Homotopy Lie algebra

There exists several different definitions of a homotopy Lie algebra, some particularly suited to certain situations more than others. The most traditional definition is via symmetric multi-linear maps, but there also exists a more succinct geometric definition using the language of formal geometry. Here the blanket assumption that the underlying field is of characteristic zero is made. Geometric definition A homotopy Lie algebra on a graded vector space V = \bigoplus V_i is a continuous derivation, m, of order >1 that squares to zero on the formal manifold \hat{S}\Sigma V^*. Here \hat{S} is the completed symmetric algebra, \Sigma is the suspension of a graded vector space, and V^* denotes the linear dual. Typically one describes (V,m) as the homotopy Lie algebra and \hat{S}\Sigma V^* with the differential m as its representing commutative differential graded algebra. Using this definition of a homotopy Lie algebra, one defines a morphism of homotopy Lie algebras, f\colon(V,m_V)\to (W,m_W), as a morphism f\colon\hat{S}\Sigma V^*\to\hat{S}\Sigma W^* of their representing commutative differential graded algebras that commutes with the vector field, i.e., f \circ m_V = m_W \circ f . Homotopy Lie algebras and their morphisms define a category. Definition via multi-linear maps The more traditional definition of a homotopy Lie algebra is through an infinite collection of symmetric multi-linear maps that is sometimes referred to as the definition via higher brackets. It should be stated that the two definitions are equivalent. A homotopy Lie algebra on a graded vector space V = \bigoplus V_i is a collection of symmetric multi-linear maps l_n \colon V^{\otimes n}\to V of degree n-2, sometimes called the n-ary bracket, for each n\in\N. Moreover, the maps l_n satisfy the generalised Jacobi identity: : \sum_{i+j=n+1} \sum_{\sigma\in \mathrm{UnShuff}(i,n-i)} \chi (\sigma ,v_1 ,\dots ,v_n ) (-1)^{i(j-1)} l_j (l_i (v_{\sigma (1)} , \dots ,v_{\sigma (i)}),v_{\sigma (i+1)}, \dots ,v_{\sigma (n)})=0, for each n. Here the inner sum runs over (i,j)-unshuffles and \chi is the signature of the permutation. The above formula have meaningful interpretations for low values of n; for instance, when n=1 it is saying that l_1 squares to zero (i.e., it is a differential on V), when n=2 it is saying that l_1 is a derivation of l_2, and when n=3 it is saying that l_2 satisfies the Jacobi identity up to an exact term of l_3 (i.e., it holds up to homotopy). Notice that when the higher brackets l_n for n\geq 3 vanish, the definition of a differential graded Lie algebra on V is recovered. Using the approach via multi-linear maps, a morphism of homotopy Lie algebras can be defined by a collection of symmetric multi-linear maps f_n\colon V^{\otimes n} \to W which satisfy certain conditions. Definition via operads There also exists a more abstract definition of a homotopy algebra using the theory of operads: that is, a homotopy Lie algebra is an algebra over an operad in the category of chain complexes over the L_\infty operad. == (Quasi) isomorphisms and minimal models ==

A morphism of homotopy Lie algebras is said to be a (quasi) isomorphism if its linear component f\colon V\to W is a (quasi) isomorphism, where the differentials of V and W are just the linear components of m_V and m_W. An important special class of homotopy Lie algebras are the so-called minimal homotopy Lie algebras, which are characterized by the vanishing of their linear component l_1. This means that any quasi isomorphism of minimal homotopy Lie algebras must be an isomorphism. Any homotopy Lie algebra is quasi-isomorphic to a minimal one, which must be unique up to isomorphism and it is therefore called its minimal model. == Examples ==

Because L_\infty-algebras have such a complex structure describing even simple cases can be a non-trivial task in most cases. Fortunately, there are the simple cases coming from differential graded Lie algebras and cases coming from finite dimensional examples. Differential graded Lie algebras One of the approachable classes of examples of L_\infty-algebras come from the embedding of differential graded Lie algebras into the category of L_\infty-algebras. This can be described by l_1 giving the derivation, l_2 the Lie algebra structure, and l_k =0 for the rest of the maps. Two term L∞ algebras In degrees 0 and 1 One notable class of examples are L_\infty-algebras which only have two nonzero underlying vector spaces V_0,V_1. Then, cranking out the definition for L_\infty-algebras this means there is a linear map :d\colon V_1 \to V_0, bilinear maps :l_2\colon V_i\times V_j \to V_{i+j}, where 0\leq i + j \leq 1, and a trilinear map :l_3\colon V_0\times V_0\times V_0 \to V_1 which satisfy a host of identities. pg 28 In particular, the map l_2 on V_0\times V_0 \to V_0 implies it has a lie algebra structure up to a homotopy. This is given by the differential of l_3 since the gives the L_\infty-algebra structure implies :dl_3(a,b,c) = -a,b],c] + a,c],b] + [a,[b,c, showing it is a higher Lie bracket. In fact, some authors write the maps l_n as [-,\cdots,-]_n: V_\bullet \to V_\bullet, so the previous equation could be read as :d[a,b,c]_3 = -a,b],c] + a,c],b] + [a,[b,c, showing that the differential of the 3-bracket gives the failure for the 2-bracket to be a Lie algebra structure. It is only a Lie algebra up to homotopy. If we took the complex H_*(V_\bullet, d) then H_0(V_\bullet, d) has a structure of a Lie algebra from the induced map of [-,-]_2. In degrees 0 and n In this case, for n \geq 2, there is no differential, so V_0 is a Lie algebra on the nose, but, there is the extra data of a vector space V_n in degree n and a higher bracket :l_{n+2}\colon \bigoplus^{n+2} V_0 \to V_n. It turns out this higher bracket is in fact a higher cocyle in Lie algebra cohomology. More specifically, if we rewrite V_0 as the Lie algebra \mathfrak{g} and V_n and a Lie algebra representation V (given by structure map \rho), then there is a bijection of quadruples :(\mathfrak{g}, V, \rho, l_{n+2}) where l_{n+2}\colon \mathfrak{g}^{\otimes n+2} \to V is an (n+2)-cocycle and the two-term L_\infty-algebras with non-zero vector spaces in degrees 0 and n. given a graded vector space V = V_0 \oplus V_1 where V_0 has basis given by the vector w and V_1 has the basis given by the vectors v_1, v_2, there is an L_\infty-algebra structure given by the following rules :\begin{align} & l_1(v_1) = l_1(v_2) = w \\ & l_2(v_1\otimes v_2) = v_1, l_2(v_1\otimes w) = w \\ & l_n(v_2\otimes w^{\otimes n-1}) = C_nw \text{ for } n \geq 3 \end{align}, where C_n = (-1)^{n-1}(n-3)C_{n-1}, C_3 = 1. Note that the first few constants are :\begin{matrix} C_3 & C_4 & C_5 & C_6 \\ 1 & -1 & -2 & 12 \end{matrix} Since l_1(w) should be of degree -1, the axioms imply that l_1(w) = 0. There are other similar examples for super Lie algebras. Furthermore, L_\infty structures on graded vector spaces whose underlying vector space is two dimensional have been completely classified. == See also ==