Exterior algebra

In mathematics, the exterior algebra or Grassmann algebra of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is an <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> that contains 
 
 
 
 V
 ,
 
 
 {\displaystyle V,}
 
 which has a product, called exterior product or wedge product and denoted with 
 
 
 
 ∧
 
 
 {\displaystyle \wedge }
 
, such that 
 
 
 
 v
 ∧
 v
 =
 0
 
 
 {\displaystyle v\wedge v=0}
 
 for every vector 
 
 
 
 v
 
 
 {\displaystyle v}
 
 in 
 
 
 
 V
 .
 
 
 {\displaystyle V.}
 
 The exterior algebra is named after <a href="/facts/Hermann_Grassmann/oHCLh2IC">Hermann Grassmann</a>, and the names of the product come from the "wedge" symbol 
 
 
 
 ∧
 
 
 {\displaystyle \wedge }
 
 and the fact that the product of two elements of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is "outside" 
 
 
 
 V
 .
 
 
 {\displaystyle V.}
 
 
The wedge product of 
 
 
 
 k
 
 
 {\displaystyle k}
 
 vectors 
 
 
 
 
 v
 
 1
 
 
 ∧
 
 v
 
 2
 
 
 ∧
 ⋯
 ∧
 
 v
 
 k
 
 
 
 
 {\displaystyle v_{1}\wedge v_{2}\wedge \dots \wedge v_{k}}
 
 is called a <a href="/facts/Blade_(geometry)/0WY05nWt">blade</a> of degree 
 
 
 
 k
 
 
 {\displaystyle k}
 
 or 
 
 
 
 k
 
 
 {\displaystyle k}
 
-blade. The wedge product was introduced originally as an algebraic construction used in <a href="/facts/Geometry/wTQf5ffQ">geometry</a> to study <a href="/facts/Area/KsNG1G9t">areas</a>, <a href="/facts/Volume/aeAXCBUd">volumes</a>, and their higher-dimensional analogues: the <a href="/facts/Magnitude_(mathematics)/KCn9PoF0">magnitude</a> of a <a href="/facts/Bivector/PBRcVE8q">2-blade</a> 
 
 
 
 v
 ∧
 w
 
 
 {\displaystyle v\wedge w}
 
 is the area of the <a href="/facts/Parallelogram/DRuktq60">parallelogram</a> defined by 
 
 
 
 v
 
 
 {\displaystyle v}
 
 and 
 
 
 
 w
 ,
 
 
 {\displaystyle w,}
 
 and, more generally, the magnitude of a 
 
 
 
 k
 
 
 {\displaystyle k}
 
-blade is the (hyper)volume of the <a href="/facts/Parallelepiped/Qn7cDgSr">parallelotope</a> defined by the constituent vectors. The <a href="/facts/Alternating_algebra/kDoTSQXn">alternating property</a> that 
 
 
 
 v
 ∧
 v
 =
 0
 
 
 {\displaystyle v\wedge v=0}
 
 implies a skew-symmetric property that 
 
 
 
 v
 ∧
 w
 =
 −
 w
 ∧
 v
 ,
 
 
 {\displaystyle v\wedge w=-w\wedge v,}
 
 and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation. 
The full exterior algebra contains objects that are not themselves blades, but <a href="/facts/Linear_combination/CVjL6kuN">linear combinations</a> of blades; a sum of blades of homogeneous degree 
 
 
 
 k
 
 
 {\displaystyle k}
 
 is called a <a href="/facts/Multivector/brM3rUny">k-vector</a>, while a more general sum of blades of arbitrary degree is called a <a href="/facts/Multivector/brM3rUny">multivector</a>. The <a href="/facts/Linear_span/vPjclEtb">linear span</a> of the 
 
 
 
 k
 
 
 {\displaystyle k}
 
-blades is called the 
 
 
 
 k
 
 
 {\displaystyle k}
 
-th exterior power of 
 
 
 
 V
 .
 
 
 {\displaystyle V.}
 
 The exterior algebra is the <a href="/facts/Direct_sum/x3ljVsP6">direct sum</a> of the 
 
 
 
 k
 
 
 {\displaystyle k}
 
-th exterior powers of 
 
 
 
 V
 ,
 
 
 {\displaystyle V,}
 
 and this makes the exterior algebra a <a href="/facts/Graded_algebra/AIUjkSaP">graded algebra</a>.
The exterior algebra is <a href="/facts/Universal_property/5XJBYMNz">universal</a> in the sense that every equation that relates elements of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 in the exterior algebra is also valid in every associative algebra that contains 
 
 
 
 V
 
 
 {\displaystyle V}
 
 and in which the square of every element of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is zero.
The definition of the exterior algebra can be extended for spaces built from vector spaces, such as <a href="/facts/Vector_field/ccFNuPPp">vector fields</a> and <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> whose <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> is a vector space. Moreover, the field of <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a> may be any field. More generally, the exterior algebra can be defined for <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a> over a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>. In particular, the algebra of <a href="/facts/Differential_forms/T9gyHtMv">differential forms</a> in 
 
 
 
 k
 
 
 {\displaystyle k}
 
 variables is an exterior algebra over the ring of the <a href="/facts/Smooth_function/mffL4ch2">smooth functions</a> in 
 
 
 
 k
 
 
 {\displaystyle k}
 
 variables.

Exterior algebra open-in-new

Exterior algebra