Representation Theory Of Algebras Research Articles

N-point locality for vertex operators: Normal ordered products, operator product expansions, twisted vertex algebras

In this paper we study fields satisfying N-point locality and their properties. We obtain residue formulae for N-point local fields in terms of derivatives of delta functions and Bell polynomials. We introduce the notion of the space of descendants of N-point local fields which includes normal ordered products and coefficients of operator product expansions. We show that examples of N-point local fields include the vertex operators generating the boson–fermion correspondences of types B, C and D-A. We apply the normal ordered products of these vertex operators to the setting of the representation theory of the double-infinite rank Lie algebras b∞, c∞, d∞. Finally, we show that the field theory generated by N-point local fields and their descendants has a structure of a twisted vertex algebra.

Algebraic Groups

Linear algebraic groups is an active research area in contemporary mathematics. It has rich connections to algebraic geometry, representation theory, algebraic combinatorics, number theory, algebraic topology, and differential equations. The foundations of this theory were laid by A. Borel, C. Chevalley, T. A. Springer and J. Tits in the second half of the 20th century. The Oberwolfach workshops on algebraic groups, led by Springer and Tits, played an important role in this effort as a forum for researchers, meeting at approximately 3 year intervals since the 1960s. The present workshop continued this tradition, featuring a number of important recent developments in the subject.

Koornwinder polynomials and the [formula omitted] spin chain

Nonsymmetric Koornwinder polynomials are multivariable extensions of nonsymmetric Askey–Wilson polynomials. They naturally arise in the representation theory of (double) affine Hecke algebras. In this paper we discuss how nonsymmetric Koornwinder polynomials naturally arise in the theory of the Heisenberg XXZ spin-12 chain with general reflecting boundary conditions. A central role in this story is played by an explicit two-parameter family of spin representations of the two-boundary Temperley–Lieb algebra. These spin representations have three different appearances. Their original definition relates them directly to the XXZ spin chain, in the form of matchmaker representations they relate to Temperley–Lieb loop models in statistical physics, while their realization as principal series representations leads to the link with nonsymmetric Koornwinder polynomials. The nonsymmetric difference Cherednik–Matsuo correspondence allows to construct for special parameter values Laurent-polynomial solutions of the associated reflection quantum KZ equations in terms of nonsymmetric Koornwinder polynomials. We discuss these aspects in detail by revisiting and extending work of De Gier, Kasatani, Nichols, Cherednik, the first author and many others.

Symplectic restriction varieties and geometric branching rules II

In this paper, we introduce combinatorially defined subvarieties of symplectic flag varieties called symplectic restriction varieties. We study their geometric properties and compute their cohomology classes. In particular, we give a positive, combinatorial, geometric branching rule for computing the map in cohomology induced by the inclusion of the symplectic partial flag variety SF(k1,…,kh;n) in the partial flag variety F(k1,…,kh;n). These rules have many applications in algebraic geometry, combinatorics, symplectic geometry and representation theory.

Geometric complexity theory: an introduction for geometers

This article is survey of recent developments in, and a tutorial on, the approach to $$\mathbf{P}$$ v. $$\mathbf{N\mathbf P}$$ and related questions called geometric complexity theory (GCT). It is written to be accessible to graduate students. Numerous open questions in algebraic geometry and representation theory relevant for GCT are presented.

Irreducible Representations of the Quantum Weyl Algebra at Roots of Unity Given by Matrices

To describe the representation theory of the quantum Weyl algebra at an lth primitive root γ of unity, Boyette, Leyk, Plunkett, Sipe, and Talley found all nonsingular irreducible matrix solutions to the equation yx − γxy = 1, assuming yx ≠ xy. In this note, we complete their result by finding and classifying, up to equivalence, all irreducible matrix solutions (X, Y), where X is singular.

Invariants of centralisers in positive characteristic

Let Q be a simple algebraic group of type A or C over a field of good positive characteristic. Let x∈q=Lie(Q) and consider the centraliser qx={y∈q:[xy]=0}. We show that the invariant algebra S(qx)qx is generated by the pth power subalgebra and the mod p reduction of the characteristic zero invariant algebra. The latter algebra is known to be polynomial [17] and we show that it remains so after reduction. Using a theory of symmetrisation in positive characteristic we prove the analogue of this result in the enveloping algebra, where the p-centre plays the role of the pth power subalgebra. In Zassenhausʼ foundational work [30], the invariant theory and representation theory of modular Lie algebras were shown to be explicitly intertwined. We exploit his theory to give a precise upper bound for the dimensions of simple qx-modules. An application to the geometry of the Zassenhaus variety is given.When g is of type A and g=k⊕p is a symmetric decomposition of orthogonal type we use similar methods to show that for every nilpotent e∈k the invariant algebra S(pe)ke is generated by the pth power subalgebra and S(pe)Ke which is also shown to be polynomial.

Category of trees in representation theory of quantum algebras

New applications of categorical methods are connected with new additional structures on categories. One of such structures in representation theory of quantum algebras, the category of Kuznetsov-Smorodinsky-Vilenkin-Smirnov (KSVS) trees, is constructed, whose objects are finite rooted KSVS trees and morphisms generated by the transition from a KSVS tree to another one.

New examples of the Green functors arising from representation theory of semisimple Hopf algebras

A general Mackey-type decomposition for representations of semisimple Hopf algebras is investigated. We show that such a decomposition occurs in the case that the module is induced from an arbitrary Hopf subalgebra and it is restricted back to a group subalgebra. Some other examples when such a decomposition occurs are also constructed. They arise from gradings on the category of corepresentations of a semisimple Hopf algebra and provide new examples of the Green functors in the literature.

When is a quantum cellular automaton (QCA) a quantum lattice gas automaton (QLGA)?

Quantum cellular automata (QCA) are models of quantum computation of particular interest from the point of view of quantum simulation. Quantum lattice gas automata (QLGA - equivalently partitioned quantum cellular automata) represent an interesting subclass of QCA. QLGA have been more deeply analyzed than QCA, whereas general QCA are likely to capture a wider range of quantum behavior. Discriminating between QLGA and QCA is therefore an important question. In spite of much prior work, classifying which QCA are QLGA has remained an open problem. In the present paper we establish necessary and sufficient conditions for unbounded, finite QCA (finitely many active cells in a quiescent background) to be QLGA. We define a local condition that classifies those QCA that are QLGA, and we show that there are QCA that are not QLGA. We use a number of tools from functional analysis of separable Hilbert spaces and representation theory of associative algebras that enable us to treat QCA on finite but unbounded configurations in full detail.

Representation theory of effect algebras

The aim of this paper is to characterize representable and weak representable effect algebras and establish a representation theory of effect algebras. An effect algebra E is said to be representable if there exists a Hilbert space H and a monomorphism π from E into the Hilbert space effect algebra e ( H ) and it is said to be weakly representable if there exists an injective morphism from E into some e ( H ). It is proved that an effect algebra E with the nonempty state space S ( E ) is representable if and only if x, y ∈ E , f ( x )+ f ( y ) ≤ 1 implies x ⊕ y is defined; it is weakly representable if and only if the state space S ( E ) separates the points of E. Some operational properties of representable effect algebras are established, and some applications of the obtained results are listed.

Conceptual problem for noncommutative inflation and the new approach for nonrelativistic inflationary equation of state

In a previous paper, we connected the phenomenological non-commutative inflation of Alexander, Brandenberger and Magueijo (2003) and Koh S and Brandenberger (2007) with the formal representation theory of groups and algebras and analyzed minimal conditions that the deformed dispersion relation should satisfy in order to lead to a successful inflation. In that paper, we showed that elementary tools of algebra allow a group like procedure in which even Hopf algebras (roughly the symmetries of non-commutative spaces) could lead to the equation of state of inflationary radiation. In this paper, we show that there exists a conceptual problem with the kind of representation that leads to the fundamental equations of the model. The problem comes from an incompatibility between one of the minimal conditions for successful inflation (the momentum of individual photons is bounded from above) and the group structure of the representation which leads to the fundamental inflationary equations of state. We show that such a group structure, although mathematically allowed, would lead to problems with the overall consistency of physics, like in scattering theory, for example. Therefore, it follows that the procedure to obtain those equations should be modified according to one of two possible proposals that we consider here. One of them relates to the general theory of Hopf algebras while the other is based on a representation theorem of Von Neumann algebras, a proposal already suggested by us to take into account interactions in the inflationary equation of state. This reopens the problem of finding inflationary deformed dispersion relations and all developments which followed the first paper of Non-commutative Inflation.

B-Construction and C-Construction

B-construction is a way of obtaining a graded algebra from the triple consisting of an additive category, an object, and an autoequivalence, while C-construction is a way of obtaining an algebra (without unity) from the pair consisting of an additive category and a set of objects. In this article, we study and compare three important classes of algebras in noncommutative algebraic geometry and representation theory of finite dimensional algebras, namely, quantum polynomial algebras, preprojetive algebras and trivial extensions, via these constructions.

Quantum corrections to Bekenstein–Hawking black hole entropy and gravity partition functions

Algebraic aspects of the computation of partition functions for quantum gravity and black holes in AdS3 are discussed. We compute the sub-leading quantum corrections to the Bekenstein–Hawking entropy. It is shown that the quantum corrections to the classical result can be included systematically by making use of the comparison with conformal field theory partition functions, via the AdS3/CFT2 correspondence. This leads to a better understanding of the role of modular and spectral functions, from the point of view of the representation theory of infinite-dimensional Lie algebras. Besides, the sum of known quantum contributions to the partition function can be presented in a closed form, involving the Patterson–Selberg spectral function. These contributions can be reproduced in a holomorphically factorized theory whose partition functions are associated with the formal characters of the Virasoro modules. We propose a spectral function formulation for quantum corrections to the elliptic genus from supergravity states.

Half-flat structures on $$ S^3\times S^3$$

We describe left-invariant half-flat SU(3)-structures on S^3xS^3 using the representation theory of SO(4) and matrix algebra. This leads to a systematic study of the associated cohomogeneity one Ricci-flat metrics with holonomy G_2 obtained on 7-manifolds with equidistant S^3xS^3 hypersurfaces. The generic case is analysed numerically.

Geometry of the analytic loop group

We introduce and study a notion of analytic loop group with a Riemann–Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity Uϵ(gˆ) with non-trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf–Poisson structure is isomorphic to the semi-classical limit of the center of Uϵ(gˆ) (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters of Uϵ(gˆ), are parameterized by certain G-bundles on an elliptic curve.

Split abelian chief factors and first degree cohomology for Lie algebras

In this paper we investigate the relation between the multiplicities of split abelian chief factors of finite-dimensional Lie algebras and first degree cohomology. In particular, we obtain a characterization of modular solvable Lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors. The analogues of these results are well known in the modular representation theory of finite groups. An important tool in the proof of these results is a refinement of a non-vanishing theorem of Seligman for the first degree cohomology of non-solvable finite-dimensional Lie algebras in prime characteristic. As an application we derive several results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module.

Decomposition numbers for the cyclotomic Brauer algebras in characteristic zero

We study the representation theory of the cyclotomic Brauer algebra via truncation to idempotent subalgebras which are isomorphic to a product of walled and classical Brauer algebras. In particular, we determine the block structure and decomposition numbers in characteristic zero.

Extremal properties for concealed-canonical algebras

Canonical algebras, introduced by C. M. Ringel in 1984, play an important role in the representation theory of nite-dimensional algebras. They also feature in many other mathematical areas like function theory, 3-manifolds, singularity theory, commuta- tive algebra, algebraic geometry and mathematical physics. We show that canonical alge- bras are characterized by a number of interesting extremal properties (among concealed- canonical algebras, that is, the endomorphism rings of tilting bundles on a weighted pro- jective line). We also investigate the corresponding class of algebras antipodal to canonical ones. Our study yields new insights into the nature of concealed-canonical algebras, and sheds a new light on an old question: Why are the canonical algebras canonical?

A Coxeter--Gram Classification of Positive Simply Laced Edge-Bipartite Graphs

Following the spectral graph theory, a graph coloring technique, and algebraic methods in graph theory, we study the category ${\cal U}{\cal B} igr_n$ of loop-free edge-bipartite (signed) graphs $\Delta$, with $n\geq 2$ vertices, up to $\mathbb{Z}$-congruences $\sim_\mathbb{Z}$ and $\approx_\mathbb{Z}$ (defined in the paper), by means of the Gram matrix $ \check G_\Delta \in \mathbb{M}_n(\mathbb{Z})$, the Coxeter--Gram matrix ${\rm Cox}_\Delta= - \check G_\Delta\cdot \check G_\Delta^{-tr} \in \mathbb{M}_n(\mathbb{Z})$, the spectrum $ {\bf specc}_\Delta$ of $\mbox{\rm Cox}_\Delta$, the Coxeter polynomial $\mbox{\rm cox}_\Delta(t)\in \mathbb{Z}[t]$, the Coxeter number ${\bf c}_\Delta \geq 2$, and the $\mathbb{Z}$-bilinear Gram form $b_\Delta: \mathbb{Z}^n\times \mathbb{Z}^n \to \mathbb{Z}$ of $\Delta$. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems. One of our aims is to compute the set ${\cal C}{\cal G} p...