Relation to tensor product of categories of modules. A package for computations in simple lie algebra representations. I have been selfstudying lie theory at a very basic level, primarily using stillwells lie theory and a bit of halls book. Tensor products of principal series representations reduction of tensor products of principal series representations of complex semisimple lie groups. One can extend the notion of tensor products to any finite number of representations. Classify irreducible representations of a given algebra a. Zolotykh, decomposition of tensor products of representations of lie algebra es.
As mentioned above, the algebra a is often given to us by generators and relations. Tensor product algebras, grassmannians and khovanov homology. The tensor algebra or free algebra of the vector space v over the eld f is the vector space. In some cases, such as su3, the decompositions are well known and there are several. Symmetry and particle physics university of surrey.
Representations of lie algebras, casimir operators and their applications libor snobl lectures presented at 5th student colloquium and school on mathematical physics, star a lesn a, august 29th september 4th, 2011. For example, the universal enveloping algebra u of the lie algebra sl2 is generated by. Unique decomposition of tensor products of irreducible. For the lie algebra of all sections, bounded unitary irreducible. We will attempt to derive the tools of such a system, but in a broader setting. Here we deal with a tensor product of representations of su2 with a specific value for the level at k 43 and a specific eigenvalue. This lie algebra is associated with, but not the same thing as, general linear group gl. These notes are an introduction to lie algebras, algebraic groups, and lie groups in characteristic zero, emphasizing the relationships between these objects visible in their categories of representations. Large deviation principle for moment map estimation. This permits one to define the tensor product of two representations of a lie algebra. Do 1 and 2 restricting to nite dimensional representations.
What i particularly want is a natural bijection from the basis in the tensor product to the product of the original bases, sending the. Lie algebras and their representations honours semester 2, 2014 as we have discussed, 30% of the mark for this course is assigned to a short project. Tensor product algebras, grassmannians and khovanov. Dec 27, 20 tensor product algebras, grassmannians and khovanov homology. A theorem is proved which indicates that completely irreducible representations enter into the expansion of the tensor product of a finite and elementary representation. In this context, might, for example, be the orbital angular momentum while is the spin angular momentum. The problem of decomposing the tensor product of principal series representations has been considered for sl2, c by g. Lie groups december 19, 2018 1 tensor operators a tensor operator is an operator that transforms in an irreducible representation of some lie algebra, in this section, we will concentrate on mathfraksu2 tensor operators, or \spherical tensors. We show that penroses work comes from the diagrammisation of the symmetric algebra. This category was introduced independently in 5 and 20. Rajan abstract we show that a tensor product of irreducible.
Representations of quantum algebras arising from noncompact quantum groups. Tensor product representations for orthosymplectic lie. Tensor product of representations of a lie algebra or lie superalgebra. Pdf tensor products, fusion rules and su2 representations. Tensor products of principal series representations. Is there a notion of tensor product of perfect bases of. In fact, gmod is more than just an abelian category. In preparing this two volume work our intention is to present to engineering and science students a modern introduction to vectors and tensors.
Tensor product decomposition and other algorithms for representations of large simple lie algebras. In section 4 we explain the idea behind coupling tensor products and describe the graphical tensor product reduction scheme. Ordered tensor categories and representations of the. 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.
Quantization of tensor representations and deformation of. Tensor product decomposition and other algorithms for. It is evident that to go any further, i have to learn somewhat about tensor products. Tensor product of representations of a lie algebra or lie. Resolving multiplicities in the tensor product of irreducible. A bialgebra equipped with an invertible antipode s is called a hopf algebra. This does not discuss the tensor product rule you asked for. Aare vector spaces with a product under which abecomes a associative ring with identity. In what sense are complex representations of a real lie algebra and complex representations of the complexified lie algebra equivalent. Does anyone know of a way of fixing this, and getting in a canonical perfect basis on the tensor product from perfect bases on the factors. Representations of lie algebras, casimir operators and their.
Jan 29, 2011 tensor product of simple algebras need not be simple. We shall occasionally suppress the tensor product sign, i. Graphical tensor product reduction scheme for the lie. Tensor products, fusion rules and su2 representations. Theres another way to build new representations via the tensor product. 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. Unique decomposition of tensor products of irreducible representations of simple algebraic groups by c. We want to study these series in a ring where convergence makes sense. In particular, the tensor product of two representations is a representation, as is the dual vector space. On the tensor product of representations of semisimple lie. The last topic is the determination of which representations can be brought into orthogonal or symplectic form. Tensor product of simple algebras need not be simple. The original reference for for the general tensor product rule is.
This was done in the frame of finite dimensional lie algebra and affine lie algebra and not in the case of fusion rules as in the. In this theory, one considers representations of the group algebra a cg of a. This lie algebra is a quite fundamental object, that crops up at. Zolotykh, a package for computations in simple lie algebra representations. Abstract tensor systems and diagrammatic representations j anis lazovskis september 28, 2012. Our intent in this note is to sketch out a new construction of khovanov homology which can be generalized to other representations of other lie algebras. I have made minor inroads into notes by elkies and gowers. Tensor product these keywords were added by machine and not by the authors. In the context of representations of the lie algebra su2, the tensor product of representations goes under the name addition of angular momentum. Thoughout, well use the name of a lie algebra, usually sl 2, to also refer to other constructions based on its cartan matrix, like quantum groups. Then the tensor product of algebras corresponds to the deligne tensor product of abelian categories ab. The basic relationship that these representations are cornmutants of each other is maintained during the quantization. When the ring is a field, the most common application of such products is to describe the product of algebra representations.
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. By corollary 1, if are simple algebras and the center of or is then is simple. Representations of quantum algebras arising from non. Pdf tensor product algebras, grassmannians and khovanov.
Journal of pure and applied algebra elsevier journal of pure and applied algebra 1998 18 tensor product representations for orthosymplectic lie superalgebras georgia benkart11, chanyoung lee shader12, arun ranr3 a department of mathematics, university of wisconsin, madison, wl 53706, usa b department of mathematics, university of wyoming, laramie, wy 82071, usa c. Section 5 is all about the graphical tensor product reduction. Journal of pure and applied algebra elsevier journal of pure and applied algebra 1998 148 tensor product representations for orthosymplectic lie superalgebras georgia benkart,, chanyoung lee shaderb, arun ramc,3 department of mathematics, university of wisconsin, madison, wi 53706, usa b department of mathematics, university of wyoming, laramie. Representations of lie algebras, casimir operators and. Eventually these notes will consist of three chapters, each about 100 pages long, and a short appendix. I meant the following if a is a semisimple algebra, say finite dimensional over. We can define their algebra tensor product as the standard tensor product of algebras with product and involution there are a variety of norms one can impose on this tensor product to make a banach algebra.
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. The tensor product of principal series representations of a complex semisimple lie group. Ordered tensor categories and representations of the mackey. Lie algebra representations and their extensions to knot theory are also discussed. The condition that is the center of or cannot be omited, as the. Hopf algebras provide a way to improve the representation theory of associative algebras, while retaining the representation theory of groups and lie algebras as special cases. Abstract tensor systems and diagrammatic representations. In mathematics, the tensor product of two algebras over a commutative ring r is also an ralgebra. Categorification of tensor product representations of slk. A rst example of such a category is t g, the category of tensor modules over a nitary simple lie algebra g. Representations of complex semisimple lie groups and lie algebras. Journal of pure and applied algebra elsevier journal of pure and applied algebra 1998 18 tensor product representations for orthosymplectic lie superalgebras georgia benkart11, chanyoung lee shader12, arun ranr3 a department of mathematics, university of wisconsin, madison, wl 53706, usa b department of mathematics, university of wyoming, laramie, wy 82071, usa c department of. It is shown that the decomposition of the tensor product of two representations from the principal series of g consists of two pieces, tc.
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