Superalgebra

In mathematics and theoretical physics, a superalgebra is a Z₂-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 supermanifolds and superschemes.

Formal definition

Let "K" be a fixed commutative ring. In most applications, "K" is a field such as R or C.

A superalgebra over "K" is an "K"-module "A" with a direct sum decomposition:$A\; =\; A\_0oplus\; A\_1$together with a bilinear multiplication "A" × "A" → "A" such that:$A\_iA\_j\; sube\; A\_\{i+j\}$where the subscripts are read modulo 2.

A superring, or Z₂-graded ring, is a superalgebra over the ring of integers Z.

The elements of "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 $|xy|\; =\; |x|\; +\; |y|.$

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 is one which satisfies a graded version of commutativity. Specifically, "A" is commutative if:$yx\; =\; (-1)^yx,$on homogeneous elements. This can be 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"::$Z(A)\; =\; \{ain\; A\; :\; [a,x]\; =0\; ext\{\; for\; all\; \}\; xin\; A\}.$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".

uper tensor product

The graded tensor product of two superalgebras may be regarded as a superalgebra with a multiplication rule determined by::$(a\_1otimes\; b\_1)(a\_2otimes\; b\_2)\; =\; (-1)^(a\_1a\_2otimes\; b\_1b\_2).$

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:$rcdot(xy)\; =\; (rcdot\; x)y\; =\; (-1)^x(rcdot\; y)$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:for which the usual diagrams commute.

References

*cite conference | authorlink=Pierre Deligne | first = Pierre | last = Deligne | coauthors = John W. Morgan | title = Notes on Supersymmetry (following Joseph Bernstein) | booktitle = Quantum Fields and Strings: A Course for Mathematicians | volume = 1 | pages = 41–97 | publisher = American Mathematical Society | year = 1999 | id = ISBN 0-8218-2012-5
*cite book | last = Manin | first = Y. I. | title = Gauge Field Theory and Complex Geometry | publisher = Springer | location = Berlin | year = 1997 | edition = (2nd ed.) | isbn = 3-540-61378-1
*cite book | first = V. S. | last = Varadarajan | year = 2004 | title = Supersymmetry for Mathematicians: An Introduction | series = Courant Lecture Notes in Mathematics 11 | publisher = American Mathematical Society | id = ISBN 0-8218-3574-2