# Exterior algebra

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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).

$r$- 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

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.