Representation Theory Of Algebras Research Articles

Holomorphic Laplacian on the Lie ball and the Penrose transform

We prove that any holomorphic function f on the Lie ball of even dimension satisfying Δf=0 is obtained uniquely by the higher-dimensional Penrose transform of a Dolbeault cohomology for a twisted line bundle of a certain domain of the Grassmannian of isotropic subspaces. To overcome the difficulties arising from our setting that the line bundle parameter is outside the good range, we use some techniques from algebraic representation theory.

On the structure and representation theory of q-deformed Clifford algebras

On the structure and representation theory of q-deformed Clifford algebras

Deformations of Pairs of Kleinian Singularities

Abstract Kleinian singularities, that is, the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory, and Singularity theory. The filtered deformations of these algebras of invariants were classified by Brieskorn (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when the smaller subgroup is normal in the larger.

Associative algebras and the representation theory of grading-restricted vertex algebras

We introduce an associative algebra [Formula: see text] using infinite matrices with entries in a grading-restricted vertex algebra [Formula: see text] such that the associated graded space [Formula: see text] of a filtration of a lower-bounded generalized [Formula: see text]-module [Formula: see text] is an [Formula: see text]-module satisfying additional properties (called a nondegenerate graded [Formula: see text]-module). We prove that a lower-bounded generalized [Formula: see text]-module [Formula: see text] is irreducible or completely reducible if and only if the nondegenerate graded [Formula: see text]-module [Formula: see text] is irreducible or completely reducible, respectively. We also prove that the set of equivalence classes of the lower-bounded generalized [Formula: see text]-modules is in bijection with the set of the equivalence classes of nondegenerate graded [Formula: see text]-modules. For [Formula: see text], there is a subalgebra [Formula: see text] of [Formula: see text] such that the subspace [Formula: see text] of [Formula: see text] is an [Formula: see text]-module satisfying additional properties (called a nondegenerate graded [Formula: see text]-module). We prove that [Formula: see text] are finite-dimensional when [Formula: see text] is of positive energy (CFT type) and [Formula: see text]-cofinite. We prove that the set of the equivalence classes of lower-bounded generalized [Formula: see text]-modules is in bijection with the set of the equivalence classes of nondegenerate graded [Formula: see text]-modules. In the case that [Formula: see text] is a Möbius vertex algebra and the differences between the real parts of the lowest weights of the irreducible lower-bounded generalized [Formula: see text]-modules are less than or equal to [Formula: see text], we prove that a lower-bounded generalized [Formula: see text]-module [Formula: see text] of finite length is irreducible or completely reducible if and only if the nondegenerate graded [Formula: see text]-module [Formula: see text] is irreducible or completely reducible, respectively.

Representation Theory of Quivers and Finite-Dimensional Algebras

Representation Theory of Quivers and Finite-Dimensional Algebras

Rational Cherednik algebras of G(ℓ,p,n) from the Coulomb perspective

We prove a number of results on the structure and representation theory of the rational Cherednik algebra of the imprimitive reflection group G(ℓ,p,n). In particular, we: (1) show a relationship to the Coulomb branch construction of Braverman, Finkelberg, and Nakajima, and 3-dimensional quantum field theory; (2) show that the spherical Cherednik algebra carries the structure of a principal Galois order; (3) construct a graded lift of category O and the larger category of Dunkl-Opdam modules, whose simple modules have the properties of a dual canonical basis and (4) give the first classification of simple Dunkl-Opdam modules for the rational Cherednik algebra of the imprimitive reflection group G(ℓ,p,n).

Tensor Products and Crossed Differential Graded Lie Algebras in the Category of Crossed Complexes

The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has resulted in many areas, including algebraic topology, representation theory, and quantum physics. In this paper, we present analogous definitions for Lie algebras within the framework of whiskered structures, bimorphisms, crossed complexes, crossed differential graded algebras, and tensor products. These definitions, given for groupoids in existing literature, have been adapted to establish a direct correspondence between these algebraic structures and Lie algebras. We show that a 2-truncation of the crossed differential graded Lie algebra, obtained from our adapted definitions, gives rise to a braided crossed module of Lie algebras. We also construct a functor to simplicial Lie algebras, enabling a systematic mapping between different Lie algebraic categories, which supports the validity of our adapted definitions and establishes their compatibility with established categories.

The maximal subgroups of the exceptional groups F_{4}(q), E_{6}(q) and ^{2}!E_{6}(q) and related almost simple groups

This article produces a complete list of all maximal subgroups of the finite simple groups of type F_{4}, E_{6} and twisted E_{6} over all finite fields. Along the way, we determine the collection of Lie primitive almost simple subgroups of the corresponding algebraic groups. We give the stabilizers under the actions of outer automorphisms, from which one can obtain complete information about the maximal subgroups of all almost simple groups with socle one of these groups. We also provide a new maximal subgroup of ^{2}!F_{4}(8), correcting the maximal subgroups for that group from the list of Malle. This provides the first new exceptional groups of Lie type to have their maximal subgroups enumerated for three decades. The techniques are a mixture of algebraic groups, representation theory, computational algebra, and use of the trilinear form on the 27-dimensional minimal module for E_{6}. We provide a collection of supplementary Magma files that prove the author’s computational claims, yielding existence and the number of conjugacy classes of all maximal subgroups mentioned in the text.

Determination of some almost split sequences in morphism categories

Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra Λ. Firstly we try to interpret the Auslander-Reiten translates of particular objects in the morphism category in terms of the Auslander-Reiten translations within the category of Λ-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the middle term of almost split sequences. As such, another part of the paper is devoted to discuss the middle term of certain almost split sequences in the morphism category of Λ. As an application, we restrict in the last part of the paper to self-injective algebras and present a structural theorem that illuminates a link between representation-finite morphism categories and Dynkin diagrams.

Kähler-Ricci flow on rational homogeneous varieties

In this work, we study the Kähler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the Kähler-Ricci flow with homogeneous initial condition. This description enables us to compute explicitly the maximal existence time for any solution starting at a homogeneous Kähler metric and obtain explicit upper and lower bounds for several geometric quantities along the flow, including curvatures, volume, diameter, and the first non-zero eigenvalue of the Laplacian. As an application of our main result, we investigate the relationship between certain numerical invariants associated with ample divisors and numerical invariants arising from solutions of the Kähler-Ricci flow in the homogeneous setting.

Geometric Structures in Group Theory

The conference was in the area of geometric group theory, the field of mathematics in which one studies infinite groups (finitely generated, or more generally locally compact, countable etc.) via actions on spaces endowed with various structures (geometric, measurable, analytic etc.). The surging current activity in the field is drawing more and more connections with other mathematical areas, and this was successfully reflected in the program of this week, during which problems in algebraic topology, representation theory and functional analysis, to name just a few, featured prominently alongside core topics in the area.

Rational Singularities of Nested Hilbert Schemes

Abstract The Hilbert scheme of points $\textrm {Hilb}^{n}(S)$ of a smooth surface $S$ is a well-studied parameter space, lying at the interface of algebraic geometry, commutative algebra, representation theory, combinatorics, and mathematical physics. The foundational result is a classical theorem of Fogarty, stating that $\textrm {Hilb}^{n}(S)$ is a smooth variety of dimension $2n$. In recent years there has been growing interest in a natural generalization of $\textrm {Hilb}^{n}(S)$, the nested Hilbert scheme$\textrm {Hilb}^{(n_{1}, n_{2})}(S)$, which parametrizes nested pairs of zero-dimensional subschemes $Z_{1} \supseteq Z_{2}$ of $S$ with $\deg Z_{i}=n_{i}$. In contrast to Fogarty’s theorem, $\textrm {Hilb}^{(n_{1}, n_{2})}(S)$ is almost always singular, and very little is known about its singularities. In this paper, we aim to advance the knowledge of the geometry of these nested Hilbert schemes. Work by Fogarty in the 70’s shows that $\textrm {Hilb}^{(n,1)}(S)$ is a normal Cohen–Macaulay variety, and Song more recently proved that it has rational singularities. In our main result, we prove that the nested Hilbert scheme $\textrm {Hilb}^{(n,2)}(S)$ has rational singularities. We employ an array of tools from commutative algebra to prove this theorem. Using Gröbner bases, we establish a connection between $\textrm {Hilb}^{(n,2)}(S)$ and a certain variety of matrices with an action of the general linear group. This variety of matrices plays a central role in our work, and we analyze it by various algebraic techniques, including the Kempf–Lascoux–Weyman technique of calculating syzygies, square-free Gröbner degenerations, and the Stanley–Reisner correspondence. Along the way, we also obtain results on classes of irreducible and reducible nested Hilbert schemes, dimension of singular loci, and $F$-singularities in positive characteristic.

New partition identities from \(C^{(1)}_\ell\)-modules

In this paper we conjecture combinatorial Rogers-Ramanu­jan type colored partition identities related to standard representations of the affine Lie algebra of type \(C^{(1)}_\ell\), \(\ell\geq2\), and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras.

Equivariant multiplicities via representations of quantum affine algebras

For any simply-laced type simple Lie algebra mathfrak {g} and any height function xi adapted to an orientation Q of the Dynkin diagram of mathfrak {g}, Hernandez–Leclerc introduced a certain category mathcal {C}^{le xi } of representations of the quantum affine algebra U_q(widehat{mathfrak {g}}), as well as a subcategory mathcal {C}_Q of mathcal {C}^{le xi } whose complexified Grothendieck ring is isomorphic to the coordinate ring mathbb {C}[textbf{N}] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism {widetilde{D}}_{xi } on a torus mathcal {Y}^{le xi } containing the image of K_0(mathcal {C}^{le xi }) under the truncated q-character morphism. We prove that the restriction of {widetilde{D}}_{xi } to K_0(mathcal {C}_Q) coincides with the morphism overline{D} recently introduced by Baumann–Kamnitzer–Knutson in their study of equivariant multiplicities of Mirković–Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov–Reshetikhin modules in mathcal {C}_Q, as well as certain results by Brundan–Kleshchev–McNamara on the representation theory of quiver Hecke algebras. This alternative description of overline{D} allows us to prove a conjecture by the first author on the distinguished values of overline{D} on the flag minors of mathbb {C}[textbf{N}]. We also provide applications of our results from the perspective of Kang–Kashiwara–Kim–Oh’s generalized Schur–Weyl duality. Finally, we use Kashiwara–Kim–Oh–Park’s recent constructions to define a cluster algebra overline{mathcal {A}}_Q as a subquotient of K_0(mathcal {C}^{le xi }) naturally containing mathbb {C}[textbf{N}], and suggest the existence of an analogue of the Mirković–Vilonen basis in overline{mathcal {A}}_Q on which the values of {widetilde{D}}_{xi } may be interpreted as certain equivariant multiplicities.

Plethysm and the algebra of uniform block permutations

We study the representation theory of the uniform block permutation algebra in the context of the representation theory of factorizable inverse monoids. The uniform block permutation algebra is a subalgebra of the partition algebra and is also known as the party algebra. We compute its characters and provide a Frobenius characteristic map to symmetric functions. This reveals connections of the characters of the uniform block permutation algebra and plethysms of Schur functions.

Coherence for Plactic Monoids via Rewriting Theory and Crystal Structures

Rewriting methods have been developed for the study of coherence for algebraic objects. This consists in starting with a convergent presentation and expliciting a family of generating confluences to obtain a coherent presentation – one with generators, generating relations, and generating relations between relations (syzygies). In this article we develop these ideas for a class of monoids which encode the representation theory of complex symmetrizable Kac-Moody algebras, called plactic monoids. The main tools for this are the crystal realization of plactic monoids due to Kashiwara, and a class of presentations compatible with a crystal structure, called crystal presentations. We show that the compatibility of the crystal structure with the presentation reduces many aspects of the study of plactic monoids via rewriting theory to computations with components of highest weight in the crystal. We thus obtain reduced versions of Newman’s Lemma and Critical Pair Lemma, which are results for verifying convergence of a presentation. Further we show that the family of generating confluences of a convergent crystal presentation is entirely determined by the components of highest weight, and show that this result applies to the plactic monoids of types An, Bn, Cn, Dn, and G2, via their convergent presentations due to Cain–Gray–Malheiro (Cain et al. J. Comb. Theory Ser. A. 162, 406–466, 6).

Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.

Gröbner Bases for Fusion Products

We provide a new approach towards the analysis of the fusion products defined by B. Feigin and S. Loktev in the representation theory of (truncated) current Lie algebras. We understand the fusion product as a degeneration using Gröbner theory of non-commutative algebras and outline a strategy on how to prove a conjecture about the defining relations for the fusion product of two evaluation modules. We conclude with following this strategy for mathfrak {sl}_2(mathbb {C}[t]) and hence provide yet another proof for the conjecture in this case.

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, J.-P. Serre, 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, covering a range of topics, with an emphasis on recent developments in the subject.

Associative Algebras and Intertwining Operators

Let $V$ be a vertex operator algebra and $A^{\infty}(V)$ and $A^{N}(V)$ for $N\in \mathbb{N}$ the associative algebras introduced by the author in [H5]. For a lower-bounded generalized $V$-module $W$, we give $W$ a structure of graded $A^{\infty}(V)$-module and we introduce an $A^{\infty}(V)$-bimodule $A^{\infty}(W)$ and an $A^{N}(V)$-bimodule $A^{N}(W)$. We prove that the space of (logarithmic) intertwining operators of type $\binom{W_{3}}{W_{1}W_{2}}$ for lower-bounded generalized $V$-modules $W_{1}$, $W_{2}$ and $W_{3}$ is isomorphic to the space $\hom_{A^{\infty}(V)}(A^{\infty}(W_{1})\otimes_{A^{\infty}(V)}W_{2}, W_{3})$. Assuming that $W_{2}$ and $W_{3}'$ are equivalent to certain universal lower-bounded generalized $V$-modules generated by their $A^{N}(V)$-submodules consisting of elements of levels less than or equal to $N\in \mathbb{N}$, we also prove that the space of (logarithmic) intertwining operators of type $\binom{W_{3}}{W_{1}W_{2}}$ is isomorphic to the space of $\hom_{A^{N}(V)}(A^{N}(W_{1})\otimes_{A^{N}(V)}\Omega_{N}^{0}(W_{2}), \Omega_{N}^{0}(W_{3}))$.