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''Grassmann algebra, of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e0370801.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e0370802.png" />''
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An associative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e0370803.png" />, the operation in which is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e0370804.png" />, with generating elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e0370805.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e0370806.png" /> is a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e0370807.png" />, and with defining relations
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e0370808.png" /></td> </tr></table>
+
''Grassmann algebra, of a vector space  $  V $
 +
over a field  $  k $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e0370809.png" /></td> </tr></table>
+
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
  
The exterior algebra does not depend on the choice of the basis and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708010.png" />. The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708012.png" />) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708013.png" /> generated by the elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708014.png" /> is said to be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708016.png" />-th exterior power of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708017.png" />. The following equalities are valid: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708021.png" />. In addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708022.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708024.png" />. The elements of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708025.png" /> are said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708027.png" />-vectors; they may also be regarded as skew-symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708028.png" />-times contravariant tensors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708029.png" /> (cf. [[Exterior product|Exterior product]]).
+
$$
 +
e _ {i} \wedge e _ {j}  = - e _ {j} \wedge e _ {i} \ \
 +
( i, j = 1 \dots n),\ \
 +
e _ {i} \wedge e _ {i}  = 0,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708030.png" />-vectors are closely connected with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708031.png" />-dimensional subspaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708032.png" />: Linearly independent systems of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708035.png" /> generate the same subspace if and only if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708036.png" />-vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708038.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708039.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708040.png" /> with identity [[#References|[4]]]. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708041.png" />-th exterior power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708043.png" />, of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708044.png" /> is defined as the quotient module of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708045.png" />-th tensor power of this module by the submodule generated by the elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708048.png" /> for certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708049.png" />. The exterior algebra for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708050.png" /> is defined as the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708052.png" />, 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]]].
+
$$
 +
1 \wedge e _ {i}  = e _ {i} \wedge 1  = e _ {i} \  ( i = 1 \dots n),\  1 \wedge 1  = 1.
 +
$$
  
The Grassmann (or Plücker) coordinates of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708053.png" />-dimensional subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708054.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708055.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708056.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708057.png" /> are defined as the coordinates of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708058.png" />-vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708059.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708060.png" />, which is defined up to proportionality. Grassmann coordinates may be used to naturally imbed the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708061.png" />-dimensional subspaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708062.png" /> into the projective space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708063.png" />, where it forms an algebraic variety (called the [[Grassmann manifold|Grassmann manifold]]). Thus one gets several important examples of projective algebraic varieties [[#References|[6]]].
+
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708064.png" /> is a finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708065.png" />-space (e.g. a Lie group), the cohomology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708066.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708067.png" /> with coefficients in a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708068.png" /> of characteristic zero is an exterior algebra with odd-degree generators. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708069.png" /> is a simply-connected compact Lie group, then the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708070.png" />, studied in [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708071.png" />-theory]], is also an exterior algebra (over the ring of integers).
+
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====
 
====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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev,   "Foundations of linear algebra" , Freeman (1963) (Translated from Russian)</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)</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)</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)</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)</TD></TR></table>
+
<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====
 
====Comments====
Anticommuting variables (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037080/e03708073.png" />) 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.
+
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====
 
====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)</TD></TR></table>
+
<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>

Latest revision as of 19:38, 5 June 2020


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).

$ 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 [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 [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 [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). Thus one gets several important examples of projective algebraic varieties [6].

Exterior algebras are employed in the calculus of exterior differential forms (cf. Differential form) as one of the basic formalisms in differential geometry [7], [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, is also an exterior algebra (over the ring of integers).

References

[1] H. Grassmann, "Gesammelte mathematische und physikalische Werke" , 1 , Teubner (1894–1896) pp. Chapt. 1; 2 MR0245419 Zbl 42.0015.01 Zbl 35.0015.01 Zbl 33.0026.01 Zbl 27.0017.01 Zbl 25.0027.03
[2] A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian) Zbl 0396.15001
[3] L.A. Kaluzhnin, "Introduction to general algebra" , Moscow (1973) (In Russian)
[4] N. Bourbaki, "Elements of mathematics. Algebra: Multilinear algebra" , Addison-Wesley (1966) pp. Chapt. 2 (Translated from French) MR0205211 MR0205210
[5] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210
[6] W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 1–3 , Cambridge Univ. Press (1947–1954) MR1288307 MR1288306 MR1288305 MR0061846 MR0048065 MR0028055 Zbl 0796.14002 Zbl 0796.14003 Zbl 0796.14001 Zbl 0157.27502 Zbl 0157.27501 Zbl 0055.38705 Zbl 0048.14502
[7] S.P. Finikov, "Cartan's method of exterior forms in differential geometry" , 1–3 , Moscow-Leningrad (1948) (In Russian)
[8] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) MR0193578 Zbl 0129.13102

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; Superalgebra). In addition the phrase fermionic variables occurs; especially in theoretical physics.

References

[a1] C. Chevalley, "The construction and study of certain important algebras" , Math. Soc. Japan (1955) MR0072867
How to Cite This Entry:
Exterior algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exterior_algebra&oldid=13734
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article