GCAlgebra

Description

An object of class GCAlgebra represents a Grassmann-Cayley algebra. The Grassmann-Cayley algebra may be viewed as an algebra of linear subspaces of $\mathbb{P}^{d−1}.$ In this algebra, there are two operations which correspond to the join and meet of subspaces. We denote these operators by * and ^, respectively. The first operator is simply multiplication in a skew-commutative polynomial ring $\mathbb{C}\langle a_1, . . . , a_n\rangle.$ An algebraic formula for the meet operator is more complicated, but it can be defined using the so-called shuffle product.

As a $k-$vector space, the Grassmann-Cayley algebra has a direct-sum decomposition $$\oplus_{k=0}^d\Lambda^k(a_1, \ldots, a_n)$$ where $\Lambda^k(a_1,\ldots, a_n)$ is the vector space of extensors of the form $a_{i_1}\cdots a_{i_k}.$ We may identify $\Lambda^d(a_1, \ldots, a_n)\cong B_{n,d}$ with the BracketRing.

i1 : G = gc(a..f, 3)

o1 = Grassmann-Cayley Algebra generated by 1-extensors a..f

o1 : GCAlgebra

i2 : a_G * b_G                -- join of two points: the line ab

o2 = a*b

o2 : GCExpression

i3 : (a_G * b_G) ^ (c_G * d_G) -- meet (intersection) of lines ab and cd in P^2

o3 = -[bcd]*a+[acd]*b

o3 : GCExpression

i4 : bracketRing G            -- the associated bracket ring B_{6,3}

o4 = B
      6,3

o4 : BracketRing

Methods that use an object of class GCAlgebra:

bracketRing(GCAlgebra) -- see bracketRing -- Constructor for bracket rings
GCExpression _ GCAlgebra (missing documentation)

For the programmer

The object GCAlgebra is a type, with ancestor classes AbstractGCRing (missing documentation) < HashTable < Thing.

The source of this document is in Brackets.m2:499:0.