Tensor product weight modules over the virasoro algebra. 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. The tensor product of highest weight modules with intermediate series modules over the virasoro algebra was discussed by zhang a class of representations over the virasoro algebra, j. An introduction to tensors for students of physics and. The third chapter extends tensor theory to spaces other than vector spaces, namely manifolds. A complete answer to this question was given by enochs and jenda in 1991. For this, we prove that the exterior tensor product of two permutation, endopermutation, endo. As usual, all modules are unital r modules over the ring r.
Let be a commutative ring, an module and a direct system of modules. Tensor product exact sequence commutative ring free module casimir. Tensor which depends on two directions one normal to the area and other that of the force on it. The decomposition of tensor products of two simple modules is nothing but the decomposition of its formal character a product of two weyl characters into the sum of weyl characters. Kolecki national aeronautics and space administration glenn research center cleveland, ohio 445 tensor analysis is the type of subject that can make even the best of students shudder. Most of the required ideas appear when we consider systems.
Notesontensorproducts rich schwartz may 3, 2014 1 modules. 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. Also assume that this order is odd and not divisible by 3 if typeg 2 occurs. Bounds on depth of tensor products of modules sciencedirect.
Decomposing tensor products of verma modules shashwat kishore mentor. We demonstrate how one can define a reduced tensor product on the familyf consisting of those finite dimensional simpleu k modules which are deformations of simplel and which have nonzero quantum dimension. The tensor product university of california, berkeley. R n,thetensor product of m and n, is an abelian group that is a z. This paper overcomes such a difficulty by exploiting the precise geometric and algebraic structure of tensor product of reproducing kernel hilbert modules.
Tensor product of lamodules now we would like to show that each or some properties of the usual tensor product hold in the new setting. As usual, all modules are unital rmodules over the ring r. Tensor products of free modules sage reference manual v9. Tensor flow basics typical modules to import for tensorflow. 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. In this paper, we determine the necessary and sufficient conditions for these tensor products to be simple. Let rbe a commutative ring with unit, and let m and n be r modules. This theory is being developed in a series of papers including hepth 9309076 and hepth 9309159. Browse other questions tagged commutativealgebra modules tensor products projective module free modules or ask your own question. 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. Tensor products of mand nover rare unique up to unique isomorphism. Let rbe a commutative ring with unit, and let m and n be rmodules. Tensor products of quantized tilting modules springerlink. The tensor product of an algebra and a module can be used for extension of scalars.
We have briefly discussed the tensor product in the setting of change of rings in sheaves, sections 6. In this way we are able to identify the right algebra maximal in some sense for endomorphisms to act. We describe a logarithmic tensor product theory for certain module categories for a conformal vertex algebra. Pureinjectivity of tensor products of modules algebra. Shyamashree upadhyay april 2015 project supervisor ii. This concept enabled him to define bimoduls, dual modules etc. Notes on tensor products and the exterior algebra for math 245 k. Hom functor and the associativity of tensor products of. Tensor products, trivial source modules and related algebras. It is this chapter that provides the foundations for tensor applications in physics. We consider whether the tilting properties of a tilting amodule t and a tilting bmodule t.
In this theory, which is a natural, although intricate, generalization of earlier work of huang and lepowsky, we do not. Tensor products and multipliers of lrctmodules by a. In this paper, we study some properties of the exterior tensor product on the category of modules. We will give the proof of this theorem at the end of this post. Pdf tensor products of modules and the rigidity of tor. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way. Motivated by these, grainger 2 defined formally left modules for left nearrings. We give a reduction formula for such multiplicities, and use this to prove that multiplicities of young. In mathematics, the tensor product of modules is a construction that allows arguments about. Differential graded algebra 7 mapsisastatementaboutrmodules. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. Tensor products of modules and the rigidity of tor article pdf available in mathematische annalen 2991.
A classical question of yoneda asks when the tensor product of two injective modules is injective. But there are examples of left nearrings acting on the left of the group. The tensor product of highest weight modules with intermediate series modules over the virasoro algebra was discussed by zhang z in 1997. Since then the irreducibility problem for the tensor products has been open. This tensor product can be generalized to the case when r r is not commutative, as long as a a is a right r r module and b b is a left r r module. In exactly the same way we define first the tensor product presheaf. The depth formula, for tensor products of finitely generated modules, is initially. We establish a set of generators and relations for the tensor product of two local weyl modules which are also level 1 demazure modules. In this paper, we obtain a class of irreducible virasoro modules by taking tensor products of the irreducible virasoro modules. It is easy to see that the hattoris r esult is also true when r is the pro duct of a.
In this paper, by using graingers work, the concept of tensor product of ring modules is generalised to the nearring. A tensor product of young modules decomposes as a direct sum of young modules, and in 14 hemmer, nakano and cohen have shown a strong connection between the tensor products of young modules for the symmetric group r and the decomposition matrices of the schur algebra sr. Gus lonergan fourth annual mit primes conference may 17, 2014. In each such case the tensor product is characterized by a similar universal property. Tensor product of quotient hilbert modules sciencedirect. The above isomorphism is an isomorphism of modules. In this section, we develop the tools needed to describe a system that contains more than one particle. Pdf a classical question due to yoneda is, when is the tensor product of any two injective modules injective. 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. Consider our action on this expansion we observe that dimv dimv. If k is a field, a and b kalgebras, m a faithful left a module, and n a faithful left bmodule, we recall the proof that the. An introduction to tensors for students of physics and engineering joseph c. More generally yet, if r r is a monoid in any monoidal category a ring being a monoid in ab with its tensor product, we can define the tensor product of a left and a right r r module in an. We also prove associativity of the tensor product of modules under certain necessary and sufficient condition.
We give a new, constructionfree proof of the associativity of tensor product for modules for rational vertex operator algebras under certain convergence. Freudenteal at the meeting of march 27, 1976 in 7 rieffel is concerned with a locally compact group g and the problem of flnding a concrete representation of. This is a very effective, representation free approach to dilation. In what follows we develop methods which link the tensor product of hilbert modules over c z 1, z n to hilbert modules over c z. In this paper the analogue question for pureinjective modules is studied. Tensor product of correspondence functors pdf free download. The tensor product of m and m is the correspondence functor m. Let mr be a right r module and rn be a leftr module. We introduce the main concepts and announce the main results in a theory of tensor products for module categories for a vertex operator algebra. Krullschmidt decomposition of some tensor products of.
We are led inevitably to the notion of tensor product systems of hilbert modules and units for. The following is an explicit construction of a module satisfying the properties of the tensor product. Tensor product of modules over a vertex algebra sciencedirect. This can be calculated by using, for example, the generalized littlewoodrichardson rule cf.
743 292 241 282 861 654 1153 49 1444 613 1449 1419 1431 593 334 1146 490 245 450 908 111 843 175 1355 757 970 988 851 905 782 906 1309 1200 865 581