Superalgebra

Sign conventions

When the Z₂ grading arises as a "rollup" of a Z- or N-graded algebra into even and odd components, then two distinct (but essentially equivalent) sign conventions can be found in the literature. These can be called the "cohomological sign convention" and the "super sign convention". They differ in how the antipode (exchange of two elements) behaves. In the first case, one has an exchange map

${\displaystyle xy\mapsto (-1{)}^{mn+pq}yx}$

where ${\displaystyle m=\mathrm{deg}x}$ is the degree (Z- or N-grading) of ${\displaystyle x}$ and ${\displaystyle p}$ the parity. Likewise, ${\displaystyle n=\mathrm{deg}y}$ is the degree of ${\displaystyle y}$ and with parity ${\displaystyle q.}$ This convention is commonly seen in conventional mathematical settings, such as differential geometry and differential topology. The other convention is to take

${\displaystyle xy\mapsto (-1{)}^{pq}yx}$

with the parities given as ${\displaystyle p=mmod2}$ and ${\displaystyle q=nmod2}$ the parity. This is more often seen in physics texts, and requires a parity functor to be judiciously employed to track isomorphisms. Detailed arguments are provided by Pierre Deligne

Examples

• Any algebra over a commutative ring K may be regarded as a purely even superalgebra over K; that is, by taking A₁ 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 ${\displaystyle \mathbf{E}\mathbf{n}\mathbf{d}(V)\equiv \mathbf{H}\mathbf{o}\mathbf{m}(V,V)}$, where the boldface ${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}}$ is referred to as internal ${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}}$, 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.
• 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 A₀, 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 A₁ is an A₀-bimodule whose scalar multiplication is just multiplication in A. The product in A equips A₁ with a bilinear form

${\displaystyle \mu :A_1{\otimes }_{A_0}{A}_1\to A_0}$

such that

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

for all x, y, and z in A₁. 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

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

and on arbitrary elements by

${\displaystyle \hat{x}=x_0-x_1}$

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:

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

Supercommutativity

The supercommutator on A is the binary operator given by

${\displaystyle [x,y]=xy-(-1{)}^{|x||y|}yx}$

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:

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

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:

${\displaystyle (a_1\otimes b_1)(a_2\otimes b_2)=(-1{)}^{|b_1||a_2|}(a_1{a}_2\otimes b_1{b}_2).}$

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

${\displaystyle r\cdot (xy)=(r\cdot x)y=(-1{)}^{|r||x|}x(r\cdot y)}$

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

${\displaystyle \begin{array}{cc}\mu & :A\otimes A\to A\\ \eta & :R\to A\end{array}}$

for which the usual diagrams commute.