exterior algebras

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.

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 variable(s)

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

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

i12 : x*y

o12 = x*y

o12 : R

i13 : matrix{{x}} * matrix{{y}}

o13 = | -xy |

              1      1
o13 : Matrix R  <-- R

You may compute Gröbner bases, syzygies, and form quotient rings of these skew commutative rings.