Some multilinear algera Multilinear forms Let V e a finite dimensional R-vector space, dim V n. For k P N 0, we denote y T k V the vector space of k-linear forms Then T 1 V V and, y convention, T 0 V R. V ˆ... ˆ V µ ÝÑ R. Let te i u e a asis of V. Then a k-linear form µ P T k V is determined y the values µpe i1,..., e ik q for 1 ď i 1,..., i k ď n, and these values can e aritrary. This follows from the representation kź µpv 1,..., v k q µpe i1,..., e ik q e i j pv j q (0.1) 1ďi 1,...,i k ďn where te i u denotes the dual asis of V. The k-linear forms e i 1...i k pv 1,..., v k q kź e i j pv j q constitute a asis of T k V which we refer to as the asis induced y the asis te i u, and (0.1) can e rewritten as: µ µpe i1,..., e ik q e i 1...i k (0.2) In particular, dim T k V pdim V q k. 1ďi 1,...,i k ďn There is a natural product of multilinear forms, called the tensor product. k, l P N 0 one has the natural ilinear map given y T k V ˆ T l V j 1 ÝÑ T k`l V j 1 pµ νqpv 1,..., v k`l q µpv 1,..., v k q νpv k`1,..., v k`l q for µ P T k V and ν P T l V. Equipped with the tensor product, T V à kě0 T k V Namely, for ecomes an associative graded R-algera, called the tensor algera of V. It can e characterized y the universal property of eing the largest associative R-algera with unity containing V. For elements of induced ases, it holds that e i 1...i k e i k`1...i k`l e i 1...i k`l and, in particular, e i 1...i k e i 1 e i k. 1
We therefore can rewrite (0.2) again as µ µpe i1,..., e ik q e i 1 e i k 1ďi 1,...,i k ďn One can pull ack multilinear forms with linear maps, and the pull-ack is compatile with the tensor product: A vector space homomorphism V Ñ A W induces a homomorphism of graded algeras A T W ÝÑ T V (0.3) y for µ P T k W. pa µqpv 1,..., v k q µpav 1,..., Av k q A reminder on permutations Let S k denote the symmetric group of k symols, realized as the group of ijective self maps of the set t1,..., ku. The sign of a permutation π P S k is defined as sgnpπq ź 1ďiăjďk πpiq πpjq i j P t 1u. It counts the parity of the numer of inversions of π, i.e. of pairs pi, jq such that i ă j and πpiq ą πpjq. The sign is positive iff the numer of inversions is even, and such permutations are called even. Transpositions are odd, and a permutation is even iff it can e written as the product of an even numer of transpositions. The sign is multiplicative, sgnpπ 1 π 2 q sgnpπ 1 q sgnpπ 2 q, i.e. the sign map is a homomorphism. S k sgn ÝÑ t 1u Alternating multilinear forms We return to our discussion of multilinear forms. There is a natural action S k ñ T k V y pσµqpv 1,..., v k q µpv σp1q,..., v σpkq q A k-linear form µ is called alternating or skew-symmetric if σµ sgnpσq µ 2
for all σ P S k, i.e. µpv σp1q,..., v σpkq q sgnpσq µpv 1,..., v k q. Note that µ then vanishes on k-tuples of linearily dependent vectors. In particular, all alternating k-forms for k ą n vanish. We denote y Λ k V Ă T k V the linear suspace of alternating k-linear forms. Then Λ 0 V R, Λ 1 V V and Λ k V 0 for k ą n. There is a natural projection y skew-symmetrization, i.e. T k V alt ÝÑ Λ k V alt µ 1 sgnpσq σµ, palt µqpv 1,..., v k q 1 sgnpσq µpv σp1q,..., v σpkq q. Fix now a asis te i u of V. Then an alternating k-linear form α P Λ k V is already determined y the values αpe i1,..., e ik q for 1 ď i 1 ă ă i k ď n, and these values can e aritrary. To see this, we consider the forms Note that they are given y determinants: alt e i 1...i k 1 sgnpσq σe i 1...i k. palt e i 1...i k qpv 1,..., v k q 1 sgnpσq e i loooooooooooomoooooooooooon 1...i k pv σp1q,..., v σpkq q ś k j 1 e i pv j σpjq q looooooooooooooooooooomooooooooooooooooooooon detpe i j pv l qq j,l The forms alt e i 1...i k constitute a asis of Λ k V. The form α can e represented as α αpe i1,..., e ik q alt e i 1...i k. (0.4) 1ďi 1 ă ăi k ďn In particular, dim Λ k V `n k. There is a natural product of alternating multilinear forms, called the wedge product, which is derived from the tensor product y skew-symmetrization: There are natural ilinear maps Λ k V ˆ Λ l V ^ ÝÑ Λ k`l V given y α ^ β pk ` lq! l! 3 altpα βq (0.5)
i.e. pα ^ βqpv 1,..., v k`l q 1 l! `l sgnpσq αpv σp1q,..., v σpkq q βpv σpk`1q,..., v σpk`lq q The wedge product is associative and (graded) anticommutative. The latter property means that β ^ α p 1q kl α ^ β. Equipped with the wedge product, Λ V nà Λ k V k 0 ecomes an anticommutative associative graded R-algera, called the exterior algera of V. It can e characterized y the universal property of eing the largest associative R-algera with unity containing V so that the elements in V anticommute. We have that dim Λ V 2 dim V. For multiple products, (0.5) generalizes to equivalently, pα 1 ^ ^ α r qpv 1,..., v k q α 1 ^ ^ α r pk 1 ` ` k r q! k 1! k r! 1 k 1! k r! alt α 1 α r, sgnpσq α 1 pv σp1q,... q α r p..., v σpkq q where k k 1 ` ` k r. For 1-forms λ 1,..., λ k P Λ 1 V V this specializes to In particular, i.e. pλ 1 ^ ^ λ k qpv 1,..., v k q detpλ j pv l qq j,l. (0.6) pe i 1 ^ ^ e i k qpv 1,..., v k q detpe i j pv l qq j,l e i 1 ^ ^ e i k alt e i 1...i k. We call the set of forms e i 1 ^ ^ e i k the asis of Λ k V induced y the asis te i u of V. The representation (0.4) can e rewritten as α αpe i1,..., e ik q e i 1 ^ ^ e i k. 1ďi 1 ă ăi k ďn For a vector v P V, the natural linear maps i v : Λ k V Ñ Λ k 1 V given y i v α αpv,... q are called interior multiplication or contraction with v. The comined linear map i v : Λ V Ñ Λ V satisfies i 2 v 0 and is an antiderivation of degree 1, i.e. one has the product rule i v pα ^ βq pi v αq ^ β ` p 1q k α ^ pi v βq (0.7) 4
for α P Λ k V and β P Λ V. To verify this, we may restrict to decomposale forms α λ 1^ ^λ k and β λ 1 1 ^ ^ λ 1 l with λ i, λ 1 i P Λ 1 V, ecause oth sides are ilinear. From (0.6), we otain that k i v pλ 1 ^... ^ λ k q p 1q i`1 lomon λ i pv 1 q pλ 1 ^... ˆλ i... ^ λ k q i 1 where ˆλ i indicates that the factor λ i is omitted. We get an analogous expression for i v pλ 1 1 ^... ^ λ 1 lq and (0.7) follows. If V Ñ A W is a linear map of vector spaces, then the pull-ack (0.3) of multilinear forms restricts to the pull-ack Λ W ÝÑ A Λ V (0.8) of alternating forms, i.e. the pull-ack of an alternating multilinear form is again alternating. The pull-ack is also compatile with the wedge product, i.e. (0.8) is a homomorphism of graded algeras. If A P EndpV q, then A Λ n V det A id Λ n V. Orientation We suppose first that n ě 1. Let BpV q denote the space of ases e pe 1,..., e n q of V. There is a natural simply transitive right action BpV q ð GLpn, Rq given y e A e 1 with e 1 j i a ij e i for A pa ij q. Note that GLpn, Rq has two (path) connected components, namely the sugroup GL`pn, Rq of matrices with positive determinant and the matrices with negative determinant. Accordingly, BpV q has two path components, which are the two GL`pn, Rq-orits. Let e, e 1 P BpV q e ases and let A P GLpn, Rq e the unique transition matrix with e A e 1. Then e and e 1 lie in the same component of BpV q iff det A ą 0. In this case one says that e and e 1 have the same orientation. One defines an orientation of V as a component of BpV q, i.e. as an equivalence class of ases with the same orientation. One then calls these ases positively oriented, and the others negatively oriented. A asis determines an orientation, namely the component of BpV q containing it. The standard orientation of R n is determined y its standard asis pe 1,..., e n q. We have that dim Λ n V 1. A non-zero element 0 ω P Λ n V is called a volume form. For ases e, e 1 with e A e 1 it holds that ωpe 1 1,..., e 1 nq det A ωpe 1,..., e n q. The volume form ω therefore determines an orientation y defining a asis e as positively oriented iff ωpe 1,..., e n q ą 0. One can alternatively define an orientation as a (ray) component of the pointed line Λ n V t0u. 5
In the case n 0, one defines an orientation of the trivial vector space V t0u as a choice of sign. In particular, there is the natural orientation `. Complex vector spaces have natural orientations as real vector spaces, ecause the groups GLpn, Cq are path connected. Indeed, if W is a C-vector space with dim C W n, then the space B C pw q of complex ases (acted upon simply transitively y GLpn, Cq) is path connected, and the image of the natural map B C pw q ÝÑ B R pw q, pe 1,..., e n q ÞÑ pe 1, ie 1,..., e n, ie n q is contained in a component of B R pw q which one defines to e the natural orientation. Symmetric multilinear forms and polynomials We continue our discussion of multilinear forms. A k-linear form µ is called symmetric if for all σ P S k, i.e. We denote y σµ µ µpv σp1q,..., v σpkq q µpv 1,..., v k q. S k V Ă T k V the linear suspace of alternating k-linear forms. Then S 0 V R and S 1 V V. There is a natural projection y symmetrization, i.e. T k V sym ÝÑ S k V sym µ 1 σµ, psym µqpv 1,..., v k q 1 µpv σp1q,..., v σpkq q. Fix now a asis te i u of V. Then a symmetric k-linear form µ P S k V is already determined y the values µpe i1,..., e ik q for 1 ď i 1 ď ď i k ď n, and these values can e aritrary. To see this, we consider the forms sym e i 1...i k 1 where e i 1...i k e i 1 e i k. They are given y: σe i 1...i k. psym e i 1...i k qpv 1,..., v k q 1 loooooooooooomoooooooooooon e i 1...i k pv σp1q,..., v σpkq q ś k j 1 e i pv j σpjq q 6
The forms sym e i 1...i k constitute a asis of S k V. In particular, dim S k V `n`k 1. k There is a natural product of symmetric multilinear forms, called the symmetric product, which is derived from the tensor product symmetrization: There are natural ilinear maps given y i.e. pµ d νqpv 1,..., v k`l q 1 pk ` lq! S k V ˆ S l V ÝÑ S k`l V µ d ν sympµ νq (0.9) `l µpv σp1q,..., v σpkq q νpv σpk`1q,..., v σpk`lq q The normalization is motivated y the isomorphism to the polynomial algera constructed elow. The symmetric product is associative and commutative. Equipped with the symmetric product, nà S V S k V k 0 ecomes a commutative associative graded R-algera, called the symmetric algera of V. It can e characterized y the universal property of eing the largest commutative associative R-algera with unity containing V. There are natural linear isomorphisms S k V ÝÑ Pol k pv q (0.10) to the spaces of homogeneous polynomials of degree k on V y restriction to the diagonal, i.e. the polynomial P µ associated to a symmetric k-linear form µ is given y P µ pvq µpv,..., vq. The asis element sym e i 1...i k of S k V corresponds to the polynomial e i 1... e i k. The inverse of (0.10) is called polarization. An explicit formula for it is given y: 1 µpw 1,..., w k q p 1qk k p 1q j j 1 i 1 ă ăi j P µ pw i1 `... ` w ij q Due to our choice of normalization, multiplication is preserved y the maps (0.10), P µdν P µ P ν, and they constitute an isomorphism of graded algeras to the algera of polynomials on V. S V ÝÑ PolpV q à kě0 Pol k pv q 1 Proof: Indeed, when replacing P µ pw i1 `... ` w ij q y µpw i1 `... ` w ij,..., w i1 `... ` w ij q for every proper suset of indices I Ĺ t1,..., ku, I l ă k, the terms µpw m1,..., w mk q with m 1,..., m k P I occur in the sums for j ě l, and hence with the multiplicity ř k p 1qj`k l j l j l 0. This leaves the terms µpwσp1q,..., w σpnq q for the permutations σ P S n, which occur with multiplicity in the sum for j k. 7
Tensor product A ilinear map U ˆ V β ÝÑ W (0.11) of R-vector spaces can e regarded as a product of vectors in U with vectors in V whose values are vectors in W. There is a natural such product with the least possile amount of linear relations etween the values: Definition 0.12 (Tensor product). The tensor product of two vector spaces U and V is a vector space U V together with a ilinear map U ˆ V ÝÑ U V satisfying the following universal property: For every ilinear map (0.11) there exists a unique linear map U V ÝÑ λ W such that β λ. In other words, the natural homomorphism HompU V, W q ÝÑ BilpU, V ; W q, λ ÞÑ λ is an isomorphism. Theorem 0.13. A tensor product exists and is unique up to natural isomorphism. We can therefore speak of the tensor product. Proof. Uniqueness follows from the universal property: Given two tensor products U ˆ V ÝÑ U V and U ˆV ÝÑ r U rv, there exist unique linear maps U V ÝÑ λ λ U rv and U rv ÝÑ r U V such that r λ and λ r r. It follows that pλ r λq and r pλ rλq r. The uniqueness part of the universal property then implies that λ r λ id UV and λ rλ id U rv. Hence, etween any two tensor products there is a natural isomorphism. Existence. We form a vector space with asis U ˆ V, namely let E e the vector space consisting of all finite linear cominations a i pu i, v i q with u i P U, v i P V and a i P R. We denote y i U ˆ V ι ÝÑ E the inclusion. It is only a map of sets. To make it ilinear, we divide out the minimal necessary amount of relations: Let R Ă E e the linear suspace generated y the elements pu 1, vq ` pu 2, vq pu 1 ` u 2, vq, pu, v 1 q ` pu, v 2 q pu, v 1 ` v 2 q, 8
pau, vq apu, vq, pu, avq apu, vq for u, u 1, u 2 P U, v, v 1, v 2 P V and a P R. Then ι descends to the ilinear map U ˆ V ÝÑ E{R : U V, pu, vq ÞÑ pu, vq ` R : u v. We must verify that it satisfies the universal property. Given a ilinear map U ˆV β Ñ W, there exists a unique linear map E ˆλÑ W extending it, β ˆλ ι. The ilinearity of β is equivalent to ˆλpRq 0. Hence, ˆλ descends to a linear map U V Ñ λ W with λpu vq ˆλ`pu, vq βpu, vq. The last property uniquely determines λ ecause the elements u v generate U V. A more concrete idea of the tensor product is provided y the following fact. Lemma 0.14. If te i : i P Iu and tf j : j P Ju are ases of U and V, then te i f j : i P I, j P Ju is a asis of U V. In particular, in the case of finite dimensions, it holds that dim U V dim U dim V. Proof. From the construction of the tensor product we know that the elements u v generate U V. Consequently, also the elements e i f j generate. In order to show that they are linearily independent, we separate them y linear forms. Namely, we use the linear forms U V λ kl ÝÑ R, u v ÞÑ e kpuqf l pvq which are induced y the ilinear forms U ˆ V β kl ÝÑ R, pu, vq ÞÑ e kpuqf l pvq. For a finite linear relation it follows y applying c ij e i f j 0 i,j c kl λ kl` c ij e i f j 0. Thus, the elements e i f j form a asis. With respect to the ases, the tensor product is given y: i,j p i a i e i q p j j f j q i,j a i j e i f j When changing the ases of the factors, ẽ k ř i g kie i and f l ř j h ljf j, the induced change of asis for the tensor product is given y: ẽ k f l i,j g ki h lj e i f j 9
Remark 0.15. It follows that if te i : i P Iu is a asis of U, then every element in U V can e written as e i v i with unique vectors v i P V. Remark 0.16. (i) There is a canonical isomorphism switching factors i U V V U (0.17) which identifies the elements u v with the elements v u. It is induced y the ilinear map U ˆ V Ñ V U sending pu, vq ÞÑ v u. (ii) There are canonical homomorphisms U V ÝÑ HompU, V q; u v ÞÑ u p qv (0.18) relating spaces of homomorphisms to tensor products. If dim U ă 8, then these are isomorphisms, i.e. the space of homomorphisms can then e represented as a tensor product, HompU, V q U V. Indeed, if te i u is a asis of U, then te i u is a asis of U (ecause of finite dimensionality), Elements of U V have unique representations as sums ř i e i v i and, correspondingly, elements of HompU, V q as sums ř i e i p qv i. If also dim V ă 8 and tf j u is a asis of V, then the homomorphism aji e i f j is given with respect to the chosen ases y the matrix pa ji q j,i. In particular, if U V, then we have the natural isomorphism EndpUq U U, and id U corresponds to the element ř i e i e i. (ii ) The homomorphism (0.18) is always injective, as one sees y restricting to finite dimensional suspaces of U. By analogy with the twofold tensor product, the multiple tensor product of an aritrary numer of vector spaces U 1,..., U n is a multilinear map U 1 ˆ... ˆ U n ÝÑ U 1... U n with the universal property that every multilinear map U 1 ˆ... ˆ U n Ñ W is the composition of with a unique linear map U 1... U n Ñ W. Existence and uniqueness of the multiple tensor product are proven in the same way. Lemma 0.19 (Associativity). There are natural isomorphisms pu 1... U n q pu n`1... U n`m q ÝÑ U 1... U n`m mapping elements pu 1... u n q pu n`1... u n`m q to elements u 1... u n`m. 10
Proof. The natural multilinear map induces a linear map U 1 ˆ... ˆ U n`m ÝÑ pu 1... U n q pu n`1... U n`m q U 1... U n`m ÝÑ pu 1... U n q pu n`1... U n`m q mapping elements u 1... u n`m to elements pu 1... u n q pu n`1... u n`m q. That it is an isomorphism, follows y choosing ases. Remark 0.20. (i) Permutations of factors. Generalizing (0.17), for permutations σ P S n there are the natural isomorphisms U 1... U n U σp1q... U σpnq mapping elements u 1... u n to elements u σp1q... u σpnq. (ii) Functoriality. Homomorphisms α i : U i Ñ V i induce a homomorphism α 1 α n : U 1... U n ÝÑ V 1... V n mapping elements u 1... u n to elements α 1 pu 1 q... α n pu n q. Indeed, it is induced y the multilinear map U 1 ˆ... ˆ U n ÝÑ V 1... V n sending pu 1,..., u n q ÞÑ α 1 pu 1 q... α n pu n q (iii) Multilinear forms. Generalizing (0.18), we have natural injective homomorphisms U 1...U nv ÝÑ MultpU 1,..., U n ; V q; u 1...u nv ÞÑ `pu 1,..., u n q ÞÑ u 1pu 1 q... u npu n qv which are isomorphisms if dim U i ă 8. Tensor algera. For a vector space U and m P N 0 we denote y T m U : m U : U... U loooooomoooooon the m-fold tensor power of U. Then T 1 U U. Convention: T 0 U R. Consider the graded vector space T U : 8 m 0T m U Due to the associativity of the tensor product, there are natural ilinear maps T m U ˆ T n U which, y ilinear extension, yield a product T U ˆ T U m ÝÑ T m`n U ÝÑ T U. Equipped with this product, T U ecomes a graded associative R-algera with unity, the (covariant) tensor algera of U. Covariant, ecause the functor U ÞÑ T U from vector spaces to 11
algeras is covariant, i.e. a linear homomorphism U Ñ V induces a homomorphism of graded algeras T U Ñ T V in the same direction. The covariant tensor algera can also e characterized y a universal property. The algera T U is the largest associative algera with unity generated y U in the sense that every linear homomorphism α : U Ñ A to an associative algera with unity uniquely extends to an algera homomorphism T U Ñ A; it maps elements u 1... u n to αpu 1 q... αpu n q. The contravariant tensor algera of U is defined as T U : T pu q. Contravariant, ecause a linear homomorphism U Ñ V induces a linear homomorphism V Ñ U of dual spaces and hence an algera homomorphism T V Ñ T U in the reverse direction. Both types of tensors can e comined to mixed tensors with covariant and contravariant components. Consider T s r U : T r U T s U for r, s P N 0, with the convention T 0 0 U R, and put T puq : T U : 8 r,s 0T s r U. Again, there is a natural product on T puq satisfying pu 1... u r1 u 1... u s 1 q pv 1... v r2 v 1... v s 2 q u 1... u r1 v 1... v r2 u 1... u s 1 v 1... v s 2 which makes T puq into a igraded associative algera with unity, the tensor algera of U. There are natural inclusions T U Ă T puq and T U Ă T puq such that T r U T 0 r U and T s U T s 0 U. The elements of T puq are called tensors, and the elements of T s r U are called homogeneous tensors of type pr, sq. Example 0.21. Examples: Tensors of type p1, 0q are vectors, tensors of type p0, 1q are linear forms, tensors of type p0, 2q are ilinear forms, tensors of type p1, 1q are endomorphisms. Trace and contraction. Suppose now that dim U ă 8. Then the natural inclusion U ãñ U is an isomorphism and induces natural isomorphisms T s r U T r s U, T pu q T puq. The natural ilinear pairing U ˆ U ÝÑ R, pu, u q ÞÑ u puq induces the linear form T 1 1 U U U EndpUq tr ÝÑ R which is nothing ut the trace. Indeed, if te i u is a asis of U, then the endomorphism A ř i,j a ij e i e j given y the matrix pa ij q is mapped to ř i,j a ij e j pe i q ř i a ii trpaq. 12
More generally, one can pair the i-th covariant factor of a homogeneous tensor with the j-th contravariant factor and thus otains the contraction homomorphisms for 1 ď i ď r and 1 ď j ď s satisfying Tr s U Cj i ÝÑ Tr 1 s 1 U u 1... u r u 1... u s ÞÑ u j pu i q u 1... û i... u r u 1... û j... u s. By composing partial contractions one otains (various) total contractions T r r U ÝÑ R, e.g. C 1 1... C r r : u 1... u r ÞÑ ź i u i pu i q. We apply contractions to construct pairings. A natural non-degenerate ilinear pairing T s r U ˆ T r s U ÝÑ R is otained y composing the tensor product T s r U ˆ T r s U Ñ T r`s r`s U with a total contraction. The pairing induces a natural isomorphism pt s r Uq T r s U T s r U. In particular, homogeneous contravariant tensors can e interpreted as multilinear forms: Mult s puq pt s Uq T s U T s U To a tensor u 1... u s corresponds the multilinear form pu 1,..., u s q ÞÑ ź i u i pu i q. If te i u is a asis of U, then te i 1... e i s u is a asis of Mult s puq. An s-linear form µ on U can e written with respect to this asis as µ µpe looooooomooooooon i1,..., e is q e i 1... e i s. i 1,...,i s PR 13