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Irfan Bagci

Publications and source records attributed to Irfan Bagci.

10 recordsLinked to original sources
arXiv2018

Weyl modules and Weyl functors for Lie superalgebras

Given an algebraically closed field $\Bbbk$ of characteristic zero, a Lie superalgebra $\mathfrak{g}$ over $\Bbbk$ and an associative, commutative $\Bbbk$-algebra $A$ with unit, a Lie superalgebra of the form $\mathfrak{g} \otimes_\Bbbk A$ is known as a map superalgebra. Map superalgebras generalize important classes of Lie superalgebras, such as, loop superalgebras (where $A=\Bbbk[t, t^{-1}]$), and current superalgebras (where $A=\Bbbk[t]$). In this paper, we define Weyl functors, global and local Weyl modules for all map superalgebras where $\mathfrak{g}$ is either $\mathfrak{sl} (n,n)$ with $n \ge 2$, or a finite-dimensional simple Lie superalgebra not of type $\mathfrak{q}(n)$. Under certain conditions on the triangular decomposition of these Lie superalgebras we prove that global and local Weyl modules satisfy certain universal and tensor product decomposition properties. We also give necessary and sufficient conditions for local (resp. global) Weyl modules to be finite dimensional (resp. finitely generated).

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arXiv2015

On the Structure of a quotient of the global Weyl module for the map superalgebra $\mathfrak{sl}(2,1)$

Let $A$ be a commutative, associative algebra with unity over $\mathbb{C}$. Using the definition of global Weyl modules for the map superalgebras given by Calixto, Lemay, and Savage we explicitly describe the structure of certain quotients of the global Weyl modules for the map superalgebra $\mathfrak{sl}(2,1)\otimes A$. We also give a nice basis for these modules. This work is an extension of a Theorem of Feigin and Loktev describing the structure of the Weyl module for the map algebra $\mathfrak{sl}_2\otimes A$. This work can naturally be extended to similar quotients of the global Weyl modules for $\mathfrak{sl}(n,m)\otimes A$.

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arXiv2013

Integral bases for the universal enveloping algebras of map superalgebras

Let $\mathfrak{g}$ be a finite dimensional complex simple classical Lie superalgebra and $A$ be a commutative, associative algebra with unity over $\mathbb{C}$. In this paper we define an integral form for the universal enveloping algebra of the map superalgebra $\mathfrak{g}\otimes A$, and exhibit an explicit integral basis for this integral form.

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arXiv2012

Whittaker categories and strongly typical Whittaker modules for Lie superalgebras

Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, a description of the strongly typical simple Whittaker modules.

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arXiv2011

Cohomology and support varieties for Lie superalgebras

Let $mathfrak{g}$ be a restricted Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$. Let $\mathfrak{u}(\mathfrak{g})$ denote the restricted enveloping algebra of $\mathfrak{g}$. In this paper we prove that the cohomology ring $\HH^\bullet(\fu(\fg), k)$ is finitely generated. This allows one to define support varieties for finite dimensional $\fu(\fg)$-supermodules. We also show that support varieties for finite dimensional $\fu(\fg)$ supermodules satisfy the desirable properties of support variety theory.

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arXiv2010

On cohomology and support varieties for Lie superalgebras

Support varieties for Lie superalgebras over the complex numbers were introduced in \cite{BKN1} using the relative cohomology. In this paper we discuss finite generation of the relative cohomology rings for Lie superalgebras, we formulate a definition for subalgebras which detect the cohomology, also discuss realizability of support varieties. In the last section as an application we compute the relative cohomology ring of the Lie superalgebra $\overline{S}(n)$ relative to the graded zero component $\overline{S}(n)_0$ and show that this ring is finitely generated. We also compute support varieties of all simple modules in the category of finite dimensional $\overline{S}(n)$-modules which are completely reducible over $\overline{S}(n)_0$.

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arXiv2008

Cohomology of quantum groups: An analog of Kostant's Theorem

We prove the analog of Kostant's Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostant's cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra $\mathfrak{g} = \mathfrak{sl}(n)$. We also show that Kostant's formula holds when $q$ is specialized to an $\ell$-th root of unity for odd $\ell \ge h-1$ (where $h$ is the Coxeter number of $\mathfrak{g}$) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an extension of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.

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arXiv2008

Cohomology and Support Varieties for Lie Superalgebras of Type W(n)

Boe, Kujawa and Nakano recently investigated relative cohomology for classical Lie superalgebras and developed a theory of support varieties. The dimensions of these support varieties give a geometric interpretation of the combinatorial notions of defect and atypicality due to Kac, Wakimoto, and Serganova. In this paper we calculate the cohomology ring of the Cartan type Lie superalgebra W(n) relative to the degree zero component W(n)_{0} and show that this ring is a finitely generated polynomial ring. This allows one to define support varieties for finite dimensional W(n)-supermodules which are completely reducible over W(n)_{0}. We calculate the support varieties of all simple supermodules in this category. Remarkably our computations coincide with the prior notion of atypicality for Cartan type superalgebras due to Serganova. We also present new results on the realizability of support varieties which hold for both classical and Cartan type superalgebras.

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