The original reference for for the general tensor product rule is. We now introduce an algebraic structure on observables. We construct the tensor product of two oneparticle representations of this algebra and study the reduction problem. A new series of representations having nonlinear mass formulas is found. Passing to the lie algebra 5 a primer on the lie group lie algebra correspondence 8. When describing physical systems with lie algebras it is vital to reduce tensor products of irreducible representations into sums of such representations. Representations of complex semisimple lie groups and lie algebras. Two other recommendable texts which only discuss lie algebras are the books \introduction to lie algebras and representation theory by j.
The nite dimensional representations can be broken in smaller representations for a semisimple lie algebra. A representation of an associative algebra aalso called a left amodule is a vector space. Tensor product decomposition in lie algebra representation. Such an algebra has been very fruitful in matrix models since it allows one to uncover computable sectors. Pdf tensor products, fusion rules and su2 representations. This chapter presents some applications of semi tensor product to differential geometry and algebra. There are several good books on lie algebras and representation theory available. Is it true that if is a finite dimensional algebra, and a separable extension, then is semisimple. We show that penroses work comes from the diagrammisation of the symmetric algebra. If k k is a field and \mathcalc svect is the category of super vector spaces over k k, then a lie algebra object is a super lie algebra. The proof of the tensor product theorem well prove the tensor product theorem, which asserts that if f is a global eld, aits adele ring, vthe places of f, and g is a redactive algebraic group over fthen every irreducible admissible representation of ga decomposes into a restricted tensor product of representations of the groups gfv.
Anyway, the general formula for the decomposition of the tensor product of two simple modules as a direct sum of irreducible ones is well known. Each lie algebra has a single root system, but many lie algebras can share the same root system. The crystal of a representation is a colored directed graph associated to that representation. What is represented, as what is it represented, and how is it represented. In the context of representations of the lie algebra su2, the tensor product of representations goes under the name addition of angular momentum. The analogous observation for lie superalgebras gives the notion of a poisson superalgebra. Complexification of a lie algebra encyclopedia of mathematics. Ellis, a nonabelian tensor product of lie algebras, glasgow math. A good knowledge of linear algebra including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector. This chapter presents some applications of semitensor product to differential geometry and algebra. By a hecke algebra we will usually mean an iwahori hecke algebra. Poisson structure over the symmetric algebra of lie nilpotent algebra, bull. Categorification of tensor product representations of slk.
This book is designed to introduce the reader to the theory of semisimple lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. For su2 but not for the higherrank lie algebras np 1. Tensor product algebras, grassmannians and khovanov. Section 5 is all about the graphical tensor product reduction. Lie algebras 3 since gl 2 is only one more dimension, choose the basis x. One approach to the general problem of decomposing a tensor product of irreducible finitedimensional representations hence any finitedimensional representations into irreducibles is to use the theory of crystals. Tensor products and duals of representations of lie algebras 37 x2. T v \displaystyle t v is an algebraic representation of g. Finally chapter 4 examines a method that is basis independent and applies to the tensor product of nite dimensional irreducible representations of any semisimple lie algebra over c. The goal of this appendix is to prove proposition 5. If is a nondegenerate representation, then there are unique nondegenerate representations and such that.
Graphical tensor product reduction scheme for the lie algebra so5. In section 4 we explain the idea behind coupling tensor products and describe the graphical tensor product reduction scheme. If a vector space is both an associative algebra and a lie algebra and the adjoint representation of the lie algebra on itself is a representation on an algebra i. Lie algebra representations and their extensions to knot theory are also discussed. We will attempt to derive the tools of such a system, but in a broader setting. The completion of with this norm is a c algebra called the minimal or spatial tensor product of and with respect to and, and is denoted. For a a an associative algebra over a field k k, write a a mod for its category of modules of finite dimension.
The condition that is the center of or cannot be omited, as the. Jul 09, 2011 this is clearly finite hence a norm and satisfies the caxiom. Chapter i develops the basic theory of lie algebras, including the fundamental theorems of engel, lie, cartan, weyl, ado, and poincarebirkhoffwitt. We first formulate a definition of tensor product for two modules for a vertex operator algebra in terms of a certain universal property and then we give a. Pdf on the nonabelian tensor product of lie algebras. An introduction to semitensor product of matrices and its.
The special linear lie algebra sln,r consists of all n. We note that many authors use the term hopf algebra for any bialgebra with an antipode. Note that a lie algebra is not an algebra \lie is not an adjective, as. In case of the adjoint representation, we have both with the same name. One can extend the notion of tensor products to any finite number of representations. In general, gl n is one more dimension than sl n, with extra basis element i. Lie algebras are an essential tool in studying both algebraic groups and lie groups.
This is clearly finite hence a norm and satisfies the caxiom. We want to study these series in a ring where convergence makes sense. Interestingly, there is a more e cient way to determine the tensor product reduction, in which the characteristic laurent polynomials are replaced by ratios of. In preparing this two volume work our intention is to present to engineering and science students a modern introduction to vectors and tensors. When the ring is a field, the most common application of such products is to describe the product of algebra representations. Graphical tensor product reduction scheme for the lie. They also showed that for each nite tensor product of nite dimensional irreducible representations of a complex semisimple lie algebra such a categori cation exists, using websters diagram algebras, web. Tensor products of finite and infinite dimensional representations of. In many mathematical problems we need to deal with multipledimensional data, or data labeled by multiindex. Resolving multiplicities in the tensor product of irreducible. Rep resentation theo ry of lie algeb ras deferential geometry.
Jan 25, 2019 a representation consists actually of three parts. Representation theory and tensor product theory for vertex. The lie algebra su2 describes rotations in 3dimensional coordinate space as well as in the isospin space of nuclear and particle physics. On the tensor product of two composition algebras patrick j. Chapter 2 is devoted to the basics of representation theory. The center of a lie algebra g is the maximal subspace zg such that rz. In this case stp could be a proper tool for formula deduction or numerical calculation. Let v be a nonzero nite dimensional representation of a semisimple lie. Then the tensor product of algebras corresponds to the deligne tensor product of abelian categories ab.
For this purpose the weight diagram of an irreducible representation is placed in a 2dimensional. There is a purely combinatorial algorithm for producing the tensor product of two crystals. In technical terms, representation theory studies representations of associative algebras. Decomposing tensor product of lie algebra representations. Abstract the basic theory of semisimple lie algebras. Tensor product algebras, grassmannians and khovanov homology this construction is a special case of that given in webb. The central algebraic fact is found to be a representation of the w tensor product z a of an abelian w algebra z lx t, p with. Zolotykh, a package for computations in simple lie algebra representations. Thus iwahori hecke algebras are involved in many diverse problems. If v is a linear representation of a group g, then with the above linear action, the tensor algebra is an algebraic representation of g. Representation type of certain tensor product algebras. Zolotykh, decomposition of tensor products of representations of lie algebra es. Several times during the preparation of this book we taught a one semester course to students with a very limited background in linear algebra and no background in tensor analysis.
By corollary 1, if are simple algebras and the center of or is then is simple. Tensor product decomposition and other algorithms for. The recently proposed infinitedimensional lie algebra as a model of a symmetry scheme is studied from the point of view of its representations. Abstract tensor systems and diagrammatic representations j anis lazovskis september 28, 2012 abstract the diagrammatic tensor calculus used by roger penrose most notably in 7 is introduced without a solid mathematical grounding. A theorem is proved which indicates that completely irreducible representations enter into the expansion of the tensor product of a finite and elementary representation. On the tensor product of representations of semisimple lie. A theorem is proved which indicates that completely irreducible representations enter into the expansion of the tensor product of a finite. Tensor product of simple algebras need not be simple. Let be a representation and be the largest subspace of such that for all and. Tensor product decomposition in lie algebra representation theory nilsson, jonathan in master thesis in mathematical science matm01 20111 mathematics faculty of sciences mark. In mathematics, the tensor product of two algebras over a commutative ring r is also an r algebra. Sometimes you can deal with lie algebras as concrete matrix lie algebras, like we did above.
Relation to tensor product of categories of modules. Humphreys book introduction to lie algebras and representation theory springer gtm series. Graphical tensor product reduction scheme for the lie algebras so. Abstract tensor systems and diagrammatic representations. Basic definitions a lie algebra is a vector space g over fwith a bracket r. In this master thesis i have looked on two different kinds of representations of the lie algebras su2 and sl2, and the tensor products of the representations. If v is a linear representation of a group g, then with the above linear action, the tensor algebra. The problem of the decomposition of the tensor product of finite and infinite representations of a complex semigroup of a lie group is examined by using the theory of characters of completely irreducible representations. Introduction to lie algebras and representation theory. A bialgebra equipped with an invertible antipode s is called a hopf algebra. The completion of with this norm is a calgebra called the minimal or spatial tensor product of and with respect to and, and is denoted.
On the nonabelian tensor product of lie algebras article pdf available in linear and multilinear algebra 583. Lecture notes introduction to representation theory. A dyad is a special tensor to be discussed later, which explains the name of this product. The lie algebra su3 describes the extension of isospin to the. Let tm be the tensor algebra generated by the module m, and let kbe the ideal in tm generated by all z. Then the connected components of the crystal graph correspond to the irreducible representations youre looking for. This does not discuss the tensor product rule you asked for. Tensor products, fusion rules and su2 representations. Because it is often denoted without a symbol between the two vectors, it is also referred to as the open product. Thus it makes a big difference whether we talk about a representation of a lie algebra or a representation on a lie algebra.
Tensor product these keywords were added by machine and not by the authors. Pdf in this master thesis i have looked on two different kinds of representations of the lie algebras su2 and sl2, and the tensor products of the. In mathematics, the tensor product of two algebras over a commutative ring r is also an ralgebra. Abstract tensor systems and diagrammatic representations j anis lazovskis september 28, 2012.
The tensor product of oneparticle representations of an. However, it is likely that teachers will wish to generate additional exercises. Notations from commutative algebra are as in my primer. But arbitrary representations of the tensor product of algebras cannot be broken into pieces. In this context, might, for example, be the orbital angular momentum while is the spin angular momentum. Complexifications of real lie algebras and the tensor product. Tensor product decomposition and other algorithms for representations of large simple lie algebras. The tensor product of two vectors represents a dyad, which is a linear vector transformation.
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