Exterior algebra

Grassmann algebra, of a vector space over a field

An associative algebra over , the operation in which is denoted by the symbol , with generating elements where is a basis of , and with defining relations

The exterior algebra does not depend on the choice of the basis and is denoted by . The subspace () in generated by the elements of the form is said to be the -th exterior power of the space . The following equalities are valid: , , , . In addition, if , . The elements of the space are said to be -vectors; they may also be regarded as skew-symmetric -times contravariant tensors in (cf. Exterior product).

-vectors are closely connected with -dimensional subspaces in : Linearly independent systems of vectors and of generate the same subspace if and only if the -vectors and 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 over a commutative ring with identity [4]. The -th exterior power , , of a module is defined as the quotient module of the -th tensor power of this module by the submodule generated by the elements of the form , where and for certain . The exterior algebra for is defined as the direct sum , where , 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 -dimensional subspace in an -dimensional space over are defined as the coordinates of the -vector in corresponding to , which is defined up to proportionality. Grassmann coordinates may be used to naturally imbed the set of all -dimensional subspaces in into the projective space of dimension , 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 is a finite-dimensional -space (e.g. a Lie group), the cohomology algebra of with coefficients in a field of characteristic zero is an exterior algebra with odd-degree generators. If is a simply-connected compact Lie group, then the ring , studied in -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 (, ) 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