This chapter on supersymmetry intends to present it WITHOUT the use of Grassmann variables, preferring to use instead the formalism of Z2 grading.
A Z2-graded vector space is a vector space together with an assignment of an even (bosoninc) (corresponding to 0 of Z2) and an odd (fermionic) (corresponding to 1) subspace such that the vector space is the direct sum of the even and odd subspaces.
An even vector is an element of the even subspace and an odd vector is an element of the odd subspace. A pure vector is either an even or an odd vector. Any vector can be decomposed uniquely as the sum of an even and an odd vector.
The tensor product of two Z2-graded vector spaces is another Z2-graded vector space.
In fact, in this book, we will take the stronger point of view that it makes no physical sense to add even and odd vectors together. From this point of view, we might as well view a Z2-graded vector space as an ordered pair <V0,V1> where V0 is the even space and V1 is the odd space.
Similarly, a Z2-graded algebra is an algebra A with a direct sum decomposition into an even and an odd part such that the product of two pure elements obeys the Z2 relations. Alternatively, we can think of it as <A0,A1>.
A Lie superalgebra is a Z2-graded algebra whose product [·, ·], called the Lie superbracket or supercommutator, satisfies
where x, y, and z are pure in the Z2-grading. Here, |x| denotes the degree of x (either 0 or 1).
Lie superalgebras are a natural generalization of normal Lie algebras to include a Z2-grading. Indeed, the above conditions on the superbracket are exactly those on the normal Lie bracket with modifications made for the grading. The last condition is sometimes called the super Jacobi identity.
Note that the even subalgebra of a Lie superalgebra forms a (normal) Lie algebra as all the funny signs disappear, and the superbracket becomes a normal Lie bracket.
One way of thinking about a Lie superalgebra—it's not the most symmetric way of looking at it—is to consider its even and odd parts, L0 and L1 separately. Then, L0 is a Lie algebra, L1 is a linear rep of L0, and there exists a symmetric L0-intertwiner such that for all x,y and z in L1,
A supermanifold is a concept in noncommutative geometry. Recall that in noncommutative geometry, we don't look at point set spaces but instead, the algebra of functions over them. If M is a (differential) manifold and H is an (smooth) algebra bundle over M with a Grassmann algebra as the fiber, then the space of (smooth) sections of M forms a supercommutative algebra under pointwise multiplication. We say that this algebra defines the supermanifold (which isn't a point set space).
If M is a real manifold and we define an involution * over the fiber turning it into a * algebra, then the resulting algebra would define a real supermanifold.