Editing Superalgebra (section)

==Further definitions and constructions==

===Even subalgebra===

Let ''A'' be a superalgebra over a commutative ring ''K''. The [[submodule]] ''A''<sub>0</sub>, 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 (ring theory)|algebra]] over ''K''.

The set of all odd elements ''A''<sub>1</sub> is an ''A''<sub>0</sub>-[[bimodule]] whose scalar multiplication is just multiplication in ''A''. The product in ''A'' equips ''A''<sub>1</sub> with a [[bilinear form]]
:<math>\mu:A_1\otimes_{A_0}A_1 \to A_0</math>
such that
:<math>\mu(x\otimes y)\cdot z = x\cdot\mu(y\otimes z)</math>
for all ''x'', ''y'', and ''z'' in ''A''<sub>1</sub>. This follows from the associativity of the product in ''A''.

===Grade involution===

There is a canonical [[Involution (mathematics)|involutive]] [[automorphism]] on any superalgebra called the '''grade involution'''. It is given on homogeneous elements by
:<math>\hat x = (-1)^{|x|}x</math>
and on arbitrary elements by
:<math>\hat x = x_0 - x_1</math>
where ''x''<sub>''i''</sub> are the homogeneous parts of ''x''. If ''A'' has no [[torsion (algebra)|2-torsion]] (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of ''A'':
:<math>A_i = \{x \in A : \hat x = (-1)^i x\}.</math>

===Supercommutativity===

The '''[[supercommutator]]''' on ''A'' is the binary operator given by
:<math>[x,y] = xy - (-1)^{|x||y|}yx</math>
on homogeneous elements, extended to all of ''A'' by linearity. Elements ''x'' and ''y'' of ''A'' are said to '''supercommute''' if {{nowrap|1=[''x'', ''y''] = 0}}.

The '''supercenter''' of ''A'' is the set of all elements of ''A'' which supercommute with all elements of ''A'':
:<math>\mathrm{Z}(A) = \{a\in A : [a,x]=0 \text{ for all } x\in A\}.</math>
The supercenter of ''A'' is, in general, different than the [[center of an algebra|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 algebras|tensor product]] of two superalgebras ''A'' and ''B'' may be regarded as a superalgebra ''A'' &otimes; ''B'' 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>
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.