Pro-reductive groups attached to irreducible representations of the General Linear Supergroup

The decomposition into irreducible modules is determined, for the tensor product of two arbitrary irreducible integrable highest weight modules, for an (untwisted) affine Kac-Moody algebra L. The result is applied to investigate full multiplicity regions and cyclic modules within the tensor product space for an affine Kac-Moody algebra. The tensor product decomposition into irreducible modules over L ⊕ Vir (Coset construction), Vir the Virasoro algebra, is also briefly investigated.

One of the central problems in the representation theory of compact groups concerns multiplicity, wherein an irreducible representation occurs more than once in the decomposition of the n-fold tensor product of irreducible representations. The problem is that there are no operators arising from the group itself whose eigenvalues can be used to label the equivalent representations occurring in the decomposition.

OF THE DISSERTATION Some results in the representation theory of strongly graded vertex algebras By JINWEI YANG Dissertation Director: Yi-Zhi Huang and James Lepowsky In the first part of this thesis, we study strongly graded vertex algebras and their strongly graded modules, which are conformal vertex algebras and their modules with a second, compatible grading by an abelian group satisfying certain grading restriction conditions. We consider a tensor product of strongly graded vertex algebras and its tensor product strongly graded modules. We prove that a tensor product of strongly graded irreducible modules for a tensor product of strongly graded vertex algebras is irreducible, and that such irreducible modules, up to equivalence, exhaust certain naturally defined strongly graded irreducible modules for a tensor product of strongly graded vertex algebras. We also prove that certain naturally defined strongly graded modules for the tensor product strongly graded vertex algebra are completely reducible if and only if every strongly graded module for each of the tensor product factors is completely reducible. These results generalize the corresponding known results for vertex operator algebras and their modules. In the second part, we derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded conformal vertex algebra under

CleGo: A package for automated computation of Clebsch–Gordan coefficients in tensor product representations for Lie algebras [formula omitted

Let $GL_M$ be general linear Lie group over the complex field. The irreducible rational representations of the group $GL_M$ are labeled by pairs of partitions $\mu$ and $\tilde\mu$ such that the total number of non-zero parts of $\mu$ and $\tilde{\mu}$ does not exceed $M$. Let $U$ be the representation of $GL_M$ corresponding to such a pair. Regard the direct product $GL_N\times GL_M$ as a subgroup of $GL_{N+M}$. Let $V$ be the irreducible rational representation of the group $GL_{N+M}$ corresponding to a pair of partitions $\lambda$ and $\tilde{\lambda}$. Consider the vector space $W=Hom_{G_M}(U,V)$. It comes with a natural action of the group $GL_N$. Let $n$ be sum of parts of $\lambda$ less the sum of parts of $\mu$. Let $\tilde{n}$ be sum of parts of $\tilde{\lambda}$ less the sum of parts of $\tilde{\mu}$. For any choice of two standard Young tableaux of skew shapes $\lambda/\mu$ and $\tilde{\lambda}/\tilde{\mu}$ respectively, we realize $W$ as a subspace in the tensor product of $n$ copies of the defining $N$-dimensional representation of $GL_N$, and of $\tilde{n}$ copies of the contragredient representation. This subspace is determined as the image of a certain linear operator $F$ in the tensor product, given by explicit multiplicative formula. When M=0 and $W=V$ is an irreducible representation of $GL_N$, we recover the classical realization of $V$ as a subspace in the space of all traceless tensors. Then the operator $F$ can be regarded as the rational analogue of the Young symmetrizer, corresponding to the chosen standard tableau of shape $\lambda$. Even in the special case M=0, our formula for the operator $F$ is new. Our results are applications of representation theory of the Yangian of the Lie algebra $gl_N$.

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.

In their article [9] on cyclic homology, Feigin and Tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra, generalizing Burghelea’s calculation [4] of the cyclic homology of a group algebra. For an analogous spectral sequence for the Hochschild homology of a crossed product algebra, see Brylinski [2], [3]. In this article, we give a new derivation of this spectral sequence, and generalize it to negative and periodic cyclic homology HC• (A) and HP•(A). The method of proof is itself of interest, since it involves a natural generalization of the notion of a cyclic module, in which the condition that the morphism τ ∈ Λ(n,n) is cyclic of order n + 1 is relaxed to the condition that it be invertible. We call this category the paracyclic category. Given a paracyclic module P , we can define a chain complex C(P ), with differentials b and B, which respectively lower and raise degree. The condition that the module P is paracyclic translates to the condition on C(P ) that 1− (bB +Bb) is invertible. Our main result is to show that there is an analogue of the Eilenberg-Zilber theorem for bi-paracyclic modules. It is then easy to obtain a new expression for the cyclic homology of a crossed product algebra which leads immediately to the spectral sequence of Feigin and Tsygan. If M is a module over a commutative ring k, we will denote by M (k) the iterated tensor product, defined by M (0) = k and M (k+1) =M (k) ⊗M . If M and N are graded modules, we will denote by M ⊗N their graded tensor product. We would like to thank J. Block, J.-L. Brylinski and C. Kassel for a number of interesting discussions on the results presented here.

LieART—A Mathematica application for Lie algebras and representation theory

LieART 2.0 – A Mathematica application for Lie Algebras and Representation Theory

Tensor products of natural modules and their centralizers are well-studied in the literature. This thesis extends classical results to tensor products of irreducible Weyl modules over a quantum group. We show that their centralizer algebras are cellular in the generic case, as pointed out in the literature, as well as under specializations. Their cellular basis, the dual canonical centralizer basis, is constructed directly from Lusztig's canonical basis of the tensor space and is given explicitly. It turns out that the cells can be derived directly from the Weyl filtration. We also describe the projective and simple modules of these centralizer algebras and describe the tensor space as module of the centralizer algebra. It is furthermore shown that the decomposition matrix of cell modules can be described in terms of Weyl filtrations of indecomposable tilting modules. The results obtained cover classical results where, for example, the centralizer is a quotient of the Hecke algebra. The latter has the Kazhdan-Lusztig basis as cellular basis. It turns out that the dual canonical centralizer basis coincides up to lower cell ideals with the Kazdan-Lusztig basis. Therefore, the dual canonical centralizer basis is a generalization of the Kazhdan-Lusztig basis to a much wider class of algebras. Tensorprodukte naturlicher Moduln und ihre Zentralisatoren sind in der Literatur gut untersucht. Diese Arbeit erweitert klassische Ergebnisse auf Tensorprodukte irreduzibler Weylmoduln fur eine Quantengruppe. Es wird gezeigt, dass diese Zentralisatoralgebren zellular sind sowohl im generischen Fall, wie bereits in der Literatur angemerkt, als auch unter Spezialisierung. Ihre zellulare Basis, die duale kanonische Zentralisatorbasis, wird direkt aus Lusztigs kanonischer Basis auf dem Tensorraum konstruiert und explizit angegeben. Es werden die projektiven und einfachen Moduln dieser Zentralisatoralgebren beschrieben ebenso wie die Struktur des Tensorraums als Modul des Zentralisators. Es wird weiterhin gezeigt, dass die Zerlegungsmatrix der Zellmoduln duch die Weylfiltrierung unzerlegbarer Kippmoduln beschrieben werden kann. Die erzielten Ergebnisse decken auch klassische Falle ab, wo z.B. der Zentralisator Quotient der Hecke Algebra ist. Letztere hat u.a. die Kazhdan-Lusztig-Basis als zellulare Basis. Es stellt sich heraus, dass die duale kanonische Zentralisatorbasis bis auf tieferliegende Zellideale mit den Kazhdan-Lusztig-Basis ubereinstimmt. Damit ist die duale kanonische Zentralisatorbasis eine Verallgemeinerung der Kazhdan-Lusztig-Basis auf eine wesentlich grosere Klasse von Zentralisatoralgebren.

Thick ideals in Deligne's category [formula omitted

We study a mixed tensor product 3⊗m⊗3‾⊗n of the three-dimensional fundamental representations of the Hopf algebra Uqsℓ(2|1), whenever q is not a root of unity. Formulas for the decomposition of tensor products of any simple and projective Uqsℓ(2|1)-module with the generating modules 3 and 3‾ are obtained. The centralizer of Uqsℓ(2|1) on the mixed tensor product is calculated. It is shown to be the quotient Xm,n of the quantum walled Brauer algebra qwBm,n. The structure of projective modules over Xm,n is written down explicitly. It is known that the walled Brauer algebras form an infinite tower. We have calculated the corresponding restriction functors on simple and projective modules over Xm,n. This result forms a crucial step in decomposition of the mixed tensor product as a bimodule over Xm,n⊠Uqsℓ(2|1). We give an explicit bimodule structure for all m,n.

A full variational calculation based on a tensor product decomposition

We give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.

When studying quantum field theories and lattice models, it is often useful to analytically continue the number of field or spin components from an integer to a real number. In spite of this, the precise meaning of such analytic continuations has never been fully clarified, and in particular the symmetry of these theories is obscure. We clarify these issues using Deligne categories and their associated Brauer algebras, and show that these provide logically satisfactory answers to these questions. Simple objects of the Deligne category generalize the notion of an irreducible representations, avoiding the need for such mathematically nonsensical notions as vector spaces of non-integer dimension. We develop a systematic theory of categorical symmetries, applying it in both perturbative and non- perturbative contexts. A partial list of our results is: categorical symmetries are preserved under RG flows; continuous categorical symmetries come equipped with conserved currents; CFTs with categorical symmetries are necessarily non-unitary.