exterior algebras

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 Macaulay 2, 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

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 = true

i8 : a*a



      2

o8 = a



o8 : S

i9 : f = a*r+b*s; f^2



o10 = 0



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 Groebner bases, syzygies, and form quotient rings of these skew commutative rings.