Supercommutative algebra

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a supercommutative (associative) algebra is a <a href="/facts/Superalgebra/96vSHP12">superalgebra</a> (i.e. a Z2-<a href="/facts/Graded_algebra/AIUjkSaP">graded algebra</a>) such that for any two <a href="/facts/Homogeneous_element/AIUjkSaP">homogeneous elements</a> x, y we have

y
        x
        =
        (
        −
        1
        
          )
          
            
              |
            
            x
            
              |
            
            
              |
            
            y
            
              |
            
          
        
        x
        y
        ,
      
    
    {\displaystyle yx=(-1)^{|x||y|}xy,}

where |x| denotes the grade of the element and is 0 or 1 (in Z2) according to whether the grade is even or odd, respectively.
Equivalently, it is a superalgebra where the <a href="/facts/Supercommutator/c1oxUoEl">supercommutator</a>

[
        x
        ,
        y
        ]
        =
        x
        y
        −
        (
        −
        1
        
          )
          
            
              |
            
            x
            
              |
            
            
              |
            
            y
            
              |
            
          
        
        y
        x
      
    
    {\displaystyle [x,y]=xy-(-1)^{|x||y|}yx}

always vanishes. Algebraic structures which supercommute in the above sense are sometimes referred to as skew-commutative associative algebras to emphasize the anti-commutation, or, to emphasize the grading, graded-commutative or, if the supercommutativity is understood, simply commutative.
Any <a href="/facts/Commutative_algebra/mKtUdPCT">commutative algebra</a> is a supercommutative algebra if given the trivial gradation (i.e. all elements are even). <a href="/facts/Grassmann_algebra/rdN4TvpR">Grassmann algebras</a> (also known as <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebras</a>) are the most common examples of nontrivial supercommutative algebras. The supercenter of any superalgebra is the set of elements that supercommute with all elements, and is a supercommutative algebra.
The <a href="/facts/Even_subalgebra/96vSHP12">even subalgebra</a> of a supercommutative algebra is always a <a href="/facts/Commutative_algebra/mKtUdPCT">commutative algebra</a>. That is, even elements always commute. Odd elements, on the other hand, always anticommute. That is,

x
        y
        +
        y
        x
        =
        0
        
      
    
    {\displaystyle xy+yx=0\,}

for odd x and y. In particular, the square of any odd element x vanishes whenever 2 is invertible:

x
          
            2
          
        
        =
        0.
      
    
    {\displaystyle x^{2}=0.}

Thus a commutative superalgebra (with 2 invertible and nonzero degree one component) always contains <a href="/facts/Nilpotent/rERm2Pp7">nilpotent</a> elements.
A Z-graded <a href="/facts/Anticommutative_algebra/crvGwsXC">anticommutative algebra</a> with the property that x2 = 0 for every element x of odd grade (irrespective of whether 2 is invertible) is called an <a href="/facts/Alternating_algebra/kDoTSQXn">alternating algebra</a>.

Supercommutative algebra open-in-new

Supercommutative algebra