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Ievgen Makedonskyi

Publications and source records attributed to Ievgen Makedonskyi.

27 records · Page 2Linked to original sources
arXiv2017

Generalized Weyl modules, alcove paths and Macdonald polynomials

Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr-Shimozono formula in order to prove that the $t=\infty$ specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.

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arXiv2016

Generalized Weyl modules for twisted current algebras

We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and connection to the theory of nonsymmetric Macdonald polynomials. As an application we compute the dimension of the classical Weyl modules in the remaining unknown case.

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arXiv2016

Generalized Weyl modules and nonsymmetric $q$-Whittaker functions

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric $q$-Whittaker functions. In particular, we show that the series part of the nonsymmetric $q$-Whittaker function is a generating function for the graded characters of generalized global Weyl modules.

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arXiv2015

Weyl modules for $\mathfrak{osp}(1,2)$ and nonsymmetric Macdonald polynomials

The main goal of our paper is to establish a connection between the Weyl modules of the current Lie superalgebras (twisted and untwisted) attached to $\mathfrak{osp}(1,2)$ and the nonsymmetric Macdonald polynomials of types $A_2^{(2)}$ and $A_2^{(2)\dagger}$. We compute the dimensions and construct bases of the Weyl modules. We also derive explicit formulas for the t=0 and t=\infty specializations of the nonsymmetric Macdonald polynomials. We show that the specializations can be described in terms of the Lie superalgebras action on the Weyl modules.

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arXiv2013

On finite-dimensional subalgebras of derivation Lie algebras

Let $\mathbb{K}$ be a field, $R$ be an associative and commutative $\mathbb{K}$-algebra and $L$ be a Lie algebra over $\mathbb{K}$. We give some descriptions of injections from $L$ to Lie algebra of $\mathbb{K}$-derivations of $R$ in the terms of $L$.

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arXiv2012

On wild Lie algebras

We give a criterion of tameness and wildness for a finite-dimensional Lie algebra over an algebraically closed field.

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arXiv2011

On noncommutative bases of the free module $W_n(\mathbb K)$

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $R=\mathbb{K}[x_1,x_2,...x_n]$ the polynomial ring in $n$ variables over $\mathbb K.$ We study bases of the free $R$-module $W_n(\mathbb{K})$ of all $\mathbb{K}$-derivations of the ring $R$, such that their linear span over $\mathbb K$ is a subalgebra of the Lie algebra $W_n(\mathbb{K})$. We proved that for any Lie algebra $L$ of dimension $n$ over $\mathbb{K}$ there exists a subalgebra $\bar{L}$ of $W_n(\mathbb{K})$ which is isomorphic to $L$ and such that every $\mathbb{K}$-basis of $\bar L$ is an $R$-basis of the $R$-module $W_n(\mathbb{K})$.

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