Definition of
        Supercommutative algebra and its dual coalgebra
        Case of
        Definition of
            Lie superalgebra structure on
            Differential of
    Representation as vector fields
    Simpler notations
    Ghost number
    Are Faddev-Popov ghosts?

Definition of

As far as we know, was introduced in Alekseev:2010gr, in the context of current algebras. We will now present it in the language of quadratic algebras.

Supercommutative algebra and its dual coalgebra

Let be a super linear space, and be the tensor algebra generated by :

Here the upper index implies that we consider tensors as forming an algebra. Let be the free super-commutative algebra generated by . We define it as the subspace of consisting of symmetric tensors. (To define the structure of algebra, we notice that this subspace is isomorphic to the factorspace by quadratiic relations of , which are antisymmetric tensors in .)

Then, the dual coalgebra , by definition, the subspace of symmetric tensors in the tensor coalgebra:

(Here the upper index means that we consider it as a coalgebra.)

For example, let be even and be odd. The following tensors belong to :

The subspace is annihilated by because of relations of . In particular, the following are elements of :

These are symmetric tensors in .

To summarize, if is the algebra of symmetric tensors in , then is the coalgebra of symmetric tensors in .

Case of

Let be a Lie superalgebra. We consider it as a graded Lie superalgebra, with all elements having grade zero. Let us apply the construction of 〚Supercommutative algebra and its dual coalgebra〛 to .

Let be the free commutative superalgebra generated by , and its Koszul dual coalgebra:

At this point, we consider only as a linear superspace. The Lie algebra structure on is forgotten.

Let us consider the cobar construction of :

Now the index in means that we consider tensors as forming an algebra, the free algebra. The overline over means that we remove the counit, see LodayVallette for precise definitions. We actually only need a subspace:

(which is denoted in Chapter 2 of LodayVallette, but we reserve for contraction of a vector field into a form). Its image belongs to . It satisfies the Maurer-Cartan equation; using the notations of Chapter 2 of LodayVallette:

where is the differential on induced by the coalgebra structure on . Since is a supercommutative algebra, actually belongs to the subspace:

This implies that preserves the subspace:

(Indeed, the of the cobar complex is , and is supercommutative; “kills the commutator”.) This means that we may write instead of .

To summarize, with is a differential graded Lie superalgebra.

Definition of

Here stands for semidirect sum of Lie superalgebras, with arrow pointing towards the ideal. The embedding of into as the first summand will be denoted :

Lie superalgebra structure on

We define the commutator of two elements of as follows:

• The commutator of two elements of is the commutator of the free Lie algebra.

• The commutator of two elements of is the commutator of .

• The commutator of elements of and elements of corresponds to the natural action of on .

Differential of

annihilating all tensors with rank (i.e. ). We define a differential on , in the following way.

We then extend Eqs. (8) and (9) to the differential of .

We consider with the differential which will be called :

The nilpotence of follows from the fact that anticommutes with :

This is proven in 〚Nilpotence of 〛.

Representation as vector fields

and its universal enveloping algebra . The Maurer-Cartan Eq. (5) implies:

This is an equation in the completion of . Let be some supermanifold, and the algebra of vector fields on it. Suppose that we are given a map of linear spaces:

Such a map defines a representation of in the space of pseudo-differential forms (PDFs) on . We want to project Eq. (11) on the space of PDFs on . It is not possible to do directly, because we do not require that act on PDFs. Instead, we will use the following version of Eq. (11):

for all .

Simpler notations

For us , and is the coalgebra of symmetric tensors in . Therefore, the space can be thought of as the space of formal Taylor series of functions on the superspace with values in a linear superspace . The subspace , where consists of rank tensors, is the space of -th coefficients of the Taylor series. In particular, we interpret as the space of functions on the supermanifold :

where means that we are not being rigorous. We ignore the question of which functions are allowed, i.e. do not explain the precise meaning of .

Let denote some basis in , and the dual basis in the space of linear functions on :

for any tensor symmetric in . To agree with Alekseev:2010gr, we will denote:

Here the notation agrees with Eq. (7).

Ghost number

In our notations, if has ghost number then has ghost number . In other words, lowers ghost number. In particular, the cone of the Lie superalgebra is:

The generator has ghost number . For example, in bosonic string theory, corresponds to where is the BV antifield for ghost and the odd Poisson bracket. As a mnemonic rule, has the same ghost number as . Elements of (the first summand in ) all have ghost number zero.

Are Faddev-Popov ghosts?

As an ideal in , the is Lie superalgebra with zero commutator. We can think of as Faddeev-Popov ghosts of the BRST complex of . Since the commutator is zero, the BRST differential annihilates .

But when we consider and not , the commutator is nonzero; we consider the free Lie superalgebra generated by . We might have introduced new Faddeev-Popov ghost for nested commutators, and the corresponding BRST differential. But this is not how the construction goes. Instead, we introduce new generators, such as , and the differential such that etc. This differential acts on elements of the Lie superalgebra, not on Faddeev-Popov ghosts. This is not the Faddeev-Popov construction. Notice that gets contracted with products of . In this sense, we may say that we replace the Faddeev-Popov with nonlinear functions of . We replace: