Σn → V ⊗n . More explicitely, setting σtw = tw(−, σ −1 ) : V ⊗n → V ⊗n , for v1 , . . . , vn homogeneous vectors and σ ∈ Σn we have: σtw (v1 ⊗ · · · ⊗ vn ) = ±(vσ−1 (1) ⊗ · · · ⊗ vσ−1 (n) ), where the sign is the signature of the restriction of σ to the subset of indices i such that vi has odd degree. Definition 1.1.1. The Koszul sign ε(σ, V ; v1 , . . . , vn ) = ±1 is defined by the relation −1 σtw (v1 ⊗ · · · ⊗ vn ) = ε(V, σ; v1 , . . . , vn )(vσ(1) ⊗ · · · ⊗ vσ(n) )
The antisymmetric Koszul sign χ(σ, V ; v1 , . . . , vn ) = ±1 is the product of the Koszul sign and the signature of the permutation: χ(σ, V ; v1 , . . . , vn ) = (−1)σ ε(σ, V ; v1 , . . . , vn ). For notational simplicity we shall write ε(σ; v1 , . . . , vn ) or ε(σ) for the Koszul sign when there is no possible confusion about V and v1 , . . . , vn ; similarly for χ(σ; v1 , . . . , vn ) and χ(σ). Notice that for σ, τ ∈ Σn we have tw(v1 ⊗ · · · ⊗ vn , στ ) = ε(σ; v1 , . . . , vn )tw(vσ(1) ⊗ · · · ⊗ vσ(n) , τ ) and therefore ε(στ ; v1 , . . . , vn ) = ε(σ; v1 , . . . , vn )ε(τ ; vσ(1) , . . . , vσ(n) ). Lemma 1.1.2. Given homogeneous vectors v1 , . . . , vn ∈ V and σ ∈ Σn we have χ(σ, sV ; sv1 , . . . , svn ) = (−1)
Pn
i=1 (n−i)(vσ(i) −vi )
ε(σ, V ; v1 , . . . , vn ).
Proof. It is sufficient to check for σ a trasposition of two consecutive elements, and this is easy. Definition 1.1.3. Let V, W be graded vector spaces, a multilinear map f : V × ··· × V → W is called (graded) symmetric if f (vσ(1) , . . . , vσ(n) ) = ε(σ)f (v1 , . . . , vn ),
for every σ ∈ Σn .
It is called (graded) skewsymmetric if f (vσ(1) , . . . , vσ(n) ) = χ(σ)f (v1 , . . . , vn ),
for every σ ∈ Σn .
Definition 1.1.4. The symmetric powers of a graded vector space V are defined as Nn Kn V n V = V = , I where I is the subspace generated by the vectors v1 ⊗ · · · ⊗ vn − ε(σ)vσ(1) ⊗ · · · ⊗ vσ(n) , vi ∈ V, Nn Jn We will denote by π : V → V the natural projection and v1 · · · vn = π(v1 ⊗ · · · ⊗ vn ). 1
σ ∈ Σn .
1.2. L∞ -ALGEBRAS
2
Definition 1.1.5. The exterior powers of a graded vector space V are defined as Nn ^n V ∧n V = V = , J where J is the subspace generated by the vectors v1 ⊗ · · · ⊗ vn − χ(σ)vσ(1) ⊗ · · · ⊗ vσ(n) , vi ∈ V, Nn Vn We will denote by π : V → V the natural projection and
σ ∈ Σn .
v1 ∧ · · · ∧ vn = π(v1 ⊗ · · · ⊗ vn ). Definition 1.1.6. The set of unshuffles of type (p, q) is the subset S(p, q) ⊂ Σp+q of permutations σ such that σ(i) < σ(i="" +="" 1)="" for="" every="" i="" 6="p." equivalently="" s(p,="" q)="{σ" ∈="" σp+q="" |="" σ(1)="">< σ(2)="">< .="" .="" .=""><>
σ(p + 1) < σ(p="" +="" 2)="">< .="" .="" .="">< σ(p="" +="">
The unshuffles are a set of representatives for the left cosets of the canonical embedding of Σp × Σq inside Σp+q . More precisely for every η ∈ Σp+q there exists a unique decomposition η = στ with σ ∈ S(p, q) and τ ∈ Σp × Σq . 1.2. L∞ -algebras Let (L, d, [, ]) be a differential graded Lie algebra. Then we have: (1) d(d(x1 )) = 0; (2) d[x1 , x2 ] − [dx1 , x2 ] − (−1)x1 x2 [dx2 , x1 ] = 0; (3) [[x1 , x2 ], x3 ] − (−1)x2 x3 [[x1 , x3 ], x2 ] + (−1)x1 (x2 +x3 ) [[x2 , x3 ], x1 ] = 0; Using the formalism of unshuffles we can write Leibniz and Jacobi identities respectively as X X χ(σ)d[xσ(1) , xσ(2) ] − χ(σ)[dxσ(1) , xσ(2) ] = 0 σ∈S(2,0)
σ∈S(1,1)
and X
χ(σ)[[xσ(1) , xσ(2) ], xσ(3) ] = 0.
σ∈S(2,1)
Definition 1.2.1. An L∞ structure on a graded vector space L is a sequence of skewsymmetric maps ^n ln : L → L, deg(ln ) = 2 − n, n > 0, such that for every n > 0 and