Superalgebra

In mathematics and theoretical physics, a **superalgebra** is a **Z**2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.

The prefix *super-* comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes.

## Formal definition

Let *K* be a commutative ring. In most applications, *K* is a field of characteristic 0, such as **R** or **C**.

A **superalgebra** over *K* is a *K*-module *A* with a direct sum decomposition

# A

A

0

⊕

A

1

{\displaystyle A=A\_{0}\oplus A\_{1}}

together with a bilinear multiplication *A* × *A* → *A* such that

A

i

A

j

⊆

A

i
j

{\displaystyle A\_{i}A\_{j}\subseteq A\_{i+j}}

where the subscripts are read modulo 2, i.e. they are thought of as elements of **Z**2.

A **superring**, or **Z**2-graded ring, is a superalgebra over the ring of integers **Z**.

The elements of each of the *A**i* are said to be **homogeneous**. The **parity** of a homogeneous element *x*, denoted by |*x*|, is 0 or 1 according to whether it is in *A*0 or *A*1. Elements of parity 0 are said to be **even** and those of parity 1 to be **odd**. If *x* and *y* are both homogeneous then so is the product *xy* and

x
y

=

x

y

.

An **associative superalgebra** is one whose multiplication is associative and a **unital superalgebra** is one with a multiplicative identity element. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital.

A **commutative superalgebra** (or supercommutative algebra) is one which satisfies a graded version of commutativity. Specifically, *A* is commutative if

y
# x
(
−
1

)

x

y

x
y

for all homogeneous elements *x* and *y* of *A*. There are superalgebras that are commutative in the ordinary sense, but not in the superalgebra sense. For this reason, commutative superalgebras are often called *supercommutative* in order to avoid confusion.

## Examples

* Any algebra over a commutative ring *K* may be regarded as a purely even superalgebra over *K*; that is, by taking *A*1 to be trivial.
* Any **Z**- or **N**-graded algebra may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as tensor algebras and polynomial rings over *K*.
* In particular, any exterior algebra over *K* is a superalgebra. The exterior algebra is the standard example of a supercommutative algebra.
* The symmetric polynomials and alternating polynomials together form a superalgebra, being the even and odd parts, respectively. Note that this is a different grading from the grading by degree.
* Clifford algebras are superalgebras. They are generally noncommutative.
* The set of all endomorphisms (denoted

E
n
d

(
V
)
≡

H
o
m

(
V
,
V
)

{\displaystyle \mathbf {End} (V)\equiv \mathbf {Hom} (V,V)}

, where the boldface

H
o
m

{\displaystyle \mathrm {Hom} }

 is referred to as *internal*

H
o
m

{\displaystyle \mathrm {Hom} }

, composed of *all* linear maps) of a super vector space forms a superalgebra under composition.
* The set of all square supermatrices with entries in *K* forms a superalgebra denoted by *M**p*|*q*(*K*). This algebra may be identified with the algebra of endomorphisms of a free supermodule over *K* of rank *p*|*q* and is the internal Hom of above for this space.
	+ (Ordinary matrices can be thought of as the coordinate representations of linear maps between vector spaces (or free modules). Likewise, supermatrices can be thought of as the coordinate representations of linear maps between super vector spaces (or free supermodules). There is an important difference in the graded case, however. A homomorphism from one super vector space to another is, by definition, one that preserves the grading (i.e. maps even elements to even elements and odd elements to odd elements). The coordinate representation of such a transformation is always an *even* supermatrix. Odd supermatrices correspond to linear transformations that reverse the grading. General supermatrices represent an arbitrary ungraded linear transformation. Such transformations are still important in the graded case, although less so than the graded (even) transformations.)
* Lie superalgebras are a graded analog of Lie algebras. Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of a universal enveloping algebra of a Lie superalgebra which is a unital, associative superalgebra.

## Further definitions and constructions

### Even subalgebra

Let *A* be a superalgebra over a commutative ring *K*. The submodule *A*0, consisting of all even elements, is closed under multiplication and contains the identity of *A* and therefore forms a subalgebra of *A*, naturally called the **even subalgebra**. It forms an ordinary algebra over *K*.

The set of all odd elements *A*1 is an *A*0-bimodule whose scalar multiplication is just multiplication in *A*. The product in *A* equips *A*1 with a bilinear form

μ
:

A

1

⊗

A

0

A

1

→

A

0

{\displaystyle \mu :A\_{1}\otimes \_{A\_{0}}A\_{1}\to A\_{0}}

such that

μ
(
x
⊗
y
)
⋅
# z
x
⋅
μ
(
y
⊗
z
)

{\displaystyle \mu (x\otimes y)\cdot z=x\cdot \mu (y\otimes z)}

for all *x*, *y*, and *z* in *A*1. This follows from the associativity of the product in *A*.

### Grade involution

There is a canonical involutive automorphism on any superalgebra called the **grade involution**. It is given on homogeneous elements by

x
^

=
(
−
1

)

x

x

{\displaystyle {\hat {x}}=(-1)^{|x|}x}

and on arbitrary elements by

x
^

=

x

0

−

x

1

{\displaystyle {\hat {x}}=x\_{0}-x\_{1}}

where *x**i* are the homogeneous parts of *x*. If *A* has no 2-torsion (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of *A*:

A

i

=
{
x
∈
A
:

x
^

=
(
−
1

)

i

x
}
.

{\displaystyle A\_{i}=\{x\in A:{\hat {x}}=(-1)^{i}x\}.}

### Supercommutativity

The **supercommutator** on *A* is the binary operator given by

[
x
,
y
# ]
x
y
−
(
−
1

)

x

y

y
x

on homogeneous elements, extended to all of *A* by linearity. Elements *x* and *y* of *A* are said to **supercommute** if [*x*, *y*] = 0.

The **supercenter** of *A* is the set of all elements of *A* which supercommute with all elements of *A*:

Z

(
A
# )
{
a
∈
A
:
[
a
,
x
# ]
0

 for all

x
∈
A
}
.

{\displaystyle \mathrm {Z} (A)=\{a\in A:[a,x]=0{\text{ for all }}x\in A\}.}

The supercenter of *A* is, in general, different than the center of *A* as an ungraded algebra. A commutative superalgebra is one whose supercenter is all of *A*.

### Super tensor product

The graded tensor product of two superalgebras *A* and *B* may be regarded as a superalgebra *A* ⊗ *B* with a multiplication rule determined by:

(

a

1

⊗

b

1

)
(

a

2

⊗

b

2

# )
(
−
1

)

b

1

a

2

(

a

1

a

2

⊗

b

1

b

2

)
.

If either *A* or *B* is purely even, this is equivalent to the ordinary ungraded tensor product (except that the result is graded). However, in general, the super tensor product is distinct from the tensor product of *A* and *B* regarded as ordinary, ungraded algebras.

## Generalizations and categorical definition

One can easily generalize the definition of superalgebras to include superalgebras over a commutative superring. The definition given above is then a specialization to the case where the base ring is purely even.

Let *R* be a commutative superring. A **superalgebra** over *R* is a *R*-supermodule *A* with a *R*-bilinear multiplication *A* × *A* → *A* that respects the grading. Bilinearity here means that

r
⋅
(
x
y
# )
(
r
⋅
x
)
# y
(
−
1

)

r

x

x
(
r
⋅
y
)

for all homogeneous elements *r* ∈ *R* and *x*, *y* ∈ *A*.

Equivalently, one may define a superalgebra over *R* as a superring *A* together with an superring homomorphism *R* → *A* whose image lies in the supercenter of *A*.

One may also define superalgebras categorically. The category of all *R*-supermodules forms a monoidal category under the super tensor product with *R* serving as the unit object. An associative, unital superalgebra over *R* can then be defined as a monoid in the category of *R*-supermodules. That is, a superalgebra is an *R*-supermodule *A* with two (even) morphisms

μ

:
A
⊗
A
→
A

η

:
R
→
A

{\displaystyle {\begin{aligned}\mu &:A\otimes A\to A\\\eta &:R\to A\end{aligned}}}

for which the usual diagrams commute.