The Exterior and Central Notions in

Gunnar Fløystad

Dedicated to Stein Arild Strømme (1951–2014) The neglect of the 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 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: and it also shows how analysis may T be extended to functions of extensive quantities. • Combinatorics •Mathematical 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 (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 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 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, , and computer

364 Notices of the AMS 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 of k is not 2, the exterior algebra The Exterior Algebra may be defined as T (V )/hS2Vi where Concrete Definition S2V = {v ⊗ w + w ⊗ v | v, w ∈ V} Given a set {e , . . . , e } with n elements, consider 1 n are the symmetric two- 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 E(n) over a field k with these expressions as elements. • How central notions in various 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 {1, 2, . . . , n} as [n]. multiplication which we also denote by ∧. The Each subset {i , . . . , i } 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, {2, 5} ⊆ [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) ∈ Z2 (where Z2 = {0, 1}), 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 {1, 2}, {3, 4}, {3, 5}, {4, 5, 6}, 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 product The point of relating these to the algebra E(n) V ⊗k V ⊗k · · · ⊗k V . The free on is that combinatorial simplicial complexes on [n] L ⊗p n V is the 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 I∆ Let R be the vector subspace of V ⊗k V generated by generated by all elements v ⊗v where v ∈ V . The exterior algebra {e ∧ · · · ∧ e | {i , . . . , i } 6∈ }. 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 ) {i1, . . . , ip} 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 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 {i1, . . . , ir } 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: 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 i6∈{i1,...,ir } ∆ the analog monomial ideal in this polynomial ring, the 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 paper by Hochster [29] and Stanley’s proof of the and of the topological space X. The Upper Bound Conjecture for simplicial spheres; cohomology is see [44]. Although one might say that E(∆) is a more natural object associated to , k[ ] has i+1 ˜i ∆ ∆ H (E(∆), d) = H (X, k) for i ≥ 0, been preferred for two reasons: (i) minimal free ˜i resolutions over k[x1, . . . , xn] are finite in contrast where H (X, k) is the reduced cohomology of to over the exterior algebra E(n), (ii) k[∆] is X. (For i > 0 this is simply the cohomology commutative and the machinery for commutative Hi(X, k), while for i = 0 this is the cokernel rings is very well developed. H0(pt, k) → H0(X, k).) Dualizing the complex ∗ ∗ Since 1975 this has been a very active of above we get E(∆) as a subcomplex of E(n) : research, with various textbooks published: [44], ∂2 ∗ ∂1 ∗ ∂0 ∗ [6], [34], and [22]. For the exterior face ring, see ··· → (E(∆) )2 → (E(∆) )1 → (E(∆) )0. [16]. ∗ Here (E(∆) )r+1 has a basis consisting of mono- mials Topology ∗ ∗ Let ui = (0,..., 0, 1, 0,..., 0) be the ith unit coor- (1) e ∧ · · · ∧ e i0 ir n dinate vector in R . To a subset {i1, . . . , ir } of [n] we may associate the (r − 1)-dimensional simplex where {i0, . . . , ir } are the r-dimensional faces of the simplicial complex . The differential ∂ is which is the convex hull of the points ui1 , . . . , uir in ∆ Rn. For instance, {2, 3, 5} ⊆ [6] gives the simplex contraction with the element u 6 consisting of all points (0, λ , λ , 0, λ , 0) in R , ∂ 2 3 5 a , u¬a, where λi ≥ 0 and λ2 + λ3 + λ5 = 1. A combinatorial simplicial complex ∆ has a sending the monomial (1) to its boundary natural topological realization X = |∆|. It is the n d union of all the simplices in R associated to the X j ∗ ∗ ∗ (−1) ee ∧ · · · ∧ eci ∧ · · · ei . sets {i1, . . . , ir } in . i0 j r ∆ j=0 Example 3. The simplicial complex given in Exam- (Here ec∗ means omitting this term.) The homology ple 2 has a topological realization which may be ij ∗ pictured as: Hi(E(∆) , ∂) computes the reduced simplicial homology of the space X (but with a shift by one in homological index i). In Example 3 ∗ we get H1(E(∆) , ∂) = k one-dimensional, one less than the number of components of ∆, and ∗ H2(E(∆) , ∂) = k one-dimensional, since there is a noncontractible 1-cycle through the points 3, 4 and 5. This is the disjoint union of a and a A good introduction to , disc with a handle. starting from simplicial complexes, is [35].

366 Notices of the AMS Volume 62, Number 4 Lie Theory g; see [40], [17] for the Koszul in the Differential graded (DGA) occur naturally differential graded setting. gives in many areas. They provide the “full story” in between the categories of these contrast to graded algebras, which often are the algebras, which on suitable quotients of these give cohomology of a DGA, like the cochain complex of an equivalence of categories. a topological space, in contrast to its cohomology ring. Combinatorics II: Hyperplane Arrangements L A DGA is graded algebra A = p≥0 Ap with a and the Orlik-Solomon Algebra differential d, i.e., d2 = 0, which is a derivation; i.e., Simplicial complexes are basic combinatorial for homogeneous elements a, b in A it satisfies structures, and we have seen in the section Combinatorics I how they are captured by the (2) d(a · b) = d(a) · b + (−1)deg(a)a · d(b). exterior face ring E(∆). One of the most successful The differential d either has degree 1 or −1 unifying abstractions in combinatorics is that of according to whether it raises degrees by one or a matroid (a term giving more associations might decreases degrees by one. be independence structures), which is a special What does it mean to give a k-linear differential type of a simplicial complex. To a matroid there d of degree 1 on E(V ) such that (E(V ), d) becomes is associated a quotient algebra of the exterior a DGA? Between degrees 1 and 2 we have a map algebra with remarkable connections to hyperplane d arrangements. V -→ ∧2V. A prime source of matroids is . By the definition of derivation above (2), it is Let us consider the vector space km and let easy to see that any between these x1, . . . , xm be coordinate functions on this space. P vector spaces extends uniquely to a derivation A v = λj xj gives a hyperplane in m m d on E(V ). Denote by g the dual vector space k : the set of all points (a1, . . . , am) ∈ k such ∗ P V = Homk(V , k). Dualizing the above map we get that j λj aj = 0. A set of linear forms v1, . . . , vn a map determines hyperplanes H1,H2,...,Hn. We call 2 this a hyperplane arrangement. It turns out that a ^ d∗ g -→ g number of essential properties of the hyperplane x ∧ y , [x, y]. arrangement are determined by the linear depen- dencies between the linear forms v1, . . . , vn. We 2 It turns out that d gives a differential, i.e., d = 0, if get a combinatorial simplicial complex M on [n] and only if the map d∗ satisfies the consisting of all subsets {i1, . . . , ir } of [n] such

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0. that vi1 , . . . , vir are linearly independent vectors. But there is more structure on this M, making Thus giving E(V ) the structure of a DGA with it a matroid. A simplicial complex M on [n] is a differential of degree 1 is precisely equivalent to matroid if the following extra condition holds: giving g = V ∗ the structure of a Lie algebra. If X and Y are independent sets of M, with the The cohomology of the complex (E(V ), d) com- cardinality of Y larger than that of X, there is y ∈ putes the of g. If g is the Lie Y \X such that X ∪ {y} is independent. algebra of a connected compact Lie G (over The elements of M are called the independent = ), it is a theorem of Cartan [1, Cor.12.4], that k R sets of the matroid, while subsets of [n] not in M the cohomology ring H∗(E(V ), d) is isomorphic ∗ are dependent. The diversity which the abstract to the cohomology ring H (G, R). A good and notion of a matroid captures is illustrated by the comprehensive introduction to Lie algebras is [31]. following examples, where we give independent The book [18] is much used as a reference book for sets of matroids: the representations of semisimple Lie groups and • Linear independent subsets of a set of vectors Lie algebras. But there are many books on this, and {v , v , . . . , v }. the above are mentioned mostly because I learned 1 2 n • Edge sets of graphs which do not contain a from these books. cycle. Now denote the dual V ∗ by W , and by S(W ) = • Partial transversals of a family of sets Sym(W ) the , i.e., the polynomial A ,A ,...,A . ring whose variables are any basis of W . The pair 1 2 N E(V ) and S(W ) is the prime example of a Koszul Example 4. Consider the hyperplane arrangement 2 dual pair of algebras; see [39], [3] for the general in C given by the two coordinate functions v1 = x1 2 framework of Koszul duality. Furthermore, when and v2 = x2. The complement C \H1 ∪ H2 consists we equip E(V ) with the differential d, the pair of the pairs (a, b) with nonzero coordinates; i.e., (E(V ), d) can be considered as the Koszul dual the complement is (C∗)2 where C∗ = C\{0}. Since of the enveloping algebra U(g) of the Lie algebra C∗ is homotopy equivalent to the circle S1, the

April 2015 Notices of the AMS 367 2 complement C \H1 ∪ H2 will be homotopy equiva- Clifford algebras are mostly applied when the lent to the S1 × S1. The cohomology ring of field k is the real numbers R. Let V = hii be a this torus is the exterior algebra E(2). one-dimensional vector space generated by a vector 2 b This example generalizes to a description of i, and let the be given by i , −1. the cohomology ring of the complement of any Then the associated Clifford algebra is the complex hyperplane arrangement in Cm. The matroid M numbers. When V = hi, ji is a two-dimensional of the hyperplane arrangement, being a simplicial space and complex, gives by the section Combinatorics I b b b i ⊗ i , −1, i ⊗ j + j ⊗ i , 0, j ⊗ j , −1, a monomial ideal IM in E(n). The dual element ∗ ∗ ∂ we obtain the . In general, for a real u = e1 + · · · + en gives a contraction a -→ u¬a symmetric form b, we may find a basis for V such sending ei1 ∧ · · · ∧ eir to X that if x1, . . . , xn are the coordinate functions, the (−1)j e ∧ · · · eˆ · · · ∧ e i1 ij ir form is j p n=p+q X X x2 − x2. (here eˆij means omitting this term). Now IM + ∂(IM ) i i also becomes an ideal in E(n). The quotient i=1 i=p+1 A(M) = E(n)/(IM + ∂(IM )) is called the Orlik- Such a Clifford algebra is denoted Clp,q. Solomon algebra associated to the hyperplane So Cl0,1 is the complex numbers and Cl0,2 is arrangement. In 1980 Peter Orlik and Louis Solomon the quaternions. Clifford algebras have interesting proved the following amazing result [36]. periodic behaviour: Clp+1,q+1 is isomorphic to the 2 × 2-matrices M (Cl ), and each of Cl + Theorem 5. Let T = Cm\ Sn H be the comple- 2 p,q p 8,q i=1 i and Cl is isomorphic to the 16 × 16-matrices ment of a complex hyperplane arrangement. Then p,q+8 M (Cl ). Thus Clifford algebras over the reals the algebra A(M) is the cohomology ring H∗(T , C). 16 p,q are essentially classified by Clp,0 and Cl0,q for p, Example 6. Consider u1 = x1−x2, u2 = x2−x3, and q ≤ 7. For a nice introduction to Clifford algebras, 3 u3 = x3 − x1 on C . There is one dependency here, see [19]. between u1, u2, and u3. Thus the Orlik-Solomon When Clp,q is a simple algebra and Clp,q → algebra is E(3) divided by the ideal generated by End(W ) is an irreducible representation of Clp,q the relation then W is called a space. These represen- tations occur a lot in . For ∂(e1 ∧ e2 ∧ e3) = e1 ∧ e2 − e1 ∧ e3 + e2 ∧ e3. instance, Cl1,3 is isomorphic to M4(R), and this The quotient algebra has 1, 3, and 2 representation on R4 is the Minkowski space with in degrees 0, 1, and 2 respectively. For the comple- one time and three space dimensions. ment T = C3\H ∪ H ∪ H we therefore have 1 2 3 A pioneer in the application of Clifford algebras H0(T , C) = C,H1(T , C) = C3,H2(T , C) = C2. in mathematical physics is David Hestenes [23], For hyperplane arrangements, or more gen- [24], [25], where he envisions the complete use of erally, for matroids, the Orlik-Solomon algebra it in classical . He calls this geometric has been much studied recently, [47], [15]. The algebra. The book [8] offers a leisurely introduction algebraic properties of the Orlik-Solomon algebra to the application of in physics. give a number of natural invariants for hyperplane is another advocate for the algebraic arrangements. approach to quantum mechanics [28]: …that quantum phenomena per se can Mathematical Physics be entirely described in terms of Clifford The Clifford algebra may be viewed as a deformation algebras taken over the reals without the of the exterior algebra. The exterior algebra E(V ) need to appeal to specific representations is defined as the quotient algebra T (V )/hS2V i. in terms of wave functions in a Hilbert Fix a symmetric b : S2V → k. Let space. This removes the necessity of using R = {r − b(r) | r ∈ S2V}. The Clifford algebra is and all the physical imagery the quotient of the tensor algebra by the relations that goes with the use of the wave function. R: Clb = T (V )/hRi. Algebraic Geometry Like the exterior algebra E(V ) = E(n) it has a Finitely generated graded modules over exterior

basis consisting of all products ei1 ····· eir for algebras seem far removed from geometry. How- subsets {i1 < ··· < ir } of {1, . . . , n} and so is also ever, we shall see that they encode perhaps the of dimension 2n as a vector space over k. Note that most significant invariants of algebraic geometry, we get the exterior algebra when the bilinear form the cohomological dimensions of twists of sheaves b = 0. on projective spaces.

368 Notices of the AMS Volume 62, Number 4 Example 7. Let n = 2 and E = E(2). Consider the Let S = Sym(W ) be the symmetric algebra (a map of free E-modules: polynomial ring). The map (6) gives rise to maps h e i (7) d= 2 e1 2 dp dp+1 dq−1 E -→ E . · · ·→S⊗kMp -→S⊗kMp+1 -→· · · -→ S⊗kMq →· · ·. 2 Writing E = Eu1 ⊕ Eu2 where u1 and u2 are gen- These maps give a (bounded) complex of S-modules, erators of this module, the cokernel of this map i.e., di+1 ◦ di = 0 (This is a correspondence within is a module M = Eu1 ⊕ Eu2/he2u1 + e1u2i. Such a the framework of Koszul duality, mentioned in map may, as we shortly explain, be completed to a the section “Lie Theory”.) Any finitely generated complex of free E-modules (we let d0 = d): graded S = Sym(W )-module may be sheafified to a coherent on the P(W ). In   e2 0= particular, we may sheafify the above complex and −2 h i −1 h i d   d = e1 e2 d = e1 ∧ e2 e1 · · · → E2 ------→ E ------→ E ------→ get a complex of coherent sheaves:   (3) e2 0 (8)   d1=e e  graded modules over E(V )  1 2 0 e1 E2 ------→ E3 → E4 → · · · ; bounded complexes of coherent sheaves

It is a complex since dp ◦ dp−1 = 0, as is easily on P(W ). verified. It is also exact at each place, meaning that This correspondence is from 1978 [4] and is the of dp equals the image of dp−1 for each the celebrated Bernstein-Gelfand-Gelfand (BGG) p. Thus it is an acyclic complex. correspondence. Somewhat more refined, it may be described as an equivalence of categories between Every finitely generated graded module M over suitable categories of the objects in (8). = E E(V ), or equivalently map d, gives rise to The amazing thing is that if M via the BGG- such an acyclic complex. In the example above the correspondence (8) gives a F ranks of the free modules follow a simple pattern, on P(W ) (this means that the sheafification of 1, 2, 3, 4,..., but in general, what are these ranks? the complex (7) has only one nonzero cohomol- Can they be given a meaningful interpretation? ogy sheaf F), then we can read off all the sheaf Indeed, a discovery from 2003, [13], tells us this is cohomology groups of all twists of F from the the case. Tate resolution T which we get via the correspon- As a module over itself E(V ) is both a projective dence (5). and an injective module. Given a finitely generated graded module M over E(V ), one can make a Theorem 8 ([13, Thm. 4.1]). If M via the BGG- minimal free (and so projective) resolution correspondence (8) gives a coherent sheaf F on the projective space P(W ), and the Tate resolution of • p M p P → M, where P = Wq ⊗k E M is (4), then the sheaf cohomology q∈Z p p+q H (P(W ), F(q)) = Wq . (the W p are vector spaces over the field k whose q Returning to the initial Example 7 in this section, elements are considered to have degree q) and a the sheaf corresponding to this module M is minimal injective resolution the structure sheaf OP1 on the projective line • p M p 1 M → I , where I = Wq ⊗k E P = P(W ). The Tate resolution (3) therefore tells q∈Z us that for d ≥ 0, the sheaf cohomology and splice these together into an acyclic complex ( d+1 0 1 k , d ≥ 0, H (P , O 1 (d)) = , (as in Example 7), P 0, d < 0, −1 0 1 2 (4) T : · · · → P → P → I → I → · · · , ( d−1 1 1 k , −d < 0, H (P , O 1 (−d)) = called the Tate resolution of M. So we get a P 0, −d ≥ 0. correspondence: An important feature of a Tate resolution T (5) graded modules over E(V ) is that it is fully determined by an arbitrary i i d i+1 ≤i ; Tate resolutions over E(V ). differential T -→ T . This is because T is a minimal projective resolution of im di, and T >i is a Now let us pass to another construction starting minimal injective resolution of im di. This gives us from the finitely generated graded module M = incredible freedom in construction. An arbitrary Lb i=a Mi over the exterior algebra E(V ). Let W homogeneous A of exterior forms gives a ∗ be the dual vector space V . The multiplication map V ⊗k Mi → Mi+1 gives a map M 0 dA M 1 (9) Wq ⊗k E -→ Wq ⊗k E. (6) Mi → W ⊗k Mi+1. q q

April 2015 Notices of the AMS 369 The module M = im dA over the exterior algebra Perwass similarly applies geometric algebra to then gives a complex of coherent sheaves F • on models occurring in engineering: camera positions, P(W ) by (8), and all such bounded complexes motion tracking, and statistics. It also considers on P(W ) do (in a suitable sense) come from numerical aspects of its implementation. Other such a homogeneous matrix A of exterior forms. books on geometric algebra and its use in computer Thus bounded complexes of coherent sheaves on modelling and engineering are [26], [27], [43], [9], projective spaces can be specified by giving a [45], and [46]. The book [11] gives a panorama of homogeneous matrix A of exterior forms, and any applications by a wide range of authors. matrix A will give some such bounded complex. The comprehensive book Grassmann Algebra [5] The Tate resolution associated to M and F • is the considers all aspects of computations concerning complex we obtain by taking a minimal projective the exterior algebra with Mathematica. It treats resolution of ker dA in (9) and a minimal injective the exterior, interior, and regressive products resolution of coker dA, and this resolution tells and geometric interpretations. A second volume us the cohomology of the complex of coherent treats the generalized Grassmann product which sheaves. constitutes an intermediate chain of products between the exterior and interior products, and Example 9. Let V be the five-dimensional vector applications to hypercomplex numbers and to space generated by {e1, e2, e3, e4, e5}. The matrix " # mechanics. Other treatises with a more purely e ∧e e ∧e e ∧e e ∧e e ∧e A = 1 2 2 3 3 4 4 5 5 1 mathematical focus are [42] and [33]. e3 ∧e5 e4 ∧e1 e5 ∧e2 e1 ∧e3 e2 ∧e4 gives a map E5 → E2. Via the BGG-correspondence References (8) this gives the celebrated Horrocks-Mumford 1. G. Bredon, Topology and geometry, Graduate Texts in Mathematics, 139, Springer-Verlag, New York, 1993. bundle on P4 discovered over forty years ago [30], 2. William E. Baylis, Clifford (Geometric) Algebras: With see also [13, Section 8]. In characteristic zero this Applications in Physics, Mathematics, and Engineering, is essentially the only known indecomposable Springer, 1996. two bundle on any projective space of dimension 3. Alexander Beilinson, Victor Ginzburg, and greater or equal to four. It is an intriguing problem Wolfgang Soergel, Koszul duality patterns in repre- to use the methods above to try to construct new sentation theory, J. Amer. Math. Soc., 9 (1996), no. 2, bundles of rank ≤ n − 2 on a projective space 473–527. 4. Alexander A. Beilinson, Coherent sheaves on Pn Pn, but to our knowledge nobody has yet been and problems of linear algebra, Funct. Anal. Appl., 12 successful. (1978), no. 3, 214–216. Tate resolutions and algebraic geometry are 5. John Browne, Grassmann Algebra, vol. 1, Create Space Independent Publishing Platform, 2012. treated in the books [14] and [12]. The software 6. Winfried Bruns and Jürgen Herzog, Cohen- program [21] contains the package BGG for doing Macaulay Rings, Cambridge University Press, 1998. computations with Tate resolutions. 7. Michael J. Crowe, A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, Courier Modelling and Computations Dover Publications, 1967. The last fifteen years have seen a flurry of books and 8. Christian Doran and Anthony Lasenby, Geomet- treatises giving applications of exterior algebras ric Algebra for Physicists, Cambridge University Press, 2007. and Clifford algebras, usually under the name 9. Leo Dorst, Chris Doran, and Joan Lasenby, Appli- “geometric algebra.” Groups at the University of cations of Geometric Algebra in Computer Science and Cambridge and the University of Amsterdam have Engineering, Springer, 2002. been particularly active in promoting geometric 10. Leo Dorst, Daniel Fontijne, and Stephen Mann, Geo- algebra. The book Geometric Algebra for Physicists metric Algebra for Computer Science (Revised Edition): [8] by C. Doran and A. Lasenby is a very well written An Object-Oriented Approach to Geometry, Morgan and readable introduction to exterior algebras, Kaufmann, 2009. 11. Leo Dorst and Joan Lasenby, Guide to Geometric Clifford algebras, and their applications in all Algebra in Practice, Springer, 2011. areas of physics, following the ideas outlined by D. 12. D. Eisenbud et al., Computations in Algebraic Geometry Hestenes. A more advanced treatment is [2]. The with Macaulay 2, vol. 8, Springer, 2002. book Geometric Algebra for Computer Scientists: 13. D. Eisenbud, G. Fløystad, and F. O. Schreyer, Sheaf An Object Oriented Approach,[10] by L. Dorst, D. cohomology and free resolutions over exterior algebras, Fontijne, and S. Mann shows geometric algebra as Trans. Amer. Math. Soc. 355 (2003), no. 11, 4397–4426. an effective tool to describe a variety of geometric 14. David Eisenbud, The Geometry of Syzygies: A Sec- ond Course in Algebraic Geometry and Commutative models involving linear spaces, circles, spheres, Algebra, vol. 229, Springer, 2005. rotations, and reflections. In particular it considers 15. Michael Falk, Combinatorial and algebraic struc- the conformal geometric model developed in ture in Orlik–Solomon algebras, European Journal of [32]. Geometric Algebra for Engineers,[37] by C. Combinatorics 22 (2001), no. 5, 687–698.

370 Notices of the AMS Volume 62, Number 4 16. G. Fløystad and J. E. Vatne, (Bi-)Cohen–Macaulay 39. Alexander Polishchuk and Leonid Positselski, simplicial complexes and their associated coherent Quadratic Algebras, University Lecture Series, vol. 37, sheaves, Comm. Algebra 33 (2005), no. 9, 3121–3136. American Mathematical Society, 2005. 17. Gunnar Fløystad, Koszul duality and equivalences 40. Leonid Efimovich Positsel’skii, Nonhomogeneous of categories, Trans. Amer. Math. Soc. 358 (2006), no. 6, quadratic duality and curvature, Funct. Anal. Appl. 27 2373–2398. (1993), no. 3, 197–204. 18. W. Fulton and J. Harris, : A 41. Gert Schubring (ed.), Hermann Günther Graßmann First Course, GTM 129, Springer, 1991. (1809–1877): Visionary Mathematician, Scientist and 19. D. J. H. Garling, Clifford Algebras: An Introduction, Neohumanist Scholar, Kluwer Academic Publishers, vol. 78, Cambridge University Press, 2011. 1997. 20. Hermann Grassmann and Lloyd C. Kannenberg, 42. William C. Schulz, Theory and Extension Theory, American Mathematical Society, application of Grassmann algebra, avail- 2000. able at http://www.cefns.nau.edu/ 21. Daniel R. Grayson and Michael E. Stillman, ~schulz/grassmann.pdf, 2011, preprint. Macaulay2, a software system for research in alge- 43. Gerald Sommer, Geometric Computing with Clifford braic geometry, available at Algebras: Theoretical Foundations and Applications in http://www.math.uiuc.edu/Macaulay2/. and Robotics, Springer, 2001. 22. Jürgen Herzog and Takayuki Hibi, Monomial Ideals, 44. Richard P. Stanley, Combinatorics and Commutative Springer, 2011. Algebra, Birkhäuser Boston, 2004. 23. David Hestenes, Space-Time Algebra, Gordon and 45. John A. Vince, Geometric Algebra for Computer Breach, 1966. Graphics, vol. 1, Springer, 2008. 24. , Clifford Algebra to Geometric Algebra: A Uni- 46. , Geometric Algebra: An Algebraic System for fied Language for Mathematics and Physics, D. Reidel Computer Games and Animation, Springer, 2009. Publishing Company, 1984. 47. S. A. Yuzvinsky, Orlik–Solomon algebras in algebra 25. , New Foundations for Classical Mechanics, and topology, Russian Mathematical Surveys 56 (2007), Springer, 1999. no. 2, 293. 26. , Old wine in new bottles: A new algebraic frame- work for computational geometry, Geometric Algebra with Applications in Science and Engineering, Springer, 2001, pp. 3–17. 27. Dietmar Hildenbrand, Foundations of Geometric Algebra Computing, vol. 8, Springer, 2012. 28. B. J. Hiley and R. E. Callaghan, Clifford algebras and the Dirac-Bohm quantum Hamilton-Jacobi equation, Foundations of Physics 42 (2012), no. 1, 192–208. 29. Melvin Hochster, Cohen-Macaulay rings, combina- torics, and simplicial complexes, Ring Theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), Lecture Notes in Pure and Appl. Math., Vol. 26, Dekker, New York, 1977, pp. 171–223. 30. G. Horrocks and D. Mumford, A rank two vector bun- dle on P4 with 15,000 symmetries, Toplogy, 12 (1973), no. 1, 63–81. 31. James E. Humphreys, Introduction to Lie Algebras and Representation Theory, vol. 1980, Springer, New York, 1972. 32. Hongbo Li, David Hestenes, and Alyn Rockwood, Spherical conformal geometry with geometric algebra, Geometric Computing with Clifford Algebras, Springer, 2001, pp. 61–75. 33. Douglas Lundholm and Lars Svensson, Clifford Algebra, Geometric Algebra, and Applications, 2009. 34. Ezra Miller and Bernd Sturmfels, Combinatorial Commutative Algebra, vol. 227, Springer, 2004. 35. James R. Munkres, Elements of Algebraic Topology, vol. 2, Addison-Wesley, Reading, 1984. 36. Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167–189. 37. Christian Perwass, Geometric Algebra with Applica- tions in Engineering, vol. 4, Springer, 2008. 38. Hans-Joachim Petsche, Albert C. Lewis, Jörg Liesen, and Steve Russ (eds.), From Past to Future: Graßmann’s Work in Context: Graßmann Bicentennial Conference, September 2009, Springer, 2010.

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