Brauer Category Research Articles

The representation theory of Brauer categories II: Curried algebra

A representation of \mathfrak{gl}(V)=V \otimes V^{\ast} is a linear map \mu \colon \mathfrak{gl}(V) \otimes M \rightarrow M satisfying a certain identity. By currying, giving a linear map \mu is equivalent to giving a linear map a \colon V \otimes M \rightarrow V \otimes M , and one can translate the condition for \mu to be a representation into a condition on a . This alternate formulation does not use the dual of V and makes sense for any object V in a tensor category \mathcal{C} . We call such objects representations of the curried general linear algebra on V . The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category is the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore.

The periplectic q-Brauer category

We introduce the periplectic q-Brauer category over an integral domain of characteristic not 2. This is a strict monoidal supercategory and can be considered as a q-analogue of the periplectic Brauer category in [18]. We prove that the periplectic q-Brauer category admits a split triangular decomposition in the sense of [6]. When the ground ring is an algebraically closed field, the category of locally finite-dimensional right modules for the periplectic q-Brauer category is an upper finite fully stratified category in the sense of [6]. We prove that periplectic q-Brauer algebras defined in [1] are isomorphic to endomorphism algebras in the periplectic q-Brauer category. Furthermore, a periplectic q-Brauer algebra is a standardly based algebra in the sense of [13]. We construct a Jucys-Murphy basis for any standard module of the periplectic q-Brauer algebra with respect to a family of commutative elements called Jucys-Murphy elements. Via them, we classify blocks for both the periplectic q-Brauer category and periplectic q-Brauer algebras in the generic case. Our result shows that both the periplectic q-Brauer category and periplectic q-Brauer algebras are always not semisimple over any algebraically closed field.

Representations of cyclotomic oriented Brauer categories

The cyclotomic Brauer category OB(u,u′) is an upper finite weakly triangular category over an arbitrary field in the sense of [14, Definition 2.2] and the category of locally finite dimensional representations of A is an upper finite fully stratified category in the sense of [10], where A is the locally unital algebra associated to OB(u,u′). The category of locally finite dimensional representations of A is used to give a direct proof of the tensor product categorification (in the general sense of Losev and Webster) for an integrable lowest weight with an integrable highest weight representation of the same level for the Lie algebra g, where g is a direct sum of copies of sl∞ (resp., slˆp) if the characteristic p of the ground field is 0 (resp., positive). Such a result was expected in [4] when k=C and proved previously in [2] when the level is 1.

On the geometry and representation theory of isomeric matrices

The space of $n \times m$ complex matrices can be regarded as an algebraic variety on which the group ${\bf GL}_n \times {\bf GL}_m$ acts. There is a rich interaction between geometry and representation theory in this example. In an important paper, de Concini, Eisenbud, and Procesi classified the equivariant ideals in the coordinate ring. More recently, we proved a noetherian result for families of equivariant modules as $n$ and $m$ vary. In this paper, we establish analogs of these results for the space of $(n|n) \times (m|m)$ isomeric matrices with respect to the action of ${\bf Q}_n \times {\bf Q}_m$, where ${\bf Q}_n$ is the automorphism group of the isomeric structure (commonly known as the "queer supergroup"). Our work is motivated by connections to the Brauer category and the theory of twisted commutative algebras.

Affine oriented Frobenius Brauer categories

To any Frobenius superalgebra A we associate an oriented Frobenius Brauer category and an affine oriented Frobenius Brauer category. We define natural actions of these categories on categories of supermodules for general linear Lie superalgebras with entries in A. These actions generalize those on module categories for general linear Lie superalgebras and queer Lie superalgebras, which correspond to the cases where A is the ground field and the two-dimensional Clifford superalgebra, respectively.

The Representation Theory of Brauer Categories I: Triangular Categories

This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure.

A basis theorem for the affine Kauffman category and its cyclotomic quotients

The affine Kauffman category is a strict monoidal category and can be considered as a q-analogue of the affine Brauer category in Rui and Song (2019) [33]. In this paper, we prove a basis theorem for the morphism spaces in the affine Kauffman category as well as for all of its cyclotomic quotients under certain assumptions.

Blocks of the Brauer Category over the Complex Field

Blocks of the Brauer Category over the Complex Field

Sandwich semigroups in diagram categories

This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley–Lieb categories. If [Formula: see text] denotes any of these categories, and if [Formula: see text] is a fixed morphism, then an associative operation [Formula: see text] may be defined on [Formula: see text] by [Formula: see text]. The resulting semigroup [Formula: see text] is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green’s relations and preorders, regularity, stability, mid-identities, ideal structure, (products of) idempotents, and minimal generation. It turns out that the Brauer category has many remarkable properties not shared by any of the other diagram categories we study. Because of these unique properties, we may completely classify isomorphism classes of sandwich semigroups in the Brauer category, calculate the rank (smallest size of a generating set) of an arbitrary sandwich semigroup, enumerate Green’s classes and idempotents, and calculate ranks (and idempotent ranks, where appropriate) of the regular subsemigroup and its ideals, as well as the idempotent-generated subsemigroup. Several illustrative examples are considered throughout, partly to demonstrate the sometimes-subtle differences between the various diagram categories.

The Chromatic Brauer Category and Its Linear Representations

The Brauer category is a symmetric strict monoidal category that arises as a (horizontal) categorification of the Brauer algebras in the context of Banagl’s framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment of the Brauer category in which the morphisms are component-wise labeled. Linear representations of the (chromatic) Brauer category are symmetric strict monoidal functors into the category of real vector spaces and linear maps equipped with the Schauenburg tensor product. We study representation theory of the (chromatic) Brauer category, and classify all its faithful linear representations. As an application, we use indices of fold lines to construct a refinement of Banagl’s concrete positive TFT based on fold maps into the plane.

Perturbative 4D conformal field theories and representation theory of diagram algebras

The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional so(4, 2) equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of non-additive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of mathcal{N} = 4 SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra U★, abstracted from Uso(d) in the limit of large integer d, which admits extension of Uso(d) representation theory to general real (or complex) d. The algebra is related, via oscillator realisations, to so(d) equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra U★, 2, related to a general d extension for Uso(d, 2) is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described.

Representations of Brauer category and categorification

We study representations of the locally unital and locally finite dimensional algebra B associated to the Brauer category B(δ0) with defining parameter δ0 over an algebraically closed field K with characteristic p≠2. The Grothendieck group K0(B-modΔ) will be used to categorify the integrable highest weight slK-module V(ϖδ0−12) with the fundamental weight ϖδ0−12 as its highest weight, where B-modΔ is a subcategory of B-lfdmod in which each object has a finite Δ-flag, and slK is either sl∞ or slˆp depending on whether p=0 or 2∤p. As g-modules, C⊗ZK0(B-modΔ) is isomorphic to V(ϖδ0−12), where g is a Lie subalgebra of slK (see Definition 4.2). When p=0, standard B-modules and projective covers of simple B-modules correspond to monomial basis and so-called quasi-canonical basis of V(ϖδ0−12), respectively.

THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$. This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$.

On geometrically defined extensions of the Temperley\u2013Lieb category in the Brauer category

We define an infinite chain of subcategories of the partition category by introducing the left-height ($l$) of a partition. For the Brauer case, the chain starts with the Temperley-Lieb ($l=-1$) and ends with the Brauer ($l=\infty$) category. The End sets are algebras, i.e., an infinite tower thereof for each $l$, whose representation theory is studied in the paper.

Frobenius Heisenberg categorification

We associate a graded monoidal supercategory $\mathcal{H}\mathit{eis}_{F,k}$ to every graded Frobenius superalgebra $F$ and integer $k$. These categories, which categorify a broad range of lattice Heisenberg algebras, recover many previously defined Heisenberg categories as special cases. In this way, the categories $\mathcal{H}\mathit{eis}_{F,k}$ serve as a unifying and generalizing framework for Heisenberg categorification. Even in the case of previously defined Heisenberg categories, we obtain new, more efficient, presentations of these categories, based on an approach of Brundan. When $k=0$, our construction yields new versions of the affine oriented Brauer category depending on a graded Frobenius superalgebra.

Affine Brauer category and parabolic category $${\mathcal {O}}$$ O in types B, C, D

A strict monoidal category referred to as affine Brauer category $$\mathcal {AB}$$ is introduced over a commutative ring $$\kappa $$ containing multiplicative identity 1 and invertible element 2. We prove that morphism spaces in $$\mathcal {AB}$$ are free over $$\kappa $$ . The cyclotomic (or level k) Brauer category $$\mathcal {CB}^f(\omega )$$ is a quotient category of $$\mathcal {AB}$$ . We prove that any morphism space in $$\mathcal {CB}^f(\omega )$$ is free over $$\kappa $$ with maximal rank if and only if the $${\mathbf {u}}$$ -admissible condition holds in the sense of (1.32). Affine Nazarov–Wenzl algebras (Nazarov in J Algebra 182(3):664–693, 1996) and cyclotomic Nazarov–Wenzl algebras (Ariki et al. in Nagoya Math J 182:47–134, 2006) will be realized as certain endomorphism algebras in $$\mathcal {AB}$$ and $$\mathcal {CB}^f(\omega )$$ , respectively. We will establish higher Schur–Weyl duality between cyclotomic Nazarov–Wenzl algebras and parabolic BGG categories $${\mathcal {O}}$$ associated to symplectic and orthogonal Lie algebras over the complex field $$\mathbb C$$ . This enables us to use standard arguments in (Anderson et al. in Pac J Math 292(1):21–59, 2018; Rui and Song in Math Zeit 280(3–4):669–689, 2015; Rui and Song in J Algebra 444:246–271, 2015), to compute decomposition matrices of cyclotomic Nazarov–Wenzl algebras. The level two case was considered by Ehrig and Stroppel in (Adv. Math. 331:58–142, 2018).

Webs and $q$-Howe dualities in types $\mathbf {BCD}$

We define web categories describing intertwiners for the orthogonal and symplectic Lie algebras, and, in the quantized setup, for certain orthogonal and symplectic coideal subalgebras. They generalize the Brauer category, and allow us to prove quantum versions of some classical type $\mathbf{B}\mathbf{C}\mathbf{D}$ Howe dualities.

Invariants of the special orthogonal group and an enhanced Brauer category

We first give a short intrinsic, diagrammatic proof of the First Fundamental Theorem of invariant theory (FFT) for the special orthogonal group SO _m(\mathbb C) , given the FFT for O _m(\mathbb C) . We then define, by means of a presentation with generators and relations, an enhanced Brauer category \widetilde{\mathcal B}(m) by adding a single generator to the usual Brauer category \mathcal B(m) , together with four relations. We prove that our category \widetilde{\mathcal B}(m) is actually (and remarkably) equivalent to the category of representations of SO _m generated by the natural representation. The FFT for SO _m amounts to the surjectivity of a certain functor \mathcal F on Hom spaces, while the Second Fundamental Theorem for SO _m says simply that \mathcal F is injective on Hom spaces. This theorem provides a diagrammatic means of computing the dimensions of spaces of homomorphisms between tensor modules for SO _m (for any m ).

On the definition of Heisenberg category

We revisit the definition of the Heisenberg category of central charge k∈ℤ. For central charge -1, this category was introduced originally by Khovanov, but with some additional cyclicity relations which we show here are unnecessary. For other negative central charges, the definition is due to Mackaay and Savage, also with some redundant relations, while central charge zero recovers the affine oriented Brauer category of Brundan, Comes, Nash and Reynolds. We also discuss cyclotomic quotients.

Rewriting in higher dimensional linear categories and application to the affine oriented Brauer category

In this paper, we introduce a rewriting theory for linear monoidal categories. Those categories are a particular case of linear (n,p)-categories that we define in this paper. We also define linear (n,p)-polygraphs, a linear adaptation of n-polygraphs, to present linear (n−1,p)-categories. We focus then on linear (3,2)-polygraphs to give presentations of linear monoidal categories. We finally give an application of this theory to prove a basis theorem on the category AOB. Our method uses decreasingness, a property introduced by van Ostroom.