Intuition given a set, we wish to give meaning to and make the collection of such objects an module. In this section, we develop the tools needed to describe a system that contains more than one particle. Tensor products of modules the construction of the concept of a tensor product of two modules yields an additive abelian group that is unique up to isomorphism. The most classical versions are for vector spaces modules over a field, more generally modules over a ring, and even more generally algebras over a commutative monad.
By f we mean the category of gmodules admitting a ltration, and, dually, by fr we mean the category of gmodules admitting a r ltration. This tensor product can be generalized to the case when is not commutative. Let mr be a right r module and rn be a leftr module. Wiegand on the tensor product of modules and the rigidity of tor. Tensors on free modules a tutorial this worksheet provides some introduction to tensors on free modules of. Kronecker product, the tensor product of matrices or vectors, which satisfies all the properties for vector spaces and allows a concrete representation. Shyamashree upadhyay april 2015 project supervisor ii. What links here related changes upload file special pages permanent link page information wikidata item cite this page. N, 1 and sis the set of all formal sums of the following type.
We give a reduction formula for such multiplicities, and use this to prove that multiplicities of young. The tensor product, as defined, is an abelian group, not an rmodule. Gill department of algebra charles university bmc, april 2012 christopher c. In exactly the same way we define first the tensor product presheaf. Let m be an rmodule, n a left rmodule and g an additive abelian group. The depth of tensor product of modules over a gorenstein local ring is studied.
Introduction in this article we study the complexity of tensor products of modules over a. We have briefly discussed the tensor product in the setting of change of rings in sheaves, sections 6. Tensor products of modules islamic university of gaza. Endomorphism algebras of tensor powers of modules for. A some basic rules of tensor calculus the tensor calculus is a powerful tool for the description of the fundamentals in continuum mechanics and the derivation of the governing equations for applied problems. Arbitrary connections on the individual abimodules induce a connection on the tensor product abimodule. Noncommutative connections on bimodules and drinfeld. Torsion properties of tensor products of modules over local rings were. The tensor product university of california, berkeley.
Pdf tensor products of modules and the rigidity of tor. For algebras over a eld, mathematicians have done a lot of work in this direction see page 229 in 5, theorem 7. The term tensor product has many different but closely related meanings in its original sense a tensor product is a representing object for a suitable sort of bilinear map and multilinear map. Note that there are two pieces of data in a tensor product. In particular, lehrer and zhang proved that in the generic case there exists a surjection from the group algebra of the braid group to the endomorphism algebra of a tensor power of any strongly multiplicity free module. The tensor product of two rmodules is built out of the examples given above. In general, there are two possibilities for the representation of the tensors and. Pdf how to define tensor products of hypermodules researchgate. One of the applications is simpli cation of the proofs of central results in a paper of c. Tensor products of modules and the rigidity of tor article pdf available in mathematische annalen 2991. The product we want to form is called the tensor product and is denoted by v w. Letu k denote the quantized enveloping algebra corresponding to a finite dimensional simple complex lie algebral. Consider our action on this expansion we observe that dimv dimv. To do this, we begin by computing the weil modules for sp43, sp63 and sp83 in order to understand how tensor products decompose for these cases.
If the ring happens to be a field, then modules are vector spaces and the tensor product of modules becomes the tensor product of vector spaces. In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. Specifically this post covers the construction of the tensor product between two modules over a ring. We give a new, constructionfree proof of the associativity of tensor product for modules for rational vertex operator algebras under certain convergence. The third chapter extends tensor theory to spaces other than vector spaces, namely manifolds. Projective modules and tensor products mathoverflow. More generally, the tensor product can be extended to other categories of mathematical objects in addition to vector spaces, such as to matrices, tensors, algebras, topological vector spaces, and modules. The usual banach module projective tensor product of central representable bimodules over an abelian c. Let r be a commutative ring with identity and m an rmodule. The assumption that the expression rmodule means right rmodule. In particular, finitely generated faithful multiplication modules are projective. This leads us to some results and hypotheses for the general case. Tensor products of young modules and pmultiples of partitions. In each such case the tensor product is characterized by a similar universal property.
R n,thetensor product of m and n, is an abelian group that is a z. In fact, they showed that if the tensor product of two injective rmodules is injective, then the injective envelope er of r is at. Heres a simple example where such a question might arise. Tensor product of vector spaces, an operation on vector spaces the original tensor product. In mathematics, the tensor product of modules is a construction that allows arguments about. Applications of a non abelian tensor product of groups. Multilinear algebra international winter school on gravity and light 2015 duration. 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. A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems. The aim of this work is to understand the decomposition, in irreducible modules, of tensor products of weil modules for sp2n3. Tensor products of modules the construction of the concept of a. Sections 46 concern the hom and tensor product functors. But before jumping in, i think nows a good time to ask, what are tensor products good for. In general, it is impossible to put an rmodule structure on the tensor product.
Hom functor and the associativity of tensor products of. Here are the main results about tensor products summarized in one theorem. Let rbe a commutative ring with unit, and let m and n be rmodules. The following is an explicit construction of a module satisfying the properties of the tensor product.
The tensor product is the coequalizer of the two maps given by the action of on and on. As usual, all modules are unital rmodules over the ring r. Definition of the tensor product throughout this note, awill denote a commutative ring. Tensor product of modules metadata this file contains additional information such as exif metadata which may have been added by the digital camera, scanner, or software program used to. The tensor product of three modules defined by the universal property of trilinear maps is isomorphic to both of these iterated tensor products. Therefore we introduce in section 4 a new tensor product x n. Most of the required ideas appear when we consider systems. Finally, a complete answer to the question of yoneda was given by enochs and jenda in 1991 by showing the converse of ishikawas result see 3, theorem 2. This is a pure algebraic subpart of sagemanifolds version 1.
On tensor products of operator modules introduction. Let rbe a commutative ring with unit, and let m and n be r modules. It is this chapter that provides the foundations for tensor applications in physics. The weheraeus international winter school on gravity and light 74,268 views.
Tensor product of modules, the same operation slightly generalized to modules over arbitrary rings. Gill prague tensor products of young modules bmc, april 2012 1 14. We note that t is the unique indecomposable module of. In fact, they showed that if the tensor product of two injective rmodules is injective, then the injective envelope er of r. Tensor products of quantized tilting modules springerlink. The construction of the concept of a tensor product of two modules yields an additive abelian group that is unique up to isomorphism. Notesontensorproducts rich schwartz may 3, 2014 1 modules. This note wants to be a proposal for defining the tensor products of hypermodules over a general hyperring. Notes on tensor products and the exterior algebra for math 245 k. If k is a field, a and b kalgebras, m a faithful left amodule, and n a faithful left bmodule, we recall the proof that the.