Grassmann Algebra

TheExteriorProduct.nb 12
Grassmann algebras
Under the foregoing axioms it may be directly shown that:
1. � 0 is a field.
2. The � m are linear spaces over � 0 .
3. The direct sum of the � m is an algebra.
This algebra is called a Grassmann algebra. Its elements are sums of elements from the �, thus
m
allowing closure over both addition and exterior multiplication. For example we can evaluate
the following product to give another element of the algebra:
���1 � 2�x � 3�x � y � 4�x � y � z���1 � 2�x � 3�x � y � 4�x � y � z��
1 � 4x� 6x� y � 8x� y � z
A Grassmann algebra is also a linear space of dimension 2n , where n is the dimension of the
underlying linear space �. We will refer to a Grassmann algebra whose underlying linear space
1
is of dimension n as an n-algebra.
Grassmann algebras will be discussed further in Chapter 9.
On the nature of scalar multiplication
The anti-commutativity axiom �10 for general elements states that:
Α m � Β k
� ��1� mk  k
� Α m
If one of these factors is a scalar (Β k
Α m � a � a � Α m
� a, say; k = 0), the axiom reduces to:
Since by axiom �6, each of these terms is an m-element, we may permit the exterior product to
subsume the normal field multiplication. Thus, if a is a scalar, a � Α is equivalent to a�Α. The
m m
latter (conventional) notation will usually be adopted.
In the usual definitions of linear spaces no discussion is given to the nature of the product of a
scalar and an element of the space. A notation is usually adopted (that is, the omission of the
product sign) that leads one to suppose this product to be of the same nature as that between two
scalars. From the axioms above it may be seen that both the product of two scalars and the
product of a scalar and an element of the linear space may be interpreted as exterior products.
2001 4 5
a � Α m � Α m � a � a�Α m
a �� 0
2.20