A super -algebra is an L-∞ algebra in the context of superalgebra: the higher category theoretical/homotopy theoretical version of a super Lie algebra. For more background see at geometry of physics -- superalgebra.
In the supergravity literature the formal dual of super -algebras of finite type came to be known as “FDA”s (see remark 2 below), a decade before plain L-∞ algebras were discussed in the mathematical literature. The key example in this context are extended super Minkowski spacetimes, which are super L-∞ algebras obtained by iterated higher central extension from the super Minkowski spacetime super Lie algebra. The super-L-∞ algebra cohomology of these (called “tau-cohomology” in the physics literature) turns out to classify super p-branes and serves as a tool for the construction of supergravity theories in the D'Auria-Fré formulation of supergravity. For more background on this see at geometry of physics -- fundamental super p-branes.
Abstractly, the definition is immediate:
Explicitly this equivalently comes down to the following definition in components:
(super graded signature of a permutation)
be an n-tuple of elements of of homogeneous degree , i.e. such that .
For a permutation of elements, write for the signature of the permutation, which is by definition equal to if is the composite of permutations that each exchange precisely one pair of neighboring elements.
We say that the super -graded signature of
is the product of the signature of the permutation with a factor of
for each interchange of neighbours to involved in the decomposition of the permuation as a sequence of swapping neighbour pairs (see at signs in supergeometry for discussion of this combination of super-grading and homological grading).
An super -algebra is
for each a multilinear map, called the -ary bracket, of the form
and of degree
such that the following conditions hold:
where is the super -graded signature of the permuation , according to def. 2;
A strict homomorphism of super -algebras
is a linear map that preserves the bidegree and all the brackets, in an evident sens.
In order to define the correct homomorphisms between super -algebras (“sh-maps”) as well as their super-L-∞ algebra cohomology, consider the following dualization of the above definition:
and extended to all of as a super-graded derivation of degree .
Notice that here the signs in supergeometry are such that for elements of homogenous bidegree, then
(see at signs in supergeometry for more on this).
A strong homotopy homomorphism (“sh-map”) between super -algbras of finite type
Let be a super L-∞ algebra.
If is concentrated in even -degree, it is called an L-∞ algebra.
If is concentrated in -degrees 0 to then it is called a super Lie n-algebra.
In particular if is concentrated in degree 0, then it is equivalently a super Lie algebra.
Combining this, if is concentrated in even -degree and in -degree 0 through , then it is called a Lie n-algebra.
In particular if is concentrated in -degree 0 and in even -degree, then it is equivalently a plain Lie algebra.
(history of the concept of (super-) algebras)
The identification of the concept of (super-)-algebras has a non-linear history:
L-∞ algebras in the incarnation of higher brackets satisfying a higher Jacobi identity, def. 3 and remark 1, were introduced in Lada-Stasheff 92, based on the example of such a structure on the BRST complex of the bosonic string that was found in the construction of closed string field theory in Zwiebach 92. Some of this history is recalled in Stasheff 16.
The observation that these systems of higher brackets are fully characterized by their Chevalley-Eilenberg dg-(co-)algebras (def. 4) is due to Lada-Markl 94. See Sati-Schreiber-Stasheff 08, around def. 13.
But in this dual incarnation, L-∞ algebras and more generally super L-∞ algebras (of finite type) had secretly been introduced within the supergravity literature already in D’Auria-Fré-Regge 80 and explicitly in van Nieuwenhuizen 82. The concept was picked up in the D'Auria-Fré formulation of supergravity (D’Auria-Fré 82) and eventually came to be referred to as “FDA”s (short for “free differential algebra”) in the supergravity literature (but beware that these dg-algebras are in general free only as graded-supercommutative superalgebras, not as differential algebras) The relation between super -algebras and the “FDA”s of the supergravity literature is made explicit in (FSS 13).
|higher Lie theory||supergravity|
|super Lie n-algebra||“FDA”|
The construction in van Nieuwenhuizen 82 in turn was motivated by the Sullivan algebras in rational homotopy theory (Sullivan 77). Indeed, their dual incarnations in rational homotopy theory are dg-Lie algebras (Quillen 69), hence a special case of -algebras (remark 1)
This close relation between rational homotopy theory and higher Lie theory might have remained less of a secret had it not been for the focus of Sullivan minimal models on the non-simply connected case, which excludes the ordinary Lie algebras from the picture. But the Quillen model of rational homotopy theory effectively says that for a rational topological space then its loop space ∞-group is reflected, infinitesimally, by an L-∞ algebra. This perspective began to receive more attention when the Sullivan construction in rational homotopy theory was concretely identified as higher Lie integration in Henriques 08. A modern review that makes this L-∞ algebra-theoretic nature of rational homotopy theory manifest is in Buijs-Félix-Murillo 12, section 2.
and its super--extensions to the
See at geometry of physics -- fundamenal super p-branes? for more on this.
The BRST complex of the superstring might form a super -algebra whose brackets give the n-point function of the string, in analogy to what happens for the bosonic string in Zwiebach’s string field theory. (…)
The concept was picked up in the D'Auria-Fré formulation of supergravity
and eventually came to be referred to as “FDA”s (short for “free differential algebra”) in the supergravity literature (where in rational homotopy theory one says “semifree dga”, since these dg-algebras are crucially not required to be free as differential algebras).
The relation between super -algebras and the “FDA”s of the supergravity literature is made explicit in