An exterior algebra is a polynomial ring where multiplication is mildly non-commutative, in that, for every x and y in the ring, y*x = (-1)^(deg(x) deg(y)) x*y, and that for every x of odd degree, x*x = 0.In Macaulay2, deg(x) is the degree of x, or the first degree of x, in case a multi-graded ring is being used. The default degree for each variable is 1, so in this case, y*x = -x*y, if x and y are variables in the ring.

All modules over exterior algebras are right modules. This means that matrices multiply from the opposite side:

You may compute GrÃ¶bner bases, syzygies, and form quotient rings of these skew commutative rings.

Create an exterior algebra with explicit generators by creating a polynomial ring with the option SkewCommutative.

i1 : R = QQ[x,y,z, SkewCommutative => true] o1 = R o1 : PolynomialRing, 3 skew commutative variables |

i2 : y*x o2 = -x*y o2 : R |

i3 : (x+y+z)^2 o3 = 0 o3 : R |

i4 : basis R o4 = | 1 x xy xyz xz y yz z | 1 8 o4 : Matrix R <--- R |

i5 : basis(2,R) o5 = | xy xz yz | 1 3 o5 : Matrix R <--- R |

i6 : S = QQ[a,b,r,s,t, SkewCommutative=>true, Degrees=>{2,2,1,1,1}]; |

i7 : r*a == a*r o7 = false |

i8 : a*a o8 = 0 o8 : S |

i9 : f = a*r+b*s; f^2 o10 = -2a*b*r*s o10 : S |

i11 : basis(2,S) o11 = | a b rs rt st | 1 5 o11 : Matrix S <--- S |

i12 : x*y o12 = x*y o12 : R |

i13 : matrix{{x}} * matrix{{y}} o13 = | -xy | 1 1 o13 : Matrix R <--- R |