Exterior algebra

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