6. Generalizations, improvements and analogues 117
6.2. The case of Lie superalgebras
The notion of a Lie superalgebra (also known under the name super Lie algebra and studied in [14], [24]) is one of the most well-understood generalizations of that of a Lie algebra. While classification results for Lie superalgebras are significantly harder than their non-super counterparts, most ”purely algebraic” properties of Lie algebras tend to have their analogues for Lie superalgebras, which usually are even proven in more or less the same manner. This has to do with the fact that Lie superalgebras are just Lie algebras in the category of super-k-modules; however there is also a much more pedestrian approach to proving properties of Lie superalgebras by re-reading the proofs of the corresponding facts about Lie algebras and adding signs via the Koszul rule.
Different sources disagree about the correct way to define the notion of a Lie su-peralgebra. This might have to do with the fact that the primary interest lies in Lie superalgebras over a field of characteristic 0 (rather than an arbitrary field, let alone a commutative ring), and all the definitions of a Lie superalgebra are equivalent to each other if we are over a field of characteristic 0. As I am interested in the general case, let me give the following definition of a Lie superalgebra (which is one of the most restrictive ones, but not as restrictive as [13, Definition 8.1.1]):
Definition 6.4. Let k be a commutative ring. A k-Lie superalgebra will mean a super-k-module g (see Definition 6.5 below) together with a k-bilinear map β : g×g→g satisfying the conditions
(β(v, v) = 0 for every v ∈g0) ; (84)
(−1)i`β(u, β(v, w)) + (−1)jiβ(v, β(w, u)) + (−1)`jβ(w, β(u, v)) = 0 for every i∈Z2Z,j ∈Z2Z and ` ∈Z2Z
and everyu∈gi, v ∈gj and w∈g`
; (85)
β(v, w) = −(−1)ijβ(w, v)
for every i∈Z2Z and j ∈Z2Z and everyv ∈gi and w∈gj
; (86)
(β(v, β(v, v)) = 0 for every v ∈g1) ; (87)
(β(gi×gj)⊆gi+j for every i∈Z2Z and j ∈Z2Z). (88)
This k-bilinear map β : g× g → g will be called the Lie bracket of the k-Lie superalgebra g. We will often use the square brackets notation for β, which means that we are going to abbreviate β(v, w) by [v, w] for any v ∈ g and w ∈ g. Using this notation, the equations (84), (85), (86), (87) and (88) rewrite as
([v, v] = 0 for every v ∈g0) ; (89)
(−1)i`[u,[v, w]] + (−1)ji[v,[w, u]] + (−1)`j[w,[u, v]] = 0 for every i∈Z2Z,j ∈Z2Z and ` ∈Z2Z
and everyu∈gi, v ∈gj and w∈g`
; (90)
[v, w] =−(−1)ij[w, v]
for every i∈Z2Zand j ∈Z2Z and every v ∈gi and w∈gj
; (91)
([v,[v, v]] = 0 for every v ∈g1) ; (92)
([gi,gj]⊆gi+j for every i∈Z2Zand j ∈Z2Z) (93) (where [gi,gj] means the k-linear spanh[v, w] | (v, w)∈gi ×gji).
The equation (85) (or its equivalent version (90)) is called thesuper-Jacobi identity.
Here we have used the following definition:
Definition 6.5. Let k be a commutative ring. A super-k-module will mean a k-module V together with a pair (V0, V1) of k-submodules ofV such that V =V0⊕V1. Here, 0 and 1 are considered not as integers, but as elements of Z2Z (so that 1 + 1 = 0). This sounds like a useless requirement, but it helps us in handling super-k-modules notationally; for example, the equation (88) would not make sense if 0 and 1 would be considered as integers (because in the case i = 1 and j = 1, we would have i+j = 2, but there is no such thing asg2 unless 2 is treated as an element of Z2Z).
The k-submodule V0 ofV is called the even part of V. Thek-submodule V1 ofV is called the odd part of V.
An element of V is said to be homogeneous if it lies inV0 or in V1.
Convention 6.6. We are going to use the notation V0 as a universal notation for the even part of a super-k-module V. This means that whenever we have some super-k-module V (it needs not be actually calledV; I only refer to it byV here in this Convention), the even part of V will be called V0.
Similarly, we are going to use the notation V1 as a universal notation for the odd part of a super-k-module V.
Remark 6.7. Our Definition 6.4 differs from some definitions of a Lie superalgebra given in literature by having the axioms (84) and (87). These axioms are indeed dispensable when one is only interested in the case ofkbeing a field of characteristic 0 (or, more generally, ofkbeing a commutative ring in which 2 and 3 are invertible)27. However, for the sake of generality, we keep these axioms in.
Just as the notion of Lie algebras gives birth to that ofg-modules, we can define the notion of a g-supermodule (or just g-module) over a Lie superalgebrag:
Definition 6.8. Let k be a commutative ring. Let g be a Lie superalgebra. (Ac-cording to Convention 6.6, this automatically entails that we denote by g0 the even part of g, and denote by g1 the odd part of g.)
LetV be ak-supermodule. (According to Convention 6.6, this automatically entails that we denote by V0 the even part of V, and denote by V1 the odd part of V.) Let µ : g×V → V be a k-bilinear map. We say that (V, µ) is a g-supermodule if and only if
µ([a, b], v) =µ(a, µ(b, v))−(−1)ijµ(b, µ(a, v))
for every i∈Z2Z, j ∈Z2Z and every a∈gi, b∈gj and v ∈V
(94) and
(µ(gi×Vj)⊆Vi+j for every i∈Z2Z and j ∈Z2Z).
If (V, µ) is a g-supermodule, then the k-bilinear map µ : g×V → V is called the Lie action of the g-supermodule V.
(This definition seems to be agreed on in most references. I have not seen any conflicting definitions as in the case of Definition 6.4.)
While I have not checked the details, I am convinced that all results of Sections 2, 3 and 4 (and Subsection 6.1) carry over to Lie superalgebras (and Lie supermodules) as long as 2 is invertible in the ground ring k. Even the invertibility of 2 might actually be redundant for most of these results (and it seems that the reason for its redundancy is the fact that most of the results still hold for Leibniz algebras - but, as I already said, this is not thoroughly checked). As for Section 5, trouble might come from the Poincar´e-Birkhoff-Witt theorem (Theorem 5.9), whose validity in the Lie superalgebra case has not been studied to the extent it has been studied in the original, Lie algebraic case. However, thereare two known results:
27In fact,
• axiom (84) follows from axiom (86) if 2 is invertible ink;
• axiom (87) follows from axiom (85) if 3 is invertible ink.
Theorem 6.9. Let k be a commutative ring. Let g be a k-Lie superalgebra. Let n ∈N.
(a) Ifg0 andg1 are freek-modules, and if 2 is invertible in the ringk, thengsatisfies the n-PBW condition.
(b) If k is a Q-algebra, then gsatisfies the n-PBW condition.
A proof of Theorem 6.9 (b) was given in [24, Part 1, Chapter 1, §1.3.7] and [22,
§2.5]; a proof of Theorem 6.9 (a) can be found in [14, §2.3, Theorem 1]. Note that whoever claims that 3 must be invertible in the ring k in order for Theorem 6.9 (a) to hold is probably using a definition of Lie superalgebra which does not contain the axiom (87). However, even having the axiom (84) does not prevent us from having to require the invertibility of 2, unless we replace our definition of a Lie superalgebra by a significantly more complicated one ([13, Definition 8.1.1]), in which case we can indeed drop the invertibility of 2 ([13, Theorem 8.2.2]). Having said this, we are not going to use [13, Definition 8.1.1] as the definition of a Lie superalgebra in this paper;
instead we will keep understanding a Lie superalgebra according to Definition 6.4. As a consequence, we will not be able to get rid of the condition that 2 be invertible in k in the Poincar´e-Birkhoff-Witt theorem and its consequences.
The correct analogue of Theorem 5.10 now says:
Theorem 6.10. Let k be a commutative ring. Let g be a k-Lie superalgebra.
Assume that 2 is invertible in the ring k. Also assume that the k-module g has a basis (ei)i∈I, whereI is a totally ordered set, and whereei is homogeneous for every i∈I. Then,
(ei1 ⊗ei2⊗...⊗ein) n∈N; (i1,i2,...,in)∈In;
i1≤i2≤...≤in;
everypwhich satisfies(eip∈g1 andeip+1∈g1)satisfiesip<ip+1 is a basis of the k-moduleU(g).
A proof of Theorem 6.10 in the case when k is a field of characteristic 6= 2 and 6= 3 can be found in [25, Theorem 6.1.1].
These changes in the formulation of the Poincar´e-Birkhoff-Witt theorem(s) don’t seem to keep Proposition 5.16 from retaining its validity in the case of g being a Lie superalgebra, at least as long as 2 is assumed to be invertible in k and we assume the even and the odd parts of h and N to be free k-modules (and not just h and N themselves). As a consequence, nothing speaks against the other results of Section 5 holding in this case, although this has yet to be verified more accurately.