# Superalgebras

# Basic Definitions

The adjective "super" with regards to mathematical structures means that all vector spaces and algebras are graded over Z/2Z\ZZ/2\ZZ. So a supervector space is a vector space VV with a Z/2Z\ZZ/2\ZZ gradation,

VV0V1, V \cong V_{0} \oplus V_{1},

where Z/2Z={0,1}\ZZ/2\ZZ = \{ 0, 1 \} in the obvious notation. An element of V0V_{0} is called even and an element of V1V_{1} is called odd. Slightly more generally, vector spaces will come with a Z\ZZ-gradation

ViZVi, V \bigoplus_{i \in \ZZ} V_{i},

where now we understand elements of V2jV_{2j} as being even,

V0:=V2j, V_{0} := \bigoplus V_{2j},

and elements of V2j+1V_{2j+1} as being odd,

V1:=V2j+1. V_{1} := \bigoplus V_{2j+1}.

An element of ViV_{i} is said to have degree ii.

Definition

A superalgebra is a supervector space AA with a multiplication satisfying

AiAjAi+j, A_{i} \cdot A_{j} \subset A_{i+j},

if AA is Z\ZZ-graded.

Examples

  • The (super)commutator of two endomorphisms of a (super)vector space is defined as

[L,M]:=LM(1)ijML,if LEndiV,MEndjV. [L,M] := LM - (-1)^{ij}ML,\quad \text{if } L \in \End_{i}V,\quad M \in \End_{j}V.

More generally, the commutator of any two elements in any associative superalgebra is defined in exactly the same way.

  • An associative algebra is called (super)commutative if the commutator of any two elements vanishes. As an example, the algebra Ω(M)\Omega(M) of all differential forms on a manifold is a commutative superalgebra.

# Lie Superalgebras

Definition

A (Z\ZZ-graded) Lie superalgebra is a Z\ZZ-graded vector space

h=iZhi h = \bigoplus_{i \in \ZZ} h_{i}

equipped with a bracket operation

[,]:hi×hjhi+j [-,-] : h_{i} \times h_{j} \longrightarrow h_{i+j}

which is (super) anti-commutative in the sense that

[u,v]+(1)ij[v,u]=0,for all uhi,vhj, [u,v] + (-1)^{ij}[v,u] = 0,\quad \text{for all } u \in h_{i},\, v \in h_{j},

and satisfies the super version of the Jacobi identity

[u,[v,w]]=[[u,v],w]+(1)ij[v,[u,w]],for all uhi,vhj. [u,[v,w]] = [[u,v],w] + (-1)^{ij}[v,[u,w]],\quad \text{for all } u \in h_{i}, v\in h_{j}.

If g\mfg is an ordinary Lie algebra in the old-fashioned sense, with a chosen basis ξ1,ξn\xi_{1},\ldots \xi_{n} of g\mfg, then we define g~\tilde{\mfg} to be the Lie superalgbera

g~:=g1g0g1 \tilde{\mfg} := \mfg_{-1} \oplus \mfg_{0} \oplus \mfg_{1}

where g1\mfg_{-1} is an nn-dimensional vector space with basis ı1,,ın\imath_{1},\ldots,\imath_{n}; where g0\mfg_{0} is an nn-dimensional vector space with basis L1,,LnL_{1},\ldots, L_{n}; and where g1\mfg_{1} is a one-dimensional vector space with basis dd. These basis elements will be defined in the next subsection.

The bracket is defined in terms of this basis by (insert - pg. 13).

A basis-free description of the Lie superalgebra g~\tilde{\mfg} is the following: the assertion

g~=g1g0g1 \tilde{\mfg} = \mfg_{-1} \oplus \mfg_{0} \oplus \mfg_{1}

as a Z\ZZ-graded algebra implies that

[g1,g1]=0and[g1,g1]=0. [\mfg_{-1},\mfg_{-1}] = 0\quad \text{and} \quad [\mfg_{1},\mfg_{1}] = 0.

The subalgebra g0\mfg_{0} is isomorphic to g\mfg; if we denote a typical element of g0\mfg_{0} by LξL_{\xi}, for ξg\xi \in \mfg, then

[Lξ,Lη]=L[ξ,η] [L_{\xi}, L_{\eta}] = L_{[\xi,\eta]}

gives the bracket [,]:g0×g0g0[-,-] : \mfg_{0} \times \mfg_{0} \ra \mfg_{0}. The space g1\mfg_{-1} is isomorphic to g\mfg as a vector spacem and [,]:g0×g1g1[-,-] : \mfg_{0} \times \mfg_{-1} \ra \mfg_{-1} is the adjoint representation: if we denote an element of g1\mfg_{-1} by ıη\imath_{\eta}, for ηg\eta \in \mfg, then

[Lξ,ıη]=ı[ξ,η]. [L_{\xi}, \imath_{\eta}] = \imath_{[\xi,\eta]}.

The bracket [,]:g0×g1g1[-,-] : \mfg_{0} \times \mfg_{1} \ra \mfg_{1} is 00, and the bracket [,]:g1×g1g0[-,-] : \mfg_{-1} \times \mfg_{1} \ra \mfg_{0} is given by

[ıξ,d]=Lξ. [\imath_{\xi}, d] = L_{\xi}.

# Derivations

Definition

If AA is a (not necessarily associative) superalgebra, then Der(A)\Der(A) is the subspace of End(A)\End(A), where

Derk(A)Endk(A) \Der_{\bfk}(A) \subset \End_{\bfk}(A)

consists of the endomorphisms DD which satisfy

D(uv)=(Du)v+(1)kmu(Dv),when uAm. D(uv) = (Du)v + (-1)^{\bfk\bfm}u(Dv),\quad \text{when } u \in A_{\bfm}.

Similarly for the Z\ZZ-graded case. An element of \Derk_{\bfk}(A) is called a derivation of degree k\bfk, regardless of whether it is even or odd.

# Other Constructions

Example

If AA and BB are (super)algebras, the product law on ABA \otimes B is defined by

(a1b1)(a2b2)=(1)ija1a2b1b2 (a_{1} \otimes b_{1}) \cdot (a_{2} \otimes b_{2}) = (-1)^{ij}a_{1}a_{2} \otimes b_{1}b_{2}

where dega2=i\deg a_{2} = i and degb1=j\deg b_{1} = j. With this definition, the tensor product of two commutative algebras is again commutative. Our definition of multiplication is the unique definition such that the maps

AAB,aa1,BAB,b1b, \begin{aligned} A \ra A \otimes B,\qquad &a \mapsto a \otimes 1, \\ B \ra A \otimes B,\qquad &b \mapsto 1 \otimes b, \end{aligned}

are algebra monomorphims and such that

(a1)(1b)=ab. (a\otimes 1) \cdot (1 \otimes b) = a \otimes b.

Example

As an example of the above, let VV and WW be ordinary vector spaces. We can choose a basis e1,,em,f1,fne_{1}, \ldots, e_{m}, f_{1},\ldots f_{n} of VWV \oplus W with the eiVe_{i} \in V and the fjW.f_{j} \in W. Thus monomials of the form

ei1eikfj1fjl e_{i_{1}} \wedge \ldots \wedge e_{i_{k}} \wedge f_{j_{1}} \wedge \ldots \wedge f_{j_{l}}

constitute a basis of (VW)\wedge(V\oplus W). This shows that in our category of superalgebras, we have that (VW)=(V)(W)\wedge(V \oplus W) = \wedge(V) \otimes \wedge(W). If MM and NN are smooth manifolds, then Ω(M)Ω(N)\W(M) \otimes \W(N) is a subalgebra of Ω(M×N)\W(M \times N), which is dense in the CC^{\infty} toplogy.

Remark

The above definition of the tensor product of two superalgebras and the respective multiplication has the following universal property: let

u:AC,v:BC, u: A \ra C,\qquad v : B \ra C,

be two morphisms of superalgebras such that

[u(a),v(b)]=0,for all aA,bB. [u(a), v(b)] = 0,\qquad \text{for all } a \in A, b \in B.

Then there exists a unique superalgebra morphism

w:ABC w : A \otimes B \ra C

such that

w(a \otimes 1) = u(a),\qquad w(1 \otimse b) = v(b).