arXiv subjects

Deniz Kus

Publications and source records attributed to Deniz Kus.

30 records · Page 2Linked to original sources

arXiv2016

The PBW filtration and convex polytopes in type $\tt B$

We study the PBW filtration on irreducible finite--dimensional representations for the Lie algebra of type $\tt B_n$. We prove in several cases, including all multiples of the adjoint representation and all irreducible finite--dimensional representations for $\tt B_3$, that there exists a normal polytope such that the lattice points of this polytope parametrize a basis of the corresponding associated graded space. As a consequence we obtain several classes of favourable modules and graded combinatorial character formulas.

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arXiv2016

Demazure flags, $q$--Fibonacci polynomials and hypergeometric series

We study a family of finite--dimensional representations of the hyperspecial parabolic subalgebra of the twisted affine Lie algebra of type $\tt A_2^{(2)}$. We prove that these modules admit a decreasing filtration whose sections are isomorphic to stable Demazure modules in an integrable highest weight module of sufficiently large level. In particular, we show that any stable level $m'$ Demazure module admits a filtration by level $m$ Demazure modules for all $m\ge m'$. We define the graded and weighted generating functions which encode the multiplicity of a given Demazure module and establish a recursive formulae. In the case when $m'=1,2$ and $m=2,3$ we determine these generating functions completely and show that they define hypergeoemetric series and that they are related to the $q$--Fibonacci polynomials defined by Carlitz.

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arXiv2015

Twisted Demazure modules, fusion product decomposition and twisted Q--systems

In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the twisted current algebras. These modules are indexed by an $|R^+|$-tuple of partitions $\bxi=(ξ^α)_{α\in R^+}$ satisfying a natural compatibility condition. We give three equivalent presentations of these modules and show that for a particular choice of $\bxi$ these modules become isomorphic to Demazure modules in various levels for the twisted affine algebras. As a consequence we see that the defining relations of twisted Demazure modules can be greatly simplified. Furthermore, we investigate the notion of fusion products for twisted modules, first defined in \cite{FL99} for untwisted modules, and use the simplified presentation to prove a fusion product decomposition of twisted Demazure modules. As a consequence we prove that twisted Demazure modules can be obtained by taking the associated graded modules of (untwisted) Demazure modules for simply-laced affine algebras. Furthermore we give a semi-infinite fusion product construction for the irreducible representations of twisted affine algebras. Finally, we prove that the twisted $Q$-sytem defined in \cite{HKOTT02} extends to a non-canonical short exact sequence of fusion products of twisted Demazure modules.

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arXiv2015

Representations of Lie superalgebras with fusion flags

We study the category of finite--dimensional representations for a basic classical Lie superalgebra $\Lg=\Lg_0\oplus \Lg_1$. For the ortho--symplectic Lie superalgebra $\Lg=\mathfrak{osp}(1,2n)$ we show that certain objects in that category admit a fusion flag, i.e. a sequence of graded $\Lg_0[t]$--modules such that the successive quotients are isomorphic to fusion products. Among these objects we find fusion products of finite--dimensional irreducible $\Lg$--modules, truncated Weyl modules and Demazure type modules. Moreover, we establish a presentation for these types of fusion products in terms of generators and relations of the enveloping algebra.

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arXiv2014

Fusion products and toroidal algebras

We study the category of finite--dimensional bi--graded representations of toroidal current algebras associated to finite--dimensional complex simple Lie algebras. Using the theory of graded representations for current algebras, we construct in different ways objects in that category and prove them to be isomorphic. As a consequence we obtain generators and relations for certain types of fusion products including the $N$--fold fusion product of $V(λ)$. This result shows that the fusion product of these types is independent of the chosen parameters, proving a special case of a conjecture by Feigin and Loktev. Moreover, we prove a conjecture by Chari, Fourier and Sagaki on truncated Weyl modules for certain classes of dominant integral weights and show that they are realizable as fusion products. In the last section we consider the case $\mathfrak{g}=\mathfrak{sl}_2$ and compute a PBW type basis for truncated Weyl modules of the associated current algebra.

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arXiv2014

Weyl modules for the hyperspecial current algebra

We develop the theory of global and local Weyl modules for the hyperspecial maximal parabolic subalgebra of type $A_{2n}^{(2)}$. We prove that the dimension of a local Weyl module depends only on its highest weight, thus establishing a freeness result for global Weyl modules. Furthermore, we show that the graded local Weyl modules are level one Demazure modules for the corresponding affine Lie algebra. In the last section we derive the same results for the special maximal parabolic subalgebras of the twisted affine Lie algebras not of type $A_{2n}^{(2)}$.

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arXiv2013

Kirillov-Reshetikhin crystals, energy function and the combinatorial R-matrix

We study the polytope model for the affine type $A$ Kirillov-Reshetikhin crystals and prove that the action of the affine Kashiwara operators can be described in a remarkable simple way. Moreover, we investigate the combinatorial $R$-matrix on a tensor product of polytopes and characterize the map explicitly on the highest weight elements. We further give a formula for the local energy function and provide an alternative proof for the perfectness. We determine for any dominant highest weight element $Λ$ of level $\ell$ the elements $b_Λ, b^Λ$ involved in the definition of perfect crystals and give an explicit description of the ground-state path in the tensor product of polytopes.

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arXiv2013

Crystal bases as tuples of integer sequences

We describe a set $\mathcal{R}^{\infty}$ consisting of tuples of integer sequences and provide certain explicit maps on it. We show that this defines a semiregular crystal for $\mathfrak{sl}_{n+1}$ and $\mathfrak{sp}_{2n}$ respectively. Furthermore we define for any dominant integral weight $λ$ a connected subcrystal $\mathcal{R}(λ)$ in $\mathcal{R}^{\infty}$, such that this crystal is isomorphic to the crystal graph $B(λ)$. Finally we provide an explicit description of these connected crystals $\mathcal{R}(λ)$.

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arXiv2012

Realization of affine type A Kirillov-Reshetikhin crystals via polytopes

On the polytope defined in Feigin, Fourier, and Littelmann (2011), associated to any rectangle highest weight, we define a structure of an type $A_n$-crystal. We show, by using the Stembridge axioms, that this crystal is isomorphic to the one obtained from Kashiwara's crystal bases theory. Further we define on this polytope a bijective map and show that this map satisfies the properties of a weak promotion operator. This implies in particular that we provide an explicit realization of Kirillov-Reshetikhin crystals for the affine type $A^{(1)}_n$ via polytopes.

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arXiv2011

Local Weyl modules for equivariant map algebras with free abelian group actions

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional representations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras where g is semisimple, X is affine of finite type, and the group is abelian and acts freely on X. We do so by defining twisting and untwisting functors, which are isomorphisms between certain categories of representations of equivariant map algebras and their untwisted analogues. We also show that other properties of local Weyl modules (e.g. their characterization by homological properties and a tensor product property) extend to the more general setting considered in the current paper.

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arXiv2011

Demazure modules and Weyl modules: The twisted current case

We study finite-dimensional respresentations of twisted current algebras and show that any graded twisted Weyl module is isomorphic to level one Demazure module for the twisted affine Kac-Moody algebra. Using the tensor product property of Demazure modules, we obtain, by analyzing the fundamental Weyl modules, dimension and character formulas. Moreover we prove that graded twisted Weyl modules can be obtained by taking the associated graded modules of Weyl modules for the loop algebra, which implies that its dimension and classical character are independent of the support and depend only on its classical highest weight. These results were known before for untwisted current algebras and are new for all twisted types.

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arXiv2011

Weyl modules for multiloop algebras

Global and local Weyl modules for the untwisted multiloop Lie algebras were defined by Chari, the first and the second author via homological properties. In this paper we extended the ideas to give a categorical definition of the Weyl modules for twisted multiloop algebras. Our methods led us to describe an identification of the finite--dimensional highest weight modules for twisted multiloop algebras with suitably chosen finite--dimensional highest weight modules for untwisted multiloop algebras.

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