arXiv subjects

Jacob Greenstein

Publications and source records attributed to Jacob Greenstein.

25 records · Page 2Linked to original sources

arXiv2007

Current algebras, highest weight categories and quivers

We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions between simple objects. The simple objects in the category are parametized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on affine the weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the injective cogenerator of the subcategory associated to a interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and tame type that arise from our study. We also prove that the path algebra of star shaped quivers are the Ext algebra of a suitable subcategory.

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arXiv2006

Graded level zero integrable representations of affine Lie algebras

We study the structure of the category of integrable level zero representations with finite dimensional weight spaces of affine Lie algebras. We show that this category possesses a weaker version of the finite length property, namely that an indecomposable object has finitely many simple constituents which are non-trivial as modules over the corresponding loop algebra. Moreover, any object in this category is a direct sum of indecomposables only finitely many of which are non-trivial. We obtain a parametrization of blocks in this category.

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arXiv2005

An application of free Lie algebras to current algebras and their representation theory

We realize the current algebra of a Kac-Moody algebra as a quotient of a semi-direct product of the Kac-Moody Lie algebra and the free Lie algebra of the Kac-Moody algebra. We use this realization to study the representations of the current alg ebra. In particular we see that every ad-invariant ideal in the symmetric algebra of the Kac-Moody algebra gives rise in a canonical way to a representation of the current algebra. These representations include certain well-known families of representations of the current algebra of a simple Lie algebra. Another family of examples, which are the classical limits of the Kirillov-Reshe tikhin modules, are also obtained explicitly by using a construction of Kostant. Finally we study extensi ons in the category of finite dimensional modules of the current algebra of a simple Lie algebra.

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arXiv2003

Filtrations and completions of certain positive level modules of affine algebras

We define a filtration indexed by the integers on the tensor product of an integrable highest weight module and a loop module for a quantum affine algebra. We prove that the filtration is either trivial or strictly decreasing and give sufficient conditions for this to happen. In the first case we prove that the module is irreducible and in the second case we prove that the intersection of all the modules is zero, thus allowing us to define the completed tensor product. In certain special cases, we identify the subsequent quotients of filtration. These are certain highest weight integrable modules and the multiplicity and the highest weight are the same as that obtained by decomposing the tensor product of the highest weight crystal bases with the crystal bases of a loop module.

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arXiv2003

Path model for quantum loop modules of fundamental type

For a certain class of simple integrable modules of level zero over a quantised affine algebra, we establish the existence of a pseudo-crystal basis and show that such a basis admits a combinatorial realisation in the framework of the path model of Littelmann.

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arXiv2002

Littelmann's path crystal and combinatorics of certain sl_{l+1}^ modules of level zero

We construct a subcrystal of the Littelmann's path crystal whose formal character coincides with that of a certain simple integrable module of level zero over the untwisted affine Lie algebra associated to sl_n. We also establish an analogue of the Littlewood-Richardson rule for the tensor product of that crystal with a highest weight crystal.

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arXiv2002

Quantum loop modules

We classify the simple infinite dimensional integrable modules with finite dimensional weight spaces over the quantized enveloping algebra of an untwisted affine algebra. We prove that these are either highest (lowest) weight integrable modules or simple submodules of a loop module of a finite-dimensional simple integrable module and describe the latter class. Their characters and crystal bases theory are discussed in a special case.

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