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