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The Exterior Algebra and Central Notions in Mathematics

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The Exterior Algebra and Central Notions in Mathematics The Exterior Algebra and Central Notions in Mathematics Gunnar Fløystad Dedicated to Stein Arild Strømme (1951–2014) The neglect of the exterior algebra is the mathematical tragedy of our century. —Gian-Carlo Rota, Indiscrete Thoughts (1997) his note surveys how the exterior algebra the somewhat lesser-known regressive product on and deformations or quotients of it the exterior algebra, which intuitively corresponds capture essences of five domains in to intersection of linear spaces. It relates this to mathematics: geometry and it also shows how analysis may T be extended to functions of extensive quantities. • Combinatorics •Mathematical physics Only in the last two decades of the 1800s did • Topology •Algebraic geometry publications inspired by Grassmann’s work achieve • Lie theory a certain mass. It may have been with some regret The exterior algebra originated in the work that Grassmann in his second version had an of Hermann Grassmann (1809–1877) in his book exclusively mathematical form, since he in the Ausdehnungslehre from 1844, and the thoroughly foreword says “[extension theory] is not simply one revised 1862 version, which now exists in an English among the other branches of mathematics, such translation [20] from 2000. Grassmann worked as algebra, combination theory or function theory, as a professor at the gymnasium in Stettin, then bur rather surpasses them, in that all fundamental Germany. Partly because Grassmann was an original elements are unified under this branch, which thinker and maybe partly because his education thus as it were forms the keystone of the entire had not focused much on mathematics, the first structure of mathematics.” edition of his book had a more philosophical The present note indicates that he was not quite than mathematical form and therefore gained little off the mark here. We do not make any further influence in the mathematical community. The connections to Grassmann’s original presentation, second (1862) version was strictly mathematical. but rather present the exterior algebra in an Nevertheless, it also gained little influence, perhaps entirely modern setting. For more on the historical because it had swung too far to the other side and context of Grassmann, see the excellent history of was scarce of motivation. Over four hundred pages vector analysis [7], as well as proceedings from it developed the exterior and interior product and conferences on Grassmann’s many-faceted legacy Gunnar Fløystad is professor of mathematics at the Univer- [41] and [38]. The last fifteen years have also seen a sity of Bergen, Norway. His email address is [email protected]. flurry of books advocating the very effective use of no. the exterior algebra and its derivation, the Clifford DOI: http://dx.doi.org/10.1090/noti1234 algebra, in physics, engineering, and computer 364 Notices of the AMS Volume 62, Number 4 science. In the last section we report briefly on we see that ei ^ ej = −ej ^ ei. In fact, we obtain this. v ^ w + w ^ v = 0 for any v; w in V. Hence when the characteristic of k is not 2, the exterior algebra The Exterior Algebra may be defined as T (V )/hS2Vi where Concrete Definition S2V = fv ⊗ w + w ⊗ v j v; w 2 Vg Given a set fe ; : : : ; e g with n elements, consider 1 n are the symmetric two-tensors in V ⊗ V. The pth the 2n expressions e ^e ^· · ·^e (here ^ is just i1 i2 ir graded piece of E(V ), which is the image of V ⊗p, a place separator), where the i ; i ; : : : ; i are strictly 1 2 r is denoted as ^pV. increasing subsequences of 1; 2; : : : ; n. From this We shall in the following indicate: we form the vector space E(n) over a field k with these expressions as basis elements. • How central notions in various areas in math- ematics arise from natural structures on the Example 1. When n = 3, the following eight ex- exterior algebra. pressions form a basis for E(3) over the field k: • How the exterior algebra or variations thereof 1; e1; e2; e3; e1 ^ e2; e1 ^ e3; e2 ^ e3; e1 ^ e2 ^ e3: are a natural tool in these areas. The element e ^ e ^ · · · ^ e is considered i1 i2 ir Combinatorics I: Simplicial Complexes and to have degree r, so we get a graded vector space Face Rings E(n). Now we equip this vector space with a For simplicity denote the set f1; 2; : : : ; ng as [n]. multiplication which we also denote by ^. The Each subset fi ; : : : ; i g of [n] corresponds to basic rules for this multiplication are 1 r a monomial ei1 ^ ei2 ^ · · · ^ eir in the exterior i) ei ^ ei = 0; ii) ei ^ ej = −ej ^ ei: algebra E(n). For instance, f2; 5g ⊆ [6] gives the These rules, together with the requirement that ^ monomial e2 ^ e5. It also gives the indicator vector 6 be associative, i.e., (0; 1; 0; 0; 1; 0) 2 Z2 (where Z2 = f0; 1g), with 1’s at positions 2 and 5. We may then consider iii) (a ^ b) ^ c = a ^ (b ^ c) e2 ^ e5 to have this multidegree. This one-to- for all a; b; c in E(n), and linear, i.e. one correspondence between subsets of [n] and iv) a ^ (βb + γc) = βa ^ b + γa ^ c monomials in E(n) suggests that it can be used to encode systems of subsets of a finite set. The for all β, γ in the field k and a; b; c in E(n), set systems naturally captured by virtue of E(n) determine the algebra structure on E(n). For being an algebra are the combinatorial simplicial instance, complexes. These are families of subsets ∆ of [n] e5 ^ (e1 ^ e3) = e5 ^ e1 ^ e3 such that if X is in ∆, then any subset Y of X is also in . = −e1 ^ e5 ^ e3 (switch e5 and e1) ∆ = e1 ^ e3 ^ e5 (switch e5 and e3). Example 2. Let n = 6. The sets f1; 2g; f3; 4g; f3; 5g; f4; 5; 6g; Abstract Definition together with all the subsets of each of these four Here we define the exterior algebra using standard sets, form a combinatorial simplicial complex. machinery from algebra. Let V be a vector space over k, and denote by V ⊗p the p-fold tensor product The point of relating these to the algebra E(n) V ⊗k V ⊗k · · · ⊗k V . The free associative algebra on is that combinatorial simplicial complexes on [n] L ⊗p n V is the tensor algebra T (V ) = p≥0 V which are in one-to-one correspondence with Z2 -graded n comes with the natural concatenation product ideals I in E(n) or equivalently with Z2 -graded quotient rings E(n)/I of E(n): To a simplicial (v1⊗·· ·⊗vr )·(w1⊗·· ·⊗ws )=v1⊗·· · vr⊗w1⊗·· ·⊗ws : complex ∆ corresponds the monomial ideal I∆ Let R be the vector subspace of V ⊗k V generated by generated by all elements v ⊗v where v 2 V . The exterior algebra fe ^ · · · ^ e j fi ; : : : ; i g 62 g: is the quotient algebra of T (V ) by the relations i1 ir 1 r ∆ R. More formally, let hRi be the two-sided ideal in Note that the monomials ei1 ^ · · · ^ eip with T (V ) generated by R. The exterior algebra E(V ) fi1; : : : ; ipg in ∆ then constitute a vector space basis is the quotient algebra T (V )/hRi. The product = for the quotient algebra E(∆) E(V )/I∆. We call in this quotient algebra is commonly denoted by this algebra the exterior face ring of ∆. ^. Let e1; : : : ; en be a basis for V. We then have For the simplicial complex in the example above, e ^ e = 0, since e ⊗ e is a relation in R. Similarly, i i i i E(∆) has a basis: + ^ + (ei ej ) (ei ej ) is zero. Expanding this • degree 0: 1, 0 = ei ^ ei + ei ^ ej + ej ^ ei + ej ^ ej ; • degree 1: e1; e2; e3; e4; e5; e6, April 2015 Notices of the AMS 365 • degree 2: e1 ^ e2; e3 ^ e4; e3 ^ e5; e4 ^ e5; e4 ^ We then say that ∆ gives a triangulation of e6; e5 ^ e6, the space X. Now we can equip E(n) with a • degree 3: e4 ^ e5 ^ e6. differential d of degree 1 by multiplying with u = e + e + · · · + e . Then d(a) = u ^ a, and this Although subsets fi1; : : : ; ir g of [n] most nat- 1 2 n 2 urally correspond to monomials in E(n), one is a differential, since d (a) = u ^ u ^ a = 0. The monomial ei1 ^ ei2 ^ · · · ^ eir of degree r is then can also consider the monomial xi1 ··· xir in the mapped to the degree (r + 1) sum: polynomial ring k[x1; : : : ; xn]. (Note, however, that monomials in this ring naturally correspond to X e ^ e ^ · · · ^ e : multisets rather than to sets.) If one associates to i i1 ir i62fi1;:::;ir g ∆ the analog monomial ideal in this polynomial ring, the quotient ring k[∆] is the Stanley-Reisner The face ring E(∆) is a quotient of E(n), and ring or simply the face ring of ∆. so we also obtain a differential on E(∆). Letting r This opens up the arsenal of algebra to study E(∆) be the degree r part, this gives a complex . The study of E( ) and k[ ] has particularly 0 1 2 ∆ ∆ ∆ 0 d 1 d 2 d centered around their minimal free resolutions and E(∆) ! E(∆) ! E(∆) !··· : all the invariants that arise from such. The study From this complex and its dual we calculate the of k[ ] was launched around 1975 with a seminal ∆ prime invariants in topology, the cohomology paper by Hochster [29] and Stanley’s proof of the and homology of the topological space X.
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