We have (in all characteristics) The image Alt(T(V)) is the alternating tensor algebra, denoted A(V). all tensors that can be expressed as the tensor product of a vector in V by itself). We thus take the two-sided ideal I in T(V) generated by all elements of the form v ⊗ v for v in V, and define Λ(V) as the quotient. 1 ( The reason is the following: given any exterior product of the form. {\displaystyle \{e_{1},\ldots ,e_{n}\}} It can also be interpreted as the vector consisting of the minors of the matrix with columns u and v. The triple product of u, v, and w is a signed scalar representing a geometric oriented volume. x where ti1⋅⋅⋅ir is completely antisymmetric in its indices. The exterior algebra was first introduced by Hermann Grassmann in 1844 under the blanket term of Ausdehnungslehre, or Theory of Extension. For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The word tensor comes from the Latin word tendere meaning "to stretch". K 0 ( . This approach is often used in differential geometry and is described in the next section. Simple Tensorflow implementation of "Unsupervised Image to Image Translation Networks" (NIPS 2017 Spotlight) - taki0112/UNIT-Tensorflow M1.2.1 Unit and alternating tensors The unit tensor, or Kronecker-G, is defined by G ij 1,i j and G ij 0,i z j. If u1, u2, ..., uk−1 are k − 1 elements of V∗, then define. Useful identities involving the Kronecker and alternating tensors 10/4/20 7. Tubal_Alt_Min (tubal-alternating minimization) is a tensor completion algorithm, based on the low-tubal-rank tensor model. x Q Similarly, if f is alternating, then we can deﬁne a skew-symmetric tensor power, n (E), and every alternating multilinear map is turned into a linear map, f ∧: n (E) → F,whichis equivalent to f in a strong sense. ( Given any unital associative K-algebra A and any K-linear map j : V → A such that j(v)j(v) = 0 for every v in V, then there exists precisely one unital algebra homomorphism f : Λ(V) → A such that j(v) = f(i(v)) for all v in V (here i is the natural inclusion of V in Λ(V), see above). As a consequence of this construction, the operation of assigning to a vector space V its exterior algebra Λ(V) is a functor from the category of vector spaces to the category of algebras. x Identities for Kronecker delta and alternating unit tensor. π If, furthermore, α can be expressed as an exterior product of k elements of V, then α is said to be decomposable. A tensor of order zero (zeroth-order tensor) is a scalar (simple number). Also, the Z-grading on the tensor and exterior algebras using the ac- tion by invertible scalars. The magnitude of the resulting k-blade is the volume of the k-dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors. ) The name orientation form comes from the fact that a choice of preferred top element determines an orientation of the whole exterior algebra, since it is tantamount to fixing an ordered basis of the vector space. i In other words, the exterior algebra has the following universal property:. The binomial coefficient produces the correct result, even for exceptional cases; in particular, Λk(V) = { 0 } for k > n . 10/4/20 11 . In applications to linear algebra, the exterior product provides an abstract algebraic manner for describing the determinant and the minors of a matrix. . In this case, one obtains. If K is a field of characteristic 0, then the exterior algebra of a vector space V can be canonically identified with the vector subspace of T(V) consisting of antisymmetric tensors. and The exterior algebra of differential forms, equipped with the exterior derivative, is a cochain complex whose cohomology is called the de Rham cohomology of the underlying manifold and plays a vital role in the algebraic topology of differentiable manifolds. In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras: Slightly more generally, if then any alternating tensor t ∈ Ar(V) ⊂ Tr(V) can be written in index notation as. The Jacobi identity holds if and only if ∂∂ = 0, and so this is a necessary and sufficient condition for an anticommutative nonassociative algebra L to be a Lie algebra.  The index subset must generally either be all covariant or all contravariant. Unless you have some preferred isomorphism between the tangent and cotangent spaces, like a metric. ⋆ It follows that the product is also anticommutative on elements of V, for supposing that x, y ∈ V, More generally, if σ is a permutation of the integers [1, ..., k], and x1, x2, ..., xk are elements of V, it follows that, where sgn(σ) is the signature of the permutation σ.. ⊗ Let W This distinction is developed in greater detail in the article on tensor algebras. Evert Jan Post, University of Houston Stan Sholar, The Boeing Company Hooman Rahimizadeh, Loyola Marymount University Michael Berg, Loyola Marymount University Follow. x The exterior product of multilinear forms defined above is dual to a coproduct defined on Λ(V), giving the structure of a coalgebra. n 2 … w v n I think the order has to be at least 2 for the definition to make sense because only then can we talk about permutation. = {\displaystyle Q(\mathbf {x} )} x {\displaystyle V} It follows that a b = b a. = The following example demonstrates the usefulness of this identity. in T(V) such that Abstract. It carries an associative graded product 0 The identity is used when two alternating tensors are present in a term, which usually arises when the term involves cross products. = The decomposable k-vectors have geometric interpretations: the bivector u ∧ v represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented parallelogram with sides u and v. Analogously, the 3-vector u ∧ v ∧ w represents the spanned 3-space weighted by the volume of the oriented parallelepiped with edges u, v, and w. Decomposable k-vectors in ΛkV correspond to weighted k-dimensional linear subspaces of V. In particular, the Grassmannian of k-dimensional subspaces of V, denoted Grk(V), can be naturally identified with an algebraic subvariety of the projective space P(ΛkV). {\displaystyle \mathbb {Z} _{2}} 1 and. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. where Observe that the coproduct preserves the grading of the algebra. A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. This is similar to the notion of tensor rank. The Alternating Unit Tensor (a) The alternating unit tensor is a rank-3 antisymmetric tensor deﬁned as follows: ijk = 1 if ijk = 123, 231, or 312 0 if any two indices are the same −1 if ijk = 132, 213, or 321 The alternating unit tensor is positive when the indices assume any clockwise cyclical progression, as shown in the ﬁgure: + 3 2 1 n β Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v, but, unlike the cross product, the exterior product is associative. holds when the tensor is antisymmetric with respect to its first three indices. , For vectors in a 3-dimensional oriented vector space with a bilinear scalar product, the exterior algebra is closely related to the cross product and triple product. 1 Tensor equal to the negative of any of its transpositions, Antisymmetric Tensor – mathworld.wolfram.com, https://en.wikipedia.org/w/index.php?title=Antisymmetric_tensor&oldid=953849792, Creative Commons Attribution-ShareAlike License, Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric), This page was last edited on 29 April 2020, at 10:58. In particular, the exterior derivative gives the exterior algebra of differential forms on a manifold the structure of a differential graded algebra. e′ i =Qe i, QQ = I. T, etc. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. The benefit of employing it is that once the epsilons are transformed into the deltas, then the substitution property of the Kronecker Deltas can be used to simplify the equation. {\displaystyle \mathbb {Z} } Any lingering doubt can be shaken by pondering the equalities (1 ⊗ v) ∧ (1 ⊗ w) = 1 ⊗ (v ∧ w) and (v ⊗ 1) ∧ (1 ⊗ w) = v ⊗ w, which follow from the definition of the coalgebra, as opposed to naive manipulations involving the tensor and wedge symbols. {\displaystyle v_{i}\in V.} k y {\displaystyle \alpha \in \wedge ^{k}(V^{*})} Some useful vector identities. {\displaystyle S(x)=(-1)^{\binom {{\text{deg}}\,x\,+1}{2}}x} The tensor product of two vectors represents a dyad, which is a linear vector transformation. x Equivalently, a differential form of degree k is a linear functional on the k-th exterior power of the tangent space. every vector vj can be written as a linear combination of the basis vectors ei; using the bilinearity of the exterior product, this can be expanded to a linear combination of exterior products of those basis vectors. 1 The fact that this may be positive or negative has the intuitive meaning that v and w may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. 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