# Grassmann algebra

## Grassmann algebra

[′gräs·mən ‚al·jə·brə]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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A Grassmann-graded K-ring [[LAMBDA].sup.*] seen as a [[LAMBDA].sup.*]-graded commutative ring [[LAMBDA].sub.*] is a Grassmann algebra (Definition 23).
A [Z.sub.2]-graded commutative K-ring [[LAMBDA].sub.*] is said to be the Grassmann algebra if it is finitely generated in degree 1 and is isomorphic to the exterior algebra [disjunction](R/[R.sup.2]) (Example 9) of a K-module R/[R.sup.2], where R is the ideal of nilpotents (19) of [[LAMBDA].sub.*].
An exterior algebra [disjunction]Q of a free K-module Q of finite rank is a Grassmann algebra. Conversely, a Grassmann algebra admits a structure of an exterior algebra [disjunction]Q by a choice of its minimal generating K-module Q [subset] [[LAMBDA].sub.1], and all these structures are mutually isomorphic if K is a field (Theorem 26).
A Grassmann algebra is local in accordance with Definition 20.
Its associated [Z.sub.2]-graded commutative ring is a Grassmann algebra [LAMBDA] (Definition 23).
We agree to call {[c.sup.i]} the generating basis for the associated Grassmann algebra [[LAMBDA].sub.*] which brings it into a Grassmann-graded ring [[LAMBDA].sup.*].
For instance, the differential calculus over a Grassmann-graded K-ring [A.sup.*] (Example 42) is the differential calculus over an associated Grassmann algebra [A.sup.*].
whose typical fibre is the Grassmann algebra A = [conjunction] [V.sup.*] in Theorem 28.
Since [Z.sub.2]-graded manifolds conventionally are sheaves in Grassmann algebras , we focus our consideration on local finitely generated N-graded commutative rings of the following type (Remark 8).
Let G be the Grassmann algebra over K generated by the elements 1, [e.sub.1], ...
When m = 0, we have that V [cross product] [C.sub.0|2n] = [[LAMBDA].sub.2n] [cross product] [W.sub.2n], with [[LAMBDA].sub.2n] being the Grassmann algebra generated by [[??].sub.j].
Vector spaces, multilinear mappings, dual spaces, tensor product spaces, tensors, symmetric and skew-symmetric tensors, and exterior or Grassmann algebra are described in the initial chapters, with definitions and examples provided.

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