A theory of quaternionic algebra, with applications to hypercomplex. A are closed under the quaternionic tensor product and lie. Is there a different construction of the tensor product. Let v and w be vector spaces over a eld k, and choose bases fe igfor v and ff jgfor w. Vector and tensor mathematics 25 atensorisdescribedassymmetricwhenttt. Examples where the tensor product occurs in physics are in quantum systems with. If v 1 and v 2 are any two vector spaces over a eld f, the tensor product is a bilinear map. How does this definition of the tensor product of hilbert spaces match up with the one we are familiar with when dealing with tensors in general relativity.
We have that qxq2 is the vector space direct sum of i and lv. If you have been confused by vectors, vector calculus, tensor analysis, or quaternions, this book is for you. The theory can be described using sheaves over cp1, an interpretation due to quillen q. B is the free rmodule generated by the cartesian product and g is the rmodule generated by the same relations as above. Remark that in this question im not interested in infinite tensor products defined in functional analysis or just colimits of finite ones. Zero tensor 0 has all its components zero in all coord systems. In this paper we completely classify the homogeneous twospheres, especially, the minimal homogeneous ones in the quaternionic projective space hpn. Pdf tensor product of quaternion hilbert modules researchgate. Tensor products rst arose for vector spaces, and this is the only setting where they occur in physics and engineering, so well describe tensor products of vector spaces rst.
Annihilationcreation operators are constructed, corresponding to the second quantization of the quaternion quantum theory with boseeinstein or fermidirac statistics. The rst two sections introduce 1 a hypermatrix, 2 a tensor as an element of a tensor product of vector spaces, its coordinate representation as a hypermatrix, and a tensor as a multilinear functional. The tensor product of vector spaces is just the special case of the tensor product of modules over some ring r r for the case that this ring happens to be a field. How to derive the expressions for gradient, divergence, laplacian, and. A dyad is a special tensor to be discussed later, which explains the name of this product. We allow k 0, in which case the tensor is just a real number. Indeed, quaternionic twosided modules permit the notion of tensor product. Under conditions that are necessary for the definition of a scalar product in the quaternionic hilbert modules. It is characterised as the vector space tsatisfying the following property.
Spanheimer ccbyndsa january 8, 2015 abstract introducing a special quaternionic vector calculus on the tangent bundle of a 4dimensional space, and by forcing a condition of holomorphism, a minkowskitype spacetime emerges. This volume contains frequent references to volume 1. This broad introduction to vector and tensor analysis is designed for the advanced undergraduate or graduate student in mathematics, physics, and engineering as well as for the practicing engineer or physicist who needs a theoretical understanding of these essential mathematical tools. A formal definition is given below there are many equivalent approaches. Roughly speaking this can be thought of as a multidimensional array. In this paper we prove a statement about the real and pseudoreal i. While we have seen that the computational molecules from chapter 1 can be written as tensor products, not all computational molecules can be written as tensor products. There are natural concepts of linear map and tensor product of ahmodules, which have many of the properties of linear maps and tensor products of vector spaces. How to prove any vector identity using cartesian tensors. How to use matlab to calculate dot and cross products, and solve linear equations.
The tensor product of two modules a and b over a commutative ring r is defined in exactly the same way as the tensor product of vector spaces over a field. In the construction of a tensor product of quaternion hilbert modules, given in a previous work real, complex, and quaternionic, inner products were defined in the vector spaces formed from the tensor product of quaternion algebras h modulo an appropriate left ideal in each case. In preparing this two volume work our intention is to present to engineering and science students a modern introduction to vectors and tensors. Quaternion algebras dartmouth mathematics dartmouth college. Ahmodules and their quaternionic tensor products are classi. A note on the reality of tensor products semantic scholar. We will start with twosided quaternion vector spaces and show in section 1 that the category of quaternion.
V is an elementary tensor if and only if cijci0j0 cij0ci0j for all i,i0,j,j0. A bilinear map is called a tensor product if it satisfies the equivalent condi. You can see that the spirit of the word tensor is there. Volume 1 is concerned with the algebra of vectors and tensors, while this volume is concerned with the geometrical. It is known that in the case of the compact lie groups there are unitary.
Inner product spaces, normed linear spaces, and bounded operators. Recall that a tensor product of v and v0 is a pait t,t consisting of a vector space t over f and a bilinear pairing t. To define it we start by defining for any set the free vector space over. The expression of its component is independent of the choice of the coordinate system. Rotations of moment of inertia tensor using quaternions mikica b kocic, 20120422, v0.
Towards a quaternionic spacetime tensor calculus preliminary draft carsten s. A physical quantity that can be completely described by a real number. A tensor product of xand yis a vector space zover k, together with a bilinear map. Relations between real, complex, and quaternionic vector spaces. The tensors of type one are naturally identi ed with the vector space v itself. The notation for each section carries on to the next. Is there a different construction of the tensor product of. V is a homomorphism of vector spaces, then there is a unique. Cartesian coords, then tensor is called a cartesian tensor. Im interested in the tensor product defined above which probably every mathematician regards as the wrong one.
This construction, together with the clebschgordan procedure, can be used to generate additional irreducible representations if one already knows a few. Introduction to the tensor product james c hateley in mathematics, a tensor refers to objects that have multiple indices. Contents tensor products if and are finite dimensional vector spaces then the cartesian product is naturally a vector space called the direct sum of and and denoted. It turns out that a unique tensor product for ordered vector spaces exists and is contained in the riesz space tensor product. Uniqueness of the scalar product in the tensor product of.
A short introduction to tensor products of vector spaces. Albert had examined the tensor product of two quaternion algebras. And then show that for finitedimensional spaces, you only need sums of boundedly many terms. We say that t satis es the characteristic property of the tensor product with respect to v and w if there is a bilinear map h. In my first encounter with the tensor product of modules in a course on representation theory by prof. It is also called kronecker product or direct product. Tensor products of archimedean partially ordered vector spaces. We note in particular that a is a vector with the same magnitude as a but p0inting in the direction opposite to a. A useful lemma about the tensor product is that it is unique, in the following sense. W is the complex vector space of states of the twoparticle system. Introduction to vectors and tensors vector and tensor analysis volume 2 ray m. W is called bilinear if for all v 1 2v 1 the map v 2 7.
Note that there are two pieces of data in a tensor product. The vector spaces seem to have been first introduced in. Tensor product of quaternionic vector spaces have been treated in the literature at various level, see e. Uxvx be a bilinear map from uxv to a vector space x. I have decided to put that into a separate question. Joyces theory develops an algebra of quaternionic vector spaces and their tensor products using the idea of an augmented hmodule, or ahmodule, which is an hmodule equipped with a special real subspace.
This means that the vector spaces must be seen as spaces carrying representations of the corresponding symmetry groups. There is a category of kvector spaces and operators between them. The next sections discuss the various generalizations of well. Real, complex, and covariant quaternion scalar products are found in the tensor product spaces. Because it is often denoted without a symbol between the two vectors, it is also referred to as the open product. Here are the main results about tensor products summarized in one theorem.
Figure 5 shows this vector, and as further examples of the multiplication of a vector by a scalar, the vectors. To volume 2 this is the second volume of a twovolume work on vectors and tensors. For algebra on vectors and tensors, an index must show up twice and only twice. Twistorial maps between paraquaternionic projective spaces. Introduction to vector and tensor analysis dover books on. Tensor products of vector spaces we can use the same process to define the tensor product of any two vector spaces. Tensor product of quaternion hilbert modules springerlink. While we do not discuss general differentiable manifolds, we do include a chapter on vector and tensor fields defined on hypersurfaces in a euclidean manifold. Quaternionic algebra described by sp1 representations.
The tensor product of two vector spaces is a new vector space with the property that bilinear maps out of the cartesian product of the two spaces are equivalently linear maps out of the tensor product the tensor product of vector spaces is just the special case of the tensor product of modules over some ring r r for the case that this ring happens to be a field. Ccsphys 191 hilbert spaces, tensor products 9203 fall 2003 lecture 3 hilbert spaces, tensor products, nqubits. A good starting point for discussion the tensor product is the notion of direct sums. A defining property of the tensor product is that there is a unique rlinear. The tensor product is just another example of a product like this. Packed with examples, including matlab examples, this book will show you. Quaternionic and clifford calculus for physicists pdf djvu. Fundamentals of tensor analysis dot product ab a b cos. If an index shows up once on the left hand side lhs of. Ps i also have a similar problem with walds definition of a direct sum of hilbert spaces. In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factorwise group action on the product. The product we want to form is called the tensor product and is denoted by v w. A basis for the tensor product is all products of basis elements in one space.
The vector space of tensors of type kis denoted tkv. The tensor product of vector spaces makes the category vect of all vector spaces into a monoidal category, in fact a distributive monoidal category. The extension of these results to maps between locally paraquaternionic grassmannian manifolds is discussed and partially proved, in line with m1 and its extension m2 to quaternionic tensor product structures. This section in particular opens the way to study a quaternionic. This lecture will formalize many of the notions introduced informally in the. Suppose that for every bilinear map f defined on uxv there is a unique linear map c defined on x such that fcb.
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