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# 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.

## 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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Brackets.m2:479:0.