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''Grassmann algebra, of a vector space  $  V $
over a field  $  k $''

An associative algebra over  $  k $, 
the operation in which is denoted by the symbol  $  \wedge $, 
with generating elements  $  1, e _ {1} \dots e _ {n} $
where  $  e _ {1} \dots e _ {n} $
is a basis of  $  V $, 
and with defining relations

$$ 
e _ {i} \wedge e _ {j}  =  - e _ {j} \wedge e _ {i} \ \ 
( i, j = 1 \dots n),\ \ 
e _ {i} \wedge e _ {i}  =  0,
$$

$$ 
1 \wedge e _ {i}  =  e _ {i} \wedge 1  =  e _ {i} \  ( i = 1 \dots n),\  1 \wedge 1  =  1.
$$

The exterior algebra does not depend on the choice of the basis and is denoted by  $  \wedge V $. 
The subspace  $  \wedge  ^ {r} V $(
$  r = 0, 1 , .  .  . $) 
in  $  \wedge V $
generated by the elements of the form  $  e _ {i _ {1}  } \wedge \dots \wedge e _ {i _ {r}  } $
is said to be the  $  r $-
th exterior power of the space  $  V $. 
The following equalities are valid:  $   \mathop{\rm dim}  \wedge  ^ {r} V = ( _ {r}  ^ {n} ) = C _ {n}  ^ {r} $, 
$  r = 0 \dots n $, 
$  \wedge  ^ {r} V = 0 $, 
$  r > n $. 
In addition,  $  v \wedge u = (- 1)  ^ {rs} u \wedge v $
if  $  u \in \wedge  ^ {r} V $, 
$  v \in \wedge  ^ {s} V $. 
The elements of the space  $  \wedge  ^ {r} V $
are said to be  $  r $-
vectors; they may also be regarded as skew-symmetric  $  r $-
times contravariant tensors in  $  V $(
cf. [[Exterior product|Exterior product]]).

$  r $-
vectors are closely connected with  $  r $-
dimensional subspaces in  $  V $: 
Linearly independent systems of vectors  $  x _ {1} \dots x _ {r} $
and  $  y _ {1} \dots y _ {r} $
of  $  V $
generate the same subspace if and only if the  $  r $-
vectors  $  x _ {1} \wedge \dots \wedge x _ {r} $
and  $  y _ {1} \wedge \dots \wedge y _ {r} $
are proportional. This fact served as one of the starting points in the studies of H. Grassmann [[#References|[1]]], who introduced exterior algebras as the algebraic apparatus to describe the generation of multi-dimensional subspaces by one-dimensional subspaces. The theory of determinants is readily constructed with the aid of exterior algebras. An exterior algebra may also be defined for more general objects, viz. for unitary modules  $  M $
over a commutative ring  $  A $
with identity [[#References|[4]]]. The  $  r $-
th exterior power  $  \wedge  ^ {r} M $, 
$  r > 0 $, 
of a module  $  M $
is defined as the quotient module of the  $  r $-
th tensor power of this module by the submodule generated by the elements of the form  $  x _ {1} \otimes \dots \otimes x _ {r} $, 
where  $  x _ {i} \in M $
and  $  x _ {j} = x _ {k} $
for certain  $  j \neq k $. 
The exterior algebra for  $  M $
is defined as the direct sum  $  \wedge M = \oplus _ {r \geq  0 }  \wedge  ^ {r} M $, 
where  $  \wedge  ^ {0} M = A $, 
with the naturally introduced multiplication. In the case of a finite-dimensional vector space this definition and the original definition are identical. The exterior algebra of a module is employed in the theory of modules over a principal ideal ring [[#References|[5]]].

The Grassmann (or Plücker) coordinates of an  $  r $-
dimensional subspace  $  L $
in an  $  n $-
dimensional space  $  V $
over  $  k $
are defined as the coordinates of the  $  r $-
vector in  $  V $
corresponding to  $  L $, 
which is defined up to proportionality. Grassmann coordinates may be used to naturally imbed the set of all  $  r $-
dimensional subspaces in  $  V $
into the projective space of dimension  $  ( _ {r}  ^ {n} ) - 1 $, 
where it forms an algebraic variety (called the [[Grassmann manifold|Grassmann manifold]]). Thus one gets several important examples of projective algebraic varieties [[#References|[6]]].

Exterior algebras are employed in the calculus of exterior differential forms (cf. [[Differential form|Differential form]]) as one of the basic formalisms in differential geometry [[#References|[7]]], [[#References|[8]]]. Many important results in algebraic topology are formulated in terms of exterior algebras.

E.g., if  $  G $
is a finite-dimensional  $  H $-
space (e.g. a Lie group), the cohomology algebra  $  H  ^ {*} ( G, k) $
of  $  G $
with coefficients in a field  $  k $
of characteristic zero is an exterior algebra with odd-degree generators. If  $  G $
is a simply-connected compact Lie group, then the ring  $  K  ^ {*} ( G) $, 
studied in [[K-theory| $  K $-
theory]], is also an exterior algebra (over the ring of integers).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Grassmann, "Gesammelte mathematische und physikalische Werke" , '''1''' , Teubner (1894–1896) pp. Chapt. 1; 2 {{MR|0245419}} {{ZBL|42.0015.01}} {{ZBL|35.0015.01}} {{ZBL|33.0026.01}} {{ZBL|27.0017.01}} {{ZBL|25.0027.03}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian) {{MR|}} {{ZBL|0396.15001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.A. Kaluzhnin, "Introduction to general algebra" , Moscow (1973) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Multilinear algebra" , Addison-Wesley (1966) pp. Chapt. 2 (Translated from French) {{MR|0205211}} {{MR|0205210}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) {{MR|2333539}} {{MR|2327161}} {{MR|2325344}} {{MR|2284892}} {{MR|2272929}} {{MR|0928386}} {{MR|0896478}} {{MR|0782297}} {{MR|0782296}} {{MR|0722608}} {{MR|0682756}} {{MR|0643362}} {{MR|0647314}} {{MR|0610795}} {{MR|0583191}} {{MR|0354207}} {{MR|0360549}} {{MR|0237342}} {{MR|0205211}} {{MR|0205210}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''1–3''' , Cambridge Univ. Press (1947–1954) {{MR|1288307}} {{MR|1288306}} {{MR|1288305}} {{MR|0061846}} {{MR|0048065}} {{MR|0028055}} {{ZBL|0796.14002}} {{ZBL|0796.14003}} {{ZBL|0796.14001}} {{ZBL|0157.27502}} {{ZBL|0157.27501}} {{ZBL|0055.38705}} {{ZBL|0048.14502}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S.P. Finikov, "Cartan's method of exterior forms in differential geometry" , '''1–3''' , Moscow-Leningrad (1948) (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) {{MR|0193578}} {{ZBL|0129.13102}} </TD></TR></table>

====Comments====
Anticommuting variables ( $  x _ {i} x _ {j} = - x _ {j} x _ {i} $, 
$  x _ {i}  ^ {2} = 0 $) 
are sometimes called Grassmann variables; especially in the context of superalgebras, super-manifolds, etc. (cf. [[Super-manifold|Super-manifold]]; [[Superalgebra|Superalgebra]]). In addition the phrase fermionic variables occurs; especially in theoretical physics.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Chevalley, "The construction and study of certain important algebras" , Math. Soc. Japan (1955) {{MR|0072867}} {{ZBL|}} </TD></TR></table>