Direct Sum Of Tensor Products Research Articles

Modular Representation Theory of Affine and Cyclotomic Yokonuma-Hecke Algebras

We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type A (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic p. Secondly, the modular branching rules for these algebras are obtained; moreover, the resulting modular branching graphs for cyclotomic Yokonuma-Hecke algebras are identified with crystal graphs of irreducible integrable representations of affine Lie algebras of type A.

On the reducibility of induced representations for classical p-adic groups and related affine Hecke algebras

Let π be an irreducible smooth complex representation of a general linear p-adic group and let σ be an irreducible complex supercuspidal representation of a classical p-adic group of a given type, so that π ⨁ σ is a representation of a standard Levi subgroup of a p-adic classical group of higher rank. We show that the reducibility of the representation of the appropriate p-adic classical group obtained by (normalized) parabolic induction from π ⨁ σ does not depend on σ, if σ is “separated” from the supercuspidal support of π. (Here, “separated” means that, for each factor ρ of a representation in the supercuspidal support of π, the representation parabolically induced from ρ ⨁ σ is irreducible.) This was conjectured by E. Lapid and M. Tadic. (In addition, they proved, using results of C. Jantzen, that this induced representation is always reducible if the supercuspidal support is not separated.) More generally, we study, for a given set I of inertial orbits of supercuspidal representations of p-adic general linear groups, the category CI,σ of smooth complex finitely generated representations of classical p-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by σ and I, and show that this category is equivalent to a category of finitely generated right modules over a direct sum of tensor products of extended affine Hecke algebras of type AB and D and establish functoriality properties, relating categories with disjoint I’s. In this way, we extend results of C. Jantzen who proved a bijection between irreducible representations corresponding to these categories. The proof of the above reducibility result is then based on Hecke algebra arguments, using Kato’s exotic geometry.

Ordered Tensor Categories and Representations of the Mackey Lie Algebra of Infinite Matrices

We introduce (partially) ordered Grothendieck categories and apply results on their structure to the study of categories of representations of the Mackey Lie algebra of infinite matrices $\mathfrak {gl}^{M}\left (V,V_{*}\right )$ . Here $\mathfrak {gl}^{M}\left (V,V_{*}\right )$ is the Lie algebra of endomorphisms of a nondegenerate pairing of countably infinite-dimensional vector spaces $V_{*}\otimes V\to \mathbb {K}$ , where $\mathbb {K}$ is the base field. Tensor representations of $\mathfrak {gl}^{M}\left (V,V_{*}\right )$ are defined as arbitrary subquotients of finite direct sums of tensor products (V∗)⊗m ⊗ (V∗)⊗n ⊗ V⊗p where V∗ denotes the algebraic dual of V. The category $\mathbb {T}^{3}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}$ which they comprise, extends a category $\mathbb {T}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}$ previously studied in Dan-Cohen et al. Adv. Math. 289, 205–278, (2016), Penkov and Serganova (2014) and Sam and Snowden Forum Math. Sigma 3(e11):108, (2015) . Our main result is that $\mathbb {T}^{3}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}$ is a finite-length, Koszul self-dual, tensor category with a certain universal property that makes it into a “categorified algebra” defined by means of a handful of generators and relations. This result uses essentially the general properties of ordered Grothendieck categories, which yield also simpler proofs of some facts about the category $\mathbb {T}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}$ established in Penkov and Serganova (2014). Finally, we discuss the extension of $\mathbb {T}^{3}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}$ obtained by adjoining the algebraic dual (V∗)∗ of V∗.

Commutators, commutativity and dimension in the socle of a Banach algebra: A generalized Wedderburn–Artin and Shoda's theorem

As a follow-up to work done in [7], some new insights to the structure of the socle of a semisimple Banach algebra are obtained. In particular, it is shown that the socle is isomorphic as an algebra to the direct sum of tensor products of corresponding left and right minimal ideals. Remarkably, the finite-dimensional case here reduces to the classical Wedderburn–Artin Theorem, and this approach does not use any continuous irreducible representations of the algebra in question. Furthermore, the structure of the socles for which the classical Shoda's Theorem for matrices can be extended, is characterized exactly as those socles which are minimal two-sided ideals. It is then shown that the set of commutators in the socle (i.e. {xy−yx:x,y∈SocA}) is a vector subspace. Finally, we characterize those socles which belong to the center of a Banach algebra and obtain results which suggest that the dimension of certain subalgebras of the socle in fact provides a measure, to some extent, of commutativity.

Systems of correlation functions, coinvariants, and the Verlinde Algebra

We study the Gaberdiel-Goddard spaces of systems of correlation functions attached to affine Kac-Moody Lie algebras . We prove that these spaces are isomorphic to spaces of coinvariants with respect to certain subalgebras of . This allows us to describe the Gaberdiel-Goddard spaces as direct sums of tensor products of irreducible g-modules with multiplicities determined by the fusion coefficients. We thus reprove and generalize the Frenkel-Zhu theorem.

Schur algebras of Brauer algebras I

A classical problem of invariant theory and of Lie theory is to determine endomorphism rings of representations of classical groups, for instance of tensor powers of the natural module (Schur–Weyl duality) or of full direct sums of tensor products of exterior powers (Ringel duality). In this article, the endomorphism rings of full direct sums of tensor products of symmetric powers over symplectic and orthogonal groups are determined. These are shown to be isomorphic to Schur algebras of Brauer algebras as defined in Henke and Koenig (Math Z 272(3–4):729–759, 2012). This implies structural properties of the endomorphism rings, such as double centraliser properties, quasi-hereditary, and a universal property, as well as a classification of simple modules.

A modular invariant bulk theory for the triplet model

A proposal for the bulk space of the logarithmic -triplet model at central charge zero is made. The construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory. The resulting bulk space is a quotient of the direct sum of projective representations, which is isomorphic, as a vector space, to the direct sum of tensor products of the irreducible representations with their projective covers. As a consistency check of our analysis, we show that the partition function of the bulk theory is modular invariant, and that the boundary state analysis is compatible with the proposed annulus partition functions of this model.

On semisimple representations of universal lattices

We study finite-dimensional semisimple complex representations of the universal lattices Γ_{n,k} = \mathrm{SL}_n(ℤ[x_1, \ldots, x_k]) ( n ≥ 3 ). One may obtain such a representation by specializing x_1, \ldots, x_k to some complex values and composing the induced homomorphism Γ_{n,k} → \mathrm{SL}_n(ℂ) with a rational representation of \mathrm{SL}_n(ℂ) . We show that any semisimple representation coincides, on a subgroup of finite index, with a direct sum of tensor products of representations obtained in this way.

On Inverting the Koszul Complex

Let V be an n-dimensional vector space. We give a direct construction of an exact sequence that gives a GL(V)-equivariant “resolution” of each symmetric power S t V in terms of direct sums of tensor products of the form ∧ i 1 V ⊗···⊗ ∧ i p V. This exact sequence corresponds to inverting the relation in the representation ring of GL(V) that is described by the Koszul complex, and has appeared before in work by B. Totaro, analogously to a construction of K. Akin involving the normalized bar resolution. Our approach yields a concrete description of the differentials, and provides an alternate direct proof that .

Lie powers and Witt vectors

In the study of Lie powers of a module V in prime characteristic p, a basic role is played by certain modules B n introduced by Bryant and Schocker. The isomorphism types of the B n are not fully understood, but these modules fall into infinite families \(\{B_{k},B_{pk},B_{p^{2}k},\dots \}\) , one family B(k) for each positive integer k not divisible by p, and there is a recursive formula for the modules within B(k). Here we use combinatorial methods and Witt vectors to show that each module in B(k) is isomorphic to a direct sum of tensor products of direct summands of the kth tensor power V ⊗k.

Convergence and Decomposition for Tensor Products of Hilbert Space Operators

It is shown that convergence of sequences of Hilbert space operators is preserved by tensor product and the converse holds in case of convergence to zero under the semigroup assumption. In particular, unlike ordinary product of operators, weak convergence is preserved by tensor product. It is also shown that a tensor product of operators is a unilateral shift if and only if it coincides with a tensor product of a unilateral shift and an isometry. These results lead to a decomposition of a tensor product of contractions into an orthogonal direct sum of tensor products of class C 00 , strongly stable tensor products, unilateral shift tensor products, and a unitary tensor product.

Some quotient algebras arising from the quantum toroidal algebra Uq(sl₂(C_γ))

Some quotient algebras arising from the quantum toroidal algebra $U_{q}(\mathit{sl}_{n+1}(\mathcal{C}_{\gamma}))$ ($n\ge 2$) are considered. They are related to integrable highest weight representations of the algebra and are shown to be isomorphic to tensor products of two algebras of symmetric Laurent polynomials and Macdonald's difference operators.

Morita equivalences of Ariki-Koike algebras

We prove that every Ariki–Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki–Koike algebras which have q–connected parameter sets. A similar result is proved for the cyclotomic q–Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki–Koike algebras defined over fields of characteristic zero are now known in principle.

𝑈(𝔤)-finite locally analytic representations

In this paper we continue our algebraic approach to the study of locally analytic representations of a p p -adic Lie group G G in vector spaces over a non-Archimedean complete field K K . We characterize the smooth representations of Langlands theory which are contained in the new category. More generally, we completely determine the structure of the representations on which the universal enveloping algebra U ( g ) U(\mathfrak g) of the Lie algebra g \mathfrak g of G G acts through a finite dimensional quotient. They are direct sums of tensor products of smooth and rational G G -representations. Finally we analyze the reducible members of the principal series of the group G = S L 2 ( Q p ) G=SL_2(\mathbb Q_p) in terms of such tensor products.

Ariki-Koike algebras with semisimple bottoms

We investigated the representation thoery of an Ariki-Koike algebra whose Poincare polynomial associated with the bottom, i.e., the subgroup on which the symmetric group acts, is non-zero in the base field. We proved that the module category of such an Ariki-Koike algebra is Morita equivalent to the module category of a direct sum of tensor products of Hecke algebras associated with certain symmetric groups. We also generalized this Morita equivalence theorem to give a Morita equivalenve between a $q$-Schur$^m$ algebra and a direct sum of tensor products of certain $q$-Schur algebras.

Direct Sums of Tensor Products and Non-commutative Independence

We point out the connection between the three known fundamental notions of non-commutative independence in quantum probability and the three basic choices for forming an associative direct sum of alternating tensor products of two vector spaces.

An explicit construction of the quantum group in chiral WZW-models

It is shown how a chiral Wess-Zumino-Witten theory with globally defined vertex operators and a one-to-one correspondence between fields and states can be constructed. The Hilbert space of this theory is the direct sum of tensor products of representations of the chiral algebra and finite dimensional internal parameter spaces. On this enlarged space there exists a natural action of Drinfeld's quasi quantum group $A_{g,t}$, which commutes with the action of the chiral algebra and plays the r\^{o}le of an internal symmetry algebra. The $R$ matrix describes the braiding of the chiral vertex operators and the coassociator $\Phi$ gives rise to a modification of the duality property. For generic $q$ the quasi quantum group is isomorphic to the coassociative quantum group $U_{q}(g)$ and thus the duality property of the chiral theory can be restored. This construction has to be modified for the physically relevant case of integer level. The quantum group has to be replaced by the corresponding truncated quasi quantum group, which is not coassociative because of the truncation. This exhibits the truncated quantum group as the internal symmetry algebra of the chiral WZW model, which therefore has only a modified duality property. The case of $g=su(2)$ is worked out in detail.

Null-plane dynamics of particles and fields

The focus of this review of null-plane dynamics is the fundamental principle of quantum theory that states must form a linear manifold with a positive scalar product. The intent is to provide an integrated overview of diverse aspects of null-plane dynamics of particles and fields. Hamiltonian particle dynamics is based on the construction of nontrivial representations of the Poincaré group on finite tensor products of single-particle spaces, and on finite direct sums of such tensor products. The structure of the space of states and the representations of a kinematic subgroup are independent of the dynamics. The dynamics is specified by Hamiltonian operators which are the Poincaré generators outside the kinematic subgroup. Fock-spaces are infinite direct sums of tensor products of single particle spaces. Fock-space representations of Lagrangean field theories can be formulated as limits of Hamiltonian many-body dynamics. The required cutoff can preserve the symmetry of the kinematic subgroup but destroys the full Poincaré invariance. The nontrivial questions involve the existence of limits as the cutoff is removed. Covariant wave functions arise either as solutions of covariant wave equations, or as matrix elements of products of covariant field operators. The linear manifold of states is a manifold of equivalence classes of covariant functions. The dynamics appear in the nontrivial inner product and all Poincaré transformations are kinematic. Null-plane restrictions of the covariant functions may provide a unitary map of the covariant constraint dynamics into null-plane Hamiltonian dynamics.

Finite resolutions of modules for reductive algebraic groups

of a suitable G&-module M by modules which are required to be direct sums of tensor products of exterior powers of the natural representation. In particular, in [Z, 31, for partitions 1 and p, resolutions are constructed for the modules L,(F) and L,,,(F) (the Schur functor corresponding to 1 and the skew Schur functor corresponding to (A, p), evaluated at F), where F is a free module over a field or the integers. The purpose of this paper is to characterise the GL,-modules which admit such a resolution as those modules which have a filtration in which each successive quotient has the form L,(F) for some partition p. By contrast with [2,3] we do not produce resolutions explicitly. Our methods are independent of those of Akin and Buchsbaum (we give a new proof of their result on the existence of a resolution for L,(F)) and are algebraic group theoretic in nature. We have therefore cast our main result (the theorem of Section 1) as a statement about resolutions for reductive algebraic groups over an algebraically closed field. One may frequently wish to work over a more general coefficient ring (see [l-3]) and so we treat in some detail Chevalley group schemes and general linear group schemes over a principal ideal domain (2.6 and 2.7), giving the appropriate versions of our main result and relating it to the work of Akin, Buchsbaum and Weyman.