What these examples have in common is that in each case, the product is a bilinear map. The center of a lie algebra g is the maximal subspace zg such that rz. The second is that you have lie algebra structures. In general, gl n is one more dimension than sl n, with extra basis element i. Paper ii uses the theory of coherent families to obtain a similar classification for uhfree modules over the symplectic lie algebras sp2nc.
The tensor product of lie soluble algebras springerlink. Tensor product of simple algebras need not be simple posted. Representation theory of lie algebras deferential geometry. Pdf tensor products and joint spectra for solvable lie. Graphical tensor product reduction scheme for the lie. It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as lie algebras or commutative algebras. Jan 29, 2011 by corollary 1, if are simple algebras and the center of or is then is simple. The condition that is the center of or cannot be omited, as the following examples show.
However, apart from the results established in 6 and 24, not much is known about the structure of closed lie ideals of tensor products of operator algebras. The problem as always in the tensorproduct business is therefore one of existence, or really of. One of the main problems, concerning the study of nonabelian tensorproductsofliealgebras,isrelatedtothesizeofthesocalleddiagonalidealsee the following section for details, because this gives a measure of how far is our lie algebra from being abelian. A c algebra is nuclear if for every c algebra, there is a unique cnorm on. Thus we have a tensor product r ono, b, the normal tensor product of r and b. In particular, a leibniz algebra becomes a lie object in lm and the universal enveloping. Pdf joint spectra of the tensor product representation of. The original reference for for the general tensor product rule is. There are a variety of norms one can impose on this tensor product to make a banach algebra. The tensor product is just another example of a product like this. However to do so, many nonobvious algebraic identities need to be verified, and various topological proofs which exist only in the group case have to be replaced by purely.
The tensor product of representations of lie algebras is bilinear up to isomor phism with respect to the direct sum of representations of lie. Graphical tensor product reduction scheme for the lie algebra so5. Relation to tensor product of categories of modules. The above theorems are a significant generalization of the work of gelbaum and natzitz in characterizing tensor products of group algebras, since here the algebra actions are arbitrary. Available formats pdf please select a format to send. Roughly speaking, given the nonabelian tensor product.
Representations of complex semisimple lie groups and lie algebras. Tensor product of irreducible representations of semisimple lie algebras hot network questions name of concept. Rep resentation theo ry of lie algeb ras clara l oh clara. Tomiyama, applications of fubini type theorem to the tensor product of calgebras, thoku math. The links to gt, ca, ag, and ags in the pdf file will work if the files are placed in the. The tensor product of two graded vector spaces aand bis again a graded vector. January 29, 2011 in noncommutative ring theory notes, simple rings tags. Vergne, poisson structure over the symmetric algebra of lie nilpotent algebra, bull. For su2 but not for the higherrank lie algebras np 1. Let be a prime number, and, where is the field of rational functions in over then is not simple.
Tensor product these keywords were added by machine and not by the authors. The tensor product of commutative algebras is of constant use in algebraic geometry. Rep resentation theo ry of lie algeb ras deferential geometry. Pdf on the nonabelian tensor product of 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 simpler girdle polynomials. Lie algebras are an essential tool in studying both algebraic groups and lie groups. Then each m3ma3 is a continuous extension of mim2 where migmai and m2e. Let a3be a locally convex algebra which is the completion of the tensor product, aia2, of two locally convex algebras in a topology not stronger than the inductive topology. We equip the category lm of linear maps of vector spaces with a tensor product which makes it suitable for various constructions related to leibniz algebras. Tomiyama, applications of fubini type theorem to the tensor product of c algebras, thoku math. Complexifications of real lie algebras and the tensor product decomposition of sl.
The centroid of a lie triple algebra liu, xiaohong and chen, liangyun, abstract and applied analysis, 20. A nonabelian tensor product of lie algebras glasgow. Pdf joint spectra of the tensor product representation. The first is that you are taking the tensor product of two vector spaces.
For affine schemes x, y, z with morphisms from x and z to y, so x spec a, y spec b, and z spec c for some commutative rings a, b, c, the fiber product scheme is the affine scheme corresponding to the tensor product of algebras. Hence in such a case, we would have, and thus denote the product c algebra by. Graphical tensor product reduction scheme for the lie algebra. Tensor product of simple algebras need not be simple. Chapter 22 tensor algebras, symmetric algebras and exterior. Tensor products of finite and infinite dimensional. Tensor products of operator algebras sciencedirect. The condition that is the center of or cannot be omited. Chapter i develops the basic theory of lie algebras, including the fundamental theorems of engel, lie, cartan, weyl, ado, and poincarebirkhoffwitt. M m n note that the three vector spaces involved arent necessarily the same. Definition of tensor product of algebras over commutative ring and existence theorem.
Sometimes you can deal with lie algebras as concrete matrix lie algebras, like we did above. Sternhaimer, deformations of poisson brackets, separate and joint analyticity in group representations, nonlinear group representations and physical application, in. Decomposition of the nonabelian tensor product of lie. Improvising their techniques, some work on the study of closed lie ideals of certain tensor products of c. For a a an associative algebra over a field k k, write a a mod for its category of modules of finite dimension. Resolving multiplicities in the tensor product of irreducible. Pdf nonabelian tensor product of lie algebras and its. Introduction to lie algebras and representation theory, graduate texts in mathematics, 9, springerverlag, new yorkberlin 1978. Borel tensor product algebras and quotient algebras.
Then the tensor product of algebras corresponds to the deligne tensor product of abelian categories ab. Ellis, a nonabelian tensor product of lie algebras, glasgow math. On closed lie ideals of certain tensor products of c. Knapp, basic algebra, digital second edition east setauket, ny. The quarterly journal of mathematics, volume 67, issue 2, june 2016, pages 303329. Chapter 22 tensor algebras, symmetric algebras and. On the tensor product of composition algebras sciencedirect. Lie algebras, algebraic groups, and lie groups james milne. Many other algebras of interest arise as twisted tensor product algebras. Lie algebras 3 since gl 2 is only one more dimension, choose the basis x. Complexifications of real lie algebras and the tensor product. By corollary 1, if are simple algebras and the center of or is then is simple. Suppose that g is the lie algebra of a lie group g. Sep 05, 2014 which generalizes the nonabelian tensor product of l ie algebras 6, and study its properties.
We also give a proof that uhfree modules do not exist for any other simple finitedimensional algebras which completes the. Tensor product lie algebras mathematics stack exchange. Tensor algebras, symmetric algebras and exterior algebras 22. A representation of an associative algebra aalso called a left amodule is a vector space v equipped with a homomorphism a endv, i. Lichnerowicz, 1differentiable cohomology of lie algebras associated with a symplectic category, c. The goal of this appendix is to prove proposition 5. G is a ktensor product of g,, g with respect to naturally induced actions of k on g. The class of nuclear c algebras includes all of the commutative ones, finite ones, and is itself closed under inductive products and quotients. For this purpose the weight diagram of an irreducible representation is placed in a 2dimensional \landscape of irreducible representations, centered at its tensor product partner. There exists a characterization of nbatalinvilkovisky algebras in terms of the product and the generator only 7, 14. On closed lie ideals of certain tensor products of calgebras ii. Feb 08, 2014 definition of tensor product of algebras over commutative ring and existence theorem.
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