Supermodule

In mathematics, a supermodule is a Z2-graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetry in theoretical physics. In mathematics, a supermodule is a Z2-graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetry in theoretical physics. Supermodules over a commutative superalgebra can be viewed as generalizations of super vector spaces over a (purely even) field K. Supermodules often play a more prominent role in super linear algebra than do super vector spaces. These reason is that it is often necessary or useful to extend the field of scalars to include odd variables. In doing so one moves from fields to commutative superalgebras and from vector spaces to modules. Let A be a fixed superalgebra. A right supermodule over A is a right module E over A with a direct sum decomposition (as an abelian group) such that multiplication by elements of A satisfies for all i and j in Z2. The subgroups Ei are then right A0-modules. The elements of Ei 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 E0 or E1. Elements of parity 0 are said to be even and those of parity 1 to be odd. If a is a homogeneous scalar and x is a homogeneous element of E then |x·a| is homogeneous and |x·a| = |x| + |a|. Likewise, left supermodules and superbimodules are defined as left modules or bimodules over A whose scalar multiplications respect the gradings in the obvious manner. If A is supercommutative, then every left or right supermodule over A may be regarded as a superbimodule by setting for homogeneous elements a ∈ A and x ∈ E, and extending by linearity. If A is purely even this reduces to the ordinary definition. A homomorphism between supermodules is a module homomorphism that preserves the grading.Let E and F be right supermodules over A. A map