Superalgebra - Examples

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 (both even and odd) 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.
• 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.