Superalgebra

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In mathematics and theoretical physics, a superalgebra over a field K is another name for a Z₂-graded algebra over K. Specifically, a superalgebra is a super vector space A = A₀ ⊕ A₁ over K together with a bilinear multiplication

<math>A\otimes A\rightarrow A</math>

which is an even morphism of super vector spaces. This means that

<math>A_iA_j \sube A_{i+j}</math>

where the subscripts are read modulo 2. As with ordinary algebras, the multiplication is usually required to be associative and unital (although there are important classes of algebras such as Lie superalgebras which are neither). The identity element is necessarily even.

Further definitions

The even subalgeba of a superalgebra A is the homogeneous subalgebra A₀ spanned by the even elements. It forms an ordinary algebra over K. By contrast, the odd subspace A₁ does not form a subalgebra since the product of any two odd elements is even.

A commutative superalgebra is one which satisfies a graded version of commutativity. Specifically, A is commutative if

<math>yx = (-1)^{|x||y|}xy.\,</math>

for all homogeneous elements x and y if A. The supercenter of A is the span of all homogeneous elements x which supercommute with all elements of A in the above sense. A commutative superalgebra is one whose supercenter is all of A. The supercenter of A is, in general, different than the center of A as an ungraded algebra.

Examples

• Any Z or N-graded algebra may be regarded as as superalgebra by reading the grading modulo 2. This includes examples such as tensor algebras and polynomial rings over K.
• In particular, any exterior algebras over K is a superalgebra. The exterior algebra is the standard example of a commutative superalgebra.
• Clifford algebras are (noncommutative) superalgebras without a Z-grading.
• 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 super vector space of dimension p|q.
• The graded tensor product of two superalgebras may be regarded as a superalgebra with a multiplication rule determined by:

  <math>(a_1\otimes b_1)(a_2\otimes b_2) = (-1)^{|b_1||a_2|}(a_1a_2\otimes b_1b_2).</math>

• Lie superalgebras are the graded analog of Lie algebras. Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of universal enveloping algebra of a Lie superalgebra which is a unital, associative superalgebra.