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R. B. Zhang

Publications and source records attributed to R. B. Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Quantum groups of Lie colour algebras fulfilling the Cartan-Weyl paradigm

Let Gamma be an additive abelian group with a commutative factor omega. We describe the simple and untwisted affine Lie colour algebras which admit a Cartan - Weyl description. For both classes, we construct the quantised universal enveloping colour algebras, and establish their quasi-triangular Hopf colour algebraic structure. These colour quantum groups generalise the Drinfeld - Jimbo quantum groups, which are recovered when Gamma is trivial. They provide an algebraic foundation for investigations in knot theory and soluble models in statistical mechanics.

math.QA↗
arXiv2026

Invariants and representations of the $Γ$-graded general linear Lie $ω$-algebras

There is considerable current interest in applications of generalised Lie algebras graded by an abelian group $Γ$ with a commutative factor $ω$. This calls for a systematic development of the theory of such algebraic structures. We treat the representation theory and invariant theory of the $Γ$-graded general linear Lie $ω$-algebra $\mathfrak{gl}(V(Γ, ω))$, where $V(Γ, ω)$ is any finite dimensional $Γ$-graded vector space. Generalised Howe dualities over symmetric $(Γ, ω)$-algebras are established, from which we derive the first and second fundamental theorems of invariant theory, and a generalised Schur-Weyl duality. The unitarisable $\mathfrak{gl}(V(Γ, ω))$-modules for two ``compact'' $\ast$-structures are classified, and it is shown that the tensor powers of $V(Γ, ω)$ and their duals are unitarisable for the two compact $\ast$-structures respectively. A Hopf $(Γ, ω)$-algebra is constructed, which gives rise to a group functor corresponding to the general linear group in the $Γ$-graded setting. Using this Hopf $(Γ, ω)$-algebra, we realise simple tensor modules and their dual modules by mimicking the classic Borel-Weil theorem. We also analyse in some detail the case with $Γ={\mathbb Z}^{\dim{V(Γ, ω)}}$ and $ω$ depending on a complex parameter $q\ne 0$, where $\mathfrak{gl}(V(Γ, ω))$ shares common features with the quantum general linear (super)group, but is better behaved especially when $q$ is a root of unity.

math.RT↗
arXiv2023

A polar Brauer category and Lie superalgebra representations

We introduce a diagram category, study its structure, and investigate some of its applications to the representation theory of Lie algebras and Lie superalgebras. The morphisms of the category, which contains a subcategory isomorphic to the Brauer category, are linear combinations of `polar enhancements' of Brauer diagrams. The endomorphism algebra of each of its objects is a quotient of an algebra of chord diagrams. Analogues of the affine Temperley-Lieb category and Temperley-Lieb category of type B, whose structures are thoroughly understood, arise from particular quotients of our category. We construct a functor from our category to the full subcategory of modules for the Lie superalgebra $\mathfrak{osp}(V; ω)$ with objects $M\otimes V^{\otimes r}$ for all $r=0, 1, \dots$, where $M$ is an arbitrary module, and $V$ is the natural module. When $M$ is the universal enveloping superalgebra $\text{U}(\mathfrak{osp}(V; ω))$, this functor provides an effective tool for the study of $\text{U}(\mathfrak{osp}(V; ω))$. An analysis of this functor leads to a diagrammatic construction of explicit generators for the centre of the universal enveloping superalgebra and, in the special cases when $V$ is purely even or purely odd (i.e. the classical cases), categorical interpretations of certain widely studied ``characteristic identities'' of the orthogonal and symplectic Lie algebras. In the case $V=\mathbb{C}^{0|2}$ so that $\mathfrak{osp}(V; ω))=\mathfrak{sp}_2(\mathbb{C})$, we prove that our type B Temperley-Lieb category is isomorphic to a full subcategory of category $\mathcal O$ for $\mathfrak{sp}_2(\mathbb{C})$.

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arXiv2022

Diagram categories and invariant theory for classical groups and supergroups

We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and super groups, with some indications concerning extensions to quantum groups and quantum super groups. Tensor functors from various diagram categories to categories of representnations are introduced and their properties investigated, leading to first and second fundamental theorems of invariant theory for classical super groups, which include the classical groups as special cases. Application of diagrammatic methods enables the constructionof a presentation for endomorphism algebras for te orthogonal and symplectic groups, leading to the solution ofproblems raised by the work of Brauer and Weyl.

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arXiv2019

Mixed cohomology of Lie superalgebras

We investigate a new cohomology of Lie superalgebras, which may be compared to a de Rham cohomology of Lie supergroups involving both differential and integral forms. It is defined by a BRST complex of Lie superalgebra modules, which is formulated in terms of a Weyl superalgebra and incorporates inequivalent representations of the bosonic Weyl subalgebra. The new cohomology includes the standard Lie superalgebra cohomology as a special case. Examples of new cohomology groups are computed.

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arXiv2017

Temperley-Lieb at roots of unity, a fusion category and the Jones quotient

When the parameter $q$ is a root of unity, the Temperley-Lieb algebra $TL_n(q)$ is non-semisimple for almost all $n$. In this work, using cellular methods, we give explicit generating functions for the dimensions of all the simple $TL_n(q)$-modules. Jones showed that if the order $|q^2|=\ell$ there is a canonical symmetric bilinear form on $TL_n(q)$, whose radical $R_n(q)$ is generated by a certain idempotent $E_\ell\in TL_{\ell-1}(q)\subseteq TL_n(q)$, which is now referred to as the Jones-Wenzl idempotent, for which an explicit formula was subsequently given by Graham and Lehrer. Although the algebras $Q_n(\ell):=TL_n(q)/R_n(q)$, which we refer to as the Jones algebras (or quotients), are not the largest semisimple quotients of the $TL_n(q)$, our results include dimension formulae for all the simple $Q_n(\ell)$-modules. This work could therefore be thought of as generalising that of Jones et al. on the algebras $Q_n(\ell)$. We also treat a fusion category $\mathcal{C}_{\rm red}$ introduced by Reshitikhin, Turaev and Andersen, whose objects are the quantum $\mathfrak{sl}_2$-tilting modules with non-zero quantum dimension, and which has an associative truncated tensor product (the fusion product). We show $Q_n(\ell)$ is the endomorphism algebra of a certain module in $\mathcal{C}_{\rm red}$ and use this fact to recover a dimension formula for $Q_n(\ell)$. We also show how to construct a "stable limit" $K(Q_\infty)$ of the corresponding fusion category of the $Q_n(\ell)$, whose structure is determined by the fusion rule of $\mathcal{C}_{\rm red}$, and observe a connection with a fusion category of affine $\mathfrak{sl}_2$ and the Virosoro algebra.

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arXiv2017

Quantum correspondences of affine Lie superalgebras

There is a surprising isomorphism between the quantised universal enveloping algebras of osp(1|2n) and so(2n+1). This same isomorphism emerged in recent work of Mikhaylov and Witten in the context of string theory as a T-duality composed with an S-duality. We construct similar Hopf superalgebra isomorphisms for families of pairs of quantum affine superalgebras. An immediate consequence is that the representation categories of the quantum affine superalgebras in each pair are equivalent as strict tensor categories.

math.QA↗
arXiv2017

The Jones quotients of the Temperley-Lieb algebras

When the parameter $q$ is a root of unity, the Temperley-Lieb algebra $TL_n(q)$ is non-semisimple for almost all $n$. Jones showed that there is a canonical symmetric bilinear form on $TL_n(q)$, whose radical $R_n(q)$ is generated by a certain idempotent $E_\ell\in TL_{\ell-1}(q)\subseteq TL_n(q)$, which is now referred to as the Jones-Wenzl idempotent, for which an explicit formula was subsequently given by Graham and Lehrer. In this work, we study the quotients $Q_n(\ell):=TL_n(q)/R_n(q)$, where $|q^2|=\ell$, which are precisely the algebras generated by Jones' projections. We give the dimensions of their simple modules, as well as $\dim(Q_n(\ell))$; en route we give generating functions and recursions for the dimensions of cell modules and associated combinatorics. When the order $|q^2|=4$, we obtain an isomorphism of $Q_n(\ell)$ with the even part of the Clifford algebra, well known to physicists through the Ising model. When $|q^2|=5$, we obtain a sequence of algebras whose dimensions are the odd-indexed Fibonacci numbers. The general case is described explicitly.

math.QA↗
arXiv2016

First fundamental theorems of invariant theory for quantum supergroups

Let $U_q(\mathfrak{g})$ be the quantum supergroup of $\mathfrak{gl}_{m|n}$ or the modified quantum supergroup of $osp_{m|2n}$ over the field of rational functions in $q$, and let $V_q$ be the natural module for $U_q(\mathfrak{g})$. There exists a unique tensor functor, associated with $V_q$, from the category of ribbon graphs to the category of finite dimensional representations of $U_q(\mathfrak{g}$, which preserves ribbon category structures. We show that this functor is full in the cases $\mathfrak{g}=\mathfrak{gl}_{m|n}$ or $osp_{2\ell+1|2n}$. For $\mathfrak{g}=osp_{2\ell|2n}$, we show that the space $Hom_{U_q(\mathfrak{g}}(V_q^{\otimes r}, V_q^{\otimes s})$ is spanned by images of ribbon graphs if $r+s< 2\ell(2n+1)$. The proofs involve an equivalence of module categories for two versions of the quantisation of $U(\mathfrak{g})$.

math.QA↗
arXiv2015

The first fundamental theorem of invariant theory for the orthosymplectic super group

We give a new proof, inspired by an argument of Atiyah, Bott and Patodi, of the first fundamental theorem of invariant theory for the orthosymplectic super group. We treat in a similar way the case of the periplectic super group. Lastly, the same method is used to explain the fact that Sergeev's super Pfaffian, an invariant for the special orthosymplectic super group, is polynomial.

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arXiv2015

Invariants of the orthosymplectic Lie superalgebra and super Pfaffians

Given a complex orthosymplectic superspace $V$, the orthosymplectic Lie superalgebra $\mathfrak {osp}(V)$ and general linear algebra ${\mathfrak {gl}}_N$ both act naturally on the coordinate super-ring $\mathcal{S}(N)$ of the dual space of $V\otimes{\mathbb C}^N$, and their actions commute. Hence the subalgebra $\mathcal{S}(N)^{\mathfrak {osp}(V)}$ of $\mathfrak {osp}(V)$-invariants in $\mathcal{S}(N)$ has a ${\mathfrak {gl}}_N$-module structure. We introduce the space of super Pfaffians as a simple ${\mathfrak {gl}}_N$-submodule of $\mathcal{S}(N)^{\mathfrak {osp}(V)}$, give an explicit formula for its highest weight vector, and show that the super Pfaffians and the elementary (or `Brauer') ${\rm OSp}$-invariants together generate $\mathcal{S}(N)^{\mathfrak {osp}(V)}$ as an algebra. The decomposition of $\mathcal{S}(N)^{\mathfrak {osp}(V)}$ as a direct sum of simple ${\mathfrak {gl}}_N$-submodules is obtained and shown to be multiplicity free. Using Howe's $({\mathfrak {gl}}(V), {\mathfrak {gl}}_N)$-duality on $\mathcal{S}(N)$, we deduce from the decomposition that the subspace of $\mathfrak{osp}(V)$-invariants in any simple ${\mathfrak {gl}}(V)$-tensor module is either $0$ or $1$-dimensional. These results also enable us to determine the $\mathfrak {osp}(V)$-invariants in the tensor powers $V^{\otimes r}$ for all $r$.

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arXiv2014

Integrable representations of the quantum affine special linear superalgebra

The simple integrable modules with finite dimensional weight spaces are classified for the quantum affine special linear superalgebra $\U_q(\hat{\mathfrak{sl}}(M|N))$ at generic $q$. Any such module is shown to be a highest weight or lowest weight module with respect to one of the two natural triangular decompositions of the quantum affine superalgebra depending on whether the level of the module is zero or not. Furthermore, integrable $\U_q(\hat{\mathfrak{sl}}(M|N))$-modules at nonzero levels exist only if $M$ or $N$ is $1$.

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arXiv2014

The second fundamental theorem of invariant theory for the orthosymplectic supergroup

In a previous work we established a super Schur-Weyl-Brauer duality between the orthosymplectic supergroup of superdimension $(m|2n)$ and the Brauer algebra with parameter $m-2n$. This led to a proof of the first fundamental theorem of invariant theory, using some elementary algebraic supergeometry, and based upon an idea of Atiyah. In this work we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. The proof uses algebraic supergeometry to reduce the problem to the case of the general linear supergroup, which is understood. The main result has a succinct formulation in terms of Brauer diagrams. Our proof includes new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues. These new proofs are independent of the Capelli identities, which are replaced by algebraic geometric arguments.

math.RT↗
arXiv2014

Integrable representations of affine A(m, n) and C(m) superalgebras

Rao and Zhao classified the irreducible integrable modules with finite dimensional weight spaces for the untwisted affine superalgebras which are not $\hat{A}(m,n)$ ($m\ne n$) or $\hat{C}(m)$. Here we treat the latter affine superalgebras to complete the classification. The problem boils down to classifying the irreducible zero-level integrable modules with finite dimensional weight spaces for these affine superalgebras, which is solved in this paper. We note in particular that such modules for $\hat{A}(m,n)$ ($m\ne n$) and $\hat{C}(m)$ must be of highest weight type, but are not necessarily loop modules. This is in sharp contrast to the cases of ordinary affine algebras and the other types of affine superalgebras.

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arXiv2013

Generalised Jantzen filtration of Lie superalgebras II: the exceptional cases

Let $g$ be an exceptional Lie superalgebra, and let $p$ be the maximal parabolic subalgebra which contains the distinguished Borel subalgebra and has a purely even Levi subalgebra. For any parabolic Verma module in the parabolic category $O^p$, it is shown that the Jantzen filtration is the unique Loewy filtration, and the decomposition numbers of the layers of the filtration are determined by the coefficients of inverse Kazhdan-Lusztig polynomials. An explicit description of the submodule lattices of the parabolic Verma modules is given, and formulae for characters and dimensions of the finite dimensional simple modules are obtained.

math.RT↗
arXiv2013

Cellularity of certain quantum endomorphism algebras

We exhibit for all positive integers r, an explicit cellular structure for the endomorphism algebra of the r'th tensor power of an integral form of the Weyl module with highest weight d of the quantised enveloping algebra of sl2. When q is specialised to a root of unity of order bigger than d, we consider the corresponding specialisation of the tensor power. We prove one general result which gives sufficient conditions for the commutativity of specialisation with the taking of endomorphism algebras, and another which relates the multiplicities of indecomposable summands to the dimensions of simple modules for an endomorphism algebra. Our cellularity result then allows us to prove that knowledge of the dimensions of the simple modules of the specialised cellular algebra above is equivalent to knowledge of the weight multiplicities of the tilting modules for the specialised quantum group. In the final section we independently determine the weight multiplicities of indecomposable tilting modules for quantum sl2, and the decomposition numbers of the endomorphism algebras. We indicate how either one of these sets of numbers determines the other.

math.GR↗