Where do alternating multilinear maps come from?

Transcription

1 Where do alternatng multlnear maps come from? Aaron Fenyes May 24, 2012 Abstract Alternatng multlnear maps, as the pontwse consttuents of dfferental forms, play a fundamental role n dfferental geometry, whle other knds of multlnear maps for example, symmetrc ones hardly show up at all. [1] In these notes, I ll try to llumnate the geometrc nature of alternatng maps by lnkng them to our ntutve concepts of volume and area. 1 What s volume? A parallelepped s one of the smplest knds of shapes. You can specfy a k- dmensonal parallelepped by gvng k vectors, lke ths: Fgure 1: 0-, 1-, 2-, and 3-dmensonal paralleleppeds. I ll call these k vectors the legs of the parallelepped. Any engneer can tell you that n an n-dmensonal vector space, every n- dmensonal parallelepped s assocated wth a non-negatve real number, called ts volume. I don t know exactly what ths volume s, but I do know t has the followng propertes: 1. If you stretch a parallelepped by multplyng one of ts legs by a scalar λ, ts volume gets multpled by λ. 1

2 2. If you shear a parallelepped by addng a multple of one of ts legs to another leg, ts volume stays the same. More formally, n an n-dmensonal vector space V over an absolute value feld 1 K, the functon ε: V n [0, ) that sends the legs of an n-parallelepped to ts volume has the followng propertes: 1. For any λ K, ε(... λv...) = λ ε(... v...). 2. For any λ K and, ε(... v... w...) = ε(... v + λw... w...). As we ll see n the next secton, these propertes are surprsngly strong for ther sze. They re strong enough, n fact, to determne the volume functon ε unquely, up to multplcaton by a constant. Moreover, they mply that ε s very nce algebracally: n Secton 4, we ll fnd that ε s the absolute value of a multlnear functon, whch s also unque up to scalng. 2 What are volume functons lke? Let s say a volume functon s any functon that fts the descrpton of the functon ε above. If you start playng wth the defnton, you ll soon dscover some general features of volume functons. Proposton 1. If ε s a volume functon, ε(... v... v...) = 0 for all. In other words, ε s alternatng. Proof. Ths follows from the shear property and a blndngly obvous consequence of the stretch property, whch I ll let you fnd for yourself. Proposton 2. If ε s a volume functon, ε(... v... w...) = ε(... w... v...) for all. In other words, ε s symmetrc. 1 That s, a feld equpped wth a map : K [0, ) such that 1. α = 0 f and only f α = αβ = α β. 3. α + β α + β. 2

3 Proof. Just keep poundng away wth the shear property. At the end, you ll need the fact that n an absolute value feld, α = α. More complcated manpulatons uncover features of deeper lnear algebrac sgnfcance. Proposton 3. Say ε s a volume functon. If v 1,..., v n are lnearly dependent, ε(v 1,..., v n ) = 0. Proof. Say α 1 v α n v n = 0, wth at least one of the α nonzero. Snce ε s symmetrc, we can assume wthout loss of generalty that α 1 s nonzero. Then, ε(v 1,..., v n ) = 1 α 1 ε(α 1v 1, v 2,..., v n ) = 1 α 1 ε(α 1v 1 + α 2 v α n v n, v 2,..., v n ) = 1 α 1 ε(0, v 2,..., v n ) = 0. The next feature a partal converse of Proposton 3 depends crucally on the fact that volumes are only defned for paralleleppeds of the same dmenson as V. Lemma 1. Say ε s a volume functon. If there are lnearly ndependent vectors v 1,..., v n for whch ε(v 1,..., v n ) = 0, then ε = 0. Proof. Suppose the proposton holds when n = m, for some m 1, and consder the case n = m + 1. Suppose there are lnearly ndependent vectors v 1,..., v m+1 V for whch ε(v 1,..., v m+1 ) = 0. Pck any vectors w 1,..., w m+1 V. Our goal s to show that ε(w 1,..., w m+1 ) = 0. If w 1,..., w m+1 are lnearly dependent, ths follows mmedately from Proposton 3, so we only need to consder the case where w 1,..., w m+1 are lnearly ndependent. Snce n = m + 1 s the dmenson of V, the vectors v 1,..., v m+1 form a bass for V. Lnear ndependence guarantees that w m+1 0, so at least one component of w m+1 n ths bass s nonzero. Snce ε s symmetrc, we can reorder the vectors v 1,..., v m+1 however we lke and stll have ε(v 1,..., v m+1 ) = 0, so we mght as well choose an order for whch the v m+1 component of w m+1 s nonzero. Let U be the subspace of V spanned by v 1,..., v m. Snce the v m+1 component of w m+1 s nonzero, we can fnd α 1,..., α m such that w = w +α w m+1 les n U for all {1... m}. Snce w 1,..., w m+1 are lnearly ndependent, so are w 1,..., w m. As a result, w 1,..., w m span U. We can therefore fnd β 1,..., β m such that w m+1 + β 1 w β m w m = λv m+1. By the shear property, ε(w 1,..., w m, w m+1 ) = ε( w 1,..., w m, w m+1 ) = ε( w 1,..., w m, λv m+1 ) = λ ε( w 1,..., w m, v m+1 ). 3

4 Defne a functon η : U m K by η(x 1,..., x m ) = ε(x 1,..., x m, v m+1 ). Because m 1, t s smple to verfy that η s a volume functon on U. Observe that v 1,..., v m are lnearly ndependent vectors for whch η(v 1,..., v m ) = 0. Snce we re assumng the proposton holds n dmenson m, t follows that η = 0, whch means ε(w 1,..., w m, w m+1 ) = λ η( w 1,..., w m ) = 0, whch s what we wanted to show. We have now proven that f the proposton holds when n = m, for some m 1, t also holds when n = m + 1. When n = 1, the proposton follows easly from the stretch property, so the proposton holds for all n 1. Snce an empty set of vectors s lnearly ndependent, the proposton also holds for n = 0. Its proof n ths case s left as a somewhat mnd-bendng exercse for the reader. (There s actually no need to consder the n = 0 case separately, but dong so allows a much clearer treatment of the other cases.) Wth Lemma 1 n hand, the man result of ths secton follows by a clever flck of the wrst. Theorem 1. If ε and η are volume functons, one s a constant multple of the other. Proof. The theorem s obvously true f ε = 0, so we only have to consder the case ε 0. Choose a bass v 1,..., v n for V. Because ε 0, Lemma 1 guarantees that ε(v 1,..., v n ) 0, so C = η(v 1,..., v n ) ε(v 1,..., v n ) s well-defned. Let δ : V n [0, ) be the functon δ(w 1,..., w n ) = η(w 1,..., w n ) Cε(w 1,..., w n ) R, where R s the standard absolute value on R. It s not hard to show that δ s a volume functon. By constructon, δ(v 1,..., v n ) = 0, so δ = 0 by Lemma 1. 3 Where does volume come from? The noton of volume s an essental part of our everyday experence what one mght call the real world. A volume functon, however, can be defned on a vector space over any absolute value feld K. The absolute value serves as an ntermedary between the K world and the real world, convertng the K scale factor λ nto the real scale factor λ n the statement of the stretch property. Metaphorcally, a volume functon looks somethng lke ths: 4

5 Fgure 2: A volume functon as a flower. Lke a flower, a volume functon has part of ts structure hdden underground, n the K world. In the next secton, we ll explore volume s K-valued roots. 4 Pre-volume functons A volume functon assocates every n-parallelepped n an n-dmensonal vector space wth a non-negatve real number. Let s try assocatng every n- parallelepped wth a scalar nstead. We ll call our assocaton a pre-volume functon f t has the followng propertes: 1. If you stretch a parallelepped by multplyng one of ts legs by a scalar λ, ts pre-volume gets multpled by λ. 2. If you shear a parallelepped by addng a multple of one of ts legs to another leg, ts pre-volume stays the same. More formally, a pre-volume on an n-dmensonal vector space V over a feld K s a functon E : V n K wth the followng propertes: 1. For any λ K, 2. For any λ K and, It s mmedately apparent that... E(... λv...) = λe(... v...). E(... v... w...) = E(... v + λw... w...). Proposton 4. If K s an absolute value feld, the absolute value of a prevolume functon s a volume functon. 5

6 We ll soon learn that the converse s also true: every volume functon s the absolute value of a pre-volume functon. Pre-volume functons act a lot lke volume functons. The followng propostons, for example, are almost dentcal to our frst three propostons about volume functons, and they re proven n essentally the same way. Proposton 5. If E s a pre-volume functon, E(... v... v...) = 0 for all. In other words, E s alternatng. Proposton 6. If E s a pre-volume functon, E(... v... w...) = E(... w... v...) for all. In other words, E s skew-symmetrc. Proposton 7. Say E s a pre-volume functon. If v 1,..., v n are lnearly dependent, E(v 1,..., v n ) = 0. For volume functons, our next step was to prove a partal converse of the thrd proposton, usng the fact that volumes are only defned for paralleleppeds of the same dmenson as V. For pre-volume functons, we can get a much stronger result. Lemma 2. Every pre-volume functon s multlnear. Proof. Say E : V n K s a pre-volume. Our goal s to show that E has the followng propertes: 1. E(... λv...) = λe(... v...) 2. E(... v + w...) = E(... v...) + E(... w...) The frst property s ust the stretch property, so t holds by defnton. Snce E s skew-symmetrc, we only have to prove the second property for = 1. Let A = E(u, v 2,..., v n ) B = E(ũ, v 2,..., v n ) C = E(u + ũ, v 2,..., v n ). We want to show that A + B = C. If v 2,..., v n are lnearly dependent, ths follows mmedately from Proposton 7, so we only need to consder the case where v 2,..., v n are lnearly ndependent. If u les n the subspace spanned by v 2,..., v n, the shear property can be used to show that A = 0 and C = B, and the desred result follows. Hence, we only need to consder the case where u, v 2,..., v n are lnearly ndependent. 6

7 Snce n s the dmenson of V, the vectors u, v 2,..., v n span V. That means we can wrte ũ as λu + x, where x les n the subspace spanned by v 2,..., v n. Notce that u + ũ = (λ + 1)u + x. By the shear property, B = E(λu, v 2,..., v n ) C = E((λ + 1)u, v 2,..., v n ), so the stretch property gves B = λa and C = (λ + 1)A. The desred result follows. We now have an extremely useful characterzaton of pre-volume functons. Theorem 2. A functon V n K s a pre-volume f and only f t s multlnear and alternatng. Proof. The only f drecton s gven by Proposton 5 and Lemma 2, and the f drecton s a straghtforward exercse n defnton-chasng. Ths characterzaton makes t easy to prove that pre-volume functons always exst, and are unque up to scalng. In fact... Corollary 1. The pre-volume functons on V form a one-dmensonal vector space. Proof. The alternatng multlnear functons V n K form a vector space under the usual addton and scalar multplcaton of functons. If v 1,..., v n s a bass for V, an alternatng multlnear functon E : V n K s completely determned by the value E(v 1,..., v n ), and any choce of E(v 1,..., v n ) gves an alternatng multlnear functon. Hence, the space of alternatng multlnear functons V n K s one-dmensonal. Together, the exstence of pre-volume functons and the unqueness of volume functons mply the converse of Proposton 4. Theorem 3. Every volume functon s the absolute value of a pre-volume functon. Proof. Say K s an absolute value feld, and ε s a volume functon on V. Corollary 1 guarantees the exstence of a nonzero pre-volume functon H on V, and Proposton 4 tells us that the absolute value η of H s a volume functon. By Theorem 1, one of the volume functons ε and η s a multple of the other. Snce η s nonzero, t follows that ε s a multple of η. Usng ths result, we can turn Theorem 2 nto a nce characterzaton of volume functons. Corollary 2. If K s an absolute value feld, a functon V n [0, ) s a volume functon f and only f t s the absolute value of an alternatng multlnear functon. 7

8 5 What s area? Imagne you wake up one mornng to fnd a two-dmensonal parallelepped floatng n the ar outsde your wndow. Naturally, you d lke to know more about t. Your frst thought s to measure ts volume usng a volume functon, but of course that doesn t make sense: n a three-dmensonal space, volumes are only defned for three-dmensonal paralleleppeds. If you could somehow wrestle the parallelepped to the ground, you could pn t flat aganst a two-dmensonal board and measure ts volume there. Unfortunately, the parallelepped s far out of reach. A few mnutes of furous ponderng brng you no closer to satsfyng your curosty, so you decde to forget about the apparton altogether. As you re leavng for work, however, you happen to glance up and notce that the mornng sun s castng a shadow of the parallelepped aganst the wall of your buldng. Now there s an dea! The shadow s a two-dmensonal parallelepped n a twodmensonal space, so you can measure ts volume wth a volume functon. As the sun moves across the sky, t ll llumnate the parallelepped from other drectons and cast ts shadow aganst other surfaces. You can use these multple perspectves to buld up a more complete pcture of your mysterous vstor. The story I ve ust told may not hold up under phlosophcal scrutny, but t suggests a reasonably natural way of generalzng the noton of volume to lower-dmensonal paralleleppeds. Ths generalzaton, whch I ll call area, wll be the subect of the next secton. 6 Area functons Let s say a k-area functon on a vector space V s a functon α: V k [0, ) of the form α(v 1,..., v k ) = ε(lv 1,..., Lv k ), where L: V W s a lnear map, W s a k-dmensonal vector space, and ε: W k [0, ) s a volume functon. (For ths to make sense, the base feld K has to be an absolute value feld.) Smlarly, we ll defne a k-pre-area functon on V as a functon A: V k K of the form A(v 1,..., v k ) = E(Lv 1,..., Lv k ), where L and W are as before, and E : W k K s a pre-volume functon. If n s the dmenson of V, an n-area s the same thng as a volume, and an n- pre-area s the same thng as a pre-volume. Hence, area really s a generalzaton of volume. In Secton 4 (Theorem 2), we used the noton of pre-volume to characterze the alternatng multlnear functons V n K. Now, more generally, we can use the noton of pre-area to characterze the alternatng multlnear functons V k K for any k. 8

9 Theorem 4. A map V k K s multlnear and alternatng f and only f t can be wrtten as a lnear combnaton of k-pre-areas. Proof. The f drecton s easy. We ll start the only f drecton by pckng a bass v 1,..., v n for V. For each functon I : {1... k} {1... n}, defne θ I : V k K as the unque multlnear functon wth the property that { 1 J = I θ I (v J(1),..., v J(k) ) = 0 J I for all J : {1... k} {1... n}. For each strctly ncreasng functon I : {1... k} {1... n}, defne χ I (w 1,..., w k ) = σ sgn(σ)θ I (w σ(1),..., w σ(k) ), where σ runs over all the permutatons of {1... k}. Any multlnear functon can be wrtten as a lnear combnaton of the θ I. Snce every alternatng multlnear functon s skew-symmetrc, any alternatng multlnear functon can be wrtten as a lnear combnaton of the χ I. It s apparent from the defnton that each χ I s alternatng and skewsymmetrc when ts arguments are restrcted to the set of bass vectors. A short calculaton then proves that χ I s alternatng for all arguments. It follows that each χ I s a k-pre-area, because χ I s a pre-volume when ts arguments are restrcted to the subspace spanned by v I(1),..., v I(k). Ths result lnks alternatng multlnear maps to the geometrc concept of area, usng the noton of pre-area as an ntermedary. In other words, t does exactly what I set out to do n the abstract of these notes. Unfortunately, the connecton I ve establshed s a lot more subtle than I d hoped t would be. The subtlety s that, although Proposton 4 and Theorem 3 let us pass from pre-areas to areas and back n a straghtforward way, they don t tell us what to make of a lnear combnaton of pre-areas. In the case of n-pre-areas, the pont s moot, because a lnear combnaton of n-pre-areas s tself an n-prearea (Corollary 1). Ths fact gves us a drect connecton between n-lnear maps and n-areas, enshrned n Corollary 2. For general k-pre-areas, however, the subtlety remans; untl t can be resolved, the story told n these notes wll be ncomplete. References [1] Matt Noonan. Antsymmetry n geometry, I. In The Everythng Semnar. antsymmetry-n-geometry-/,