[Paper Review] On representations of Cartan map Lie superalgebras

Irfan Bagci|arXiv (Cornell University)|Aug 4, 2015

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper classifies finite-dimensional irreducible representations of Cartan map Lie superalgebras $χ\otimes A$ and their twisted counterparts $(χ\otimes A)^T$, where $χ$ is a simple Cartan-type Lie superalgebra and $A$ is a finitely generated commutative algebra over $\mathbb{C}$. It introduces Generalized Kac Modules as induced modules from the even part $\mathfrak{g}_0\otimes A$, showing all irreducible representations arise as their irreducible quotients, and proves that for twisted algebras under free abelian group actions, all irreducible representations are restrictions of those from the untwisted algebra.

ABSTRACT

This paper describes finite dimensionall irreducible representations of both twisted and untwisted Cartan map Lie superalgebras.

Motivation & Objective

• To classify finite-dimensional irreducible representations of Lie superalgebras of the form $\mathfrak{g}\otimes A$, where $\mathfrak{g}$ is a simple Cartan-type Lie superalgebra and $A$ is a finitely generated commutative algebra over $\mathbb{C}$.
• To extend this classification to twisted Cartan map Lie superalgebras $(\mathfrak{g}\otimes A)^T$, where a finite abelian group $T$ acts on both $\mathfrak{g}$ and $A$ by automorphisms.
• To establish that all irreducible finite-dimensional representations of $(\mathfrak{g}\otimes A)^T$ arise as restrictions of irreducible representations of $\mathfrak{g}\otimes A$ under the assumption that $T$ acts freely on $\operatorname{MaxSpec}(A)$.
• To provide a uniform framework using Generalized Kac Modules for constructing and analyzing irreducible representations in the Cartan-type setting.

Proposed method

• Introduce Generalized Kac Modules as induced modules from the even part $\mathfrak{g}_0\otimes A$, generalizing Kac’s original construction for $\mathfrak{g}$.
• Use the ${\mathbb{Z}}$-grading of $\mathfrak{g}$ to define $\mathfrak{g}_0$ as the degree-zero component, which is a simple Lie algebra.
• Construct representations via evaluation maps $\varphi_{\mathfrak{m}_i^{n_i}}(\pi_i)$, where $\pi_i$ are irreducible representations of $\mathfrak{g}$ and $\mathfrak{m}_i$ are maximal ideals of $A$.
• Apply group action techniques: decompose $\mathfrak{g}$ and $A$ into isotypic components under the action of a finite abelian group $T$.
• Use the isomorphism $\operatorname{MaxSpec}(A)^T \cong \operatorname{MaxSpec}(A)/T$ under free action to classify orbits of maximal ideals.
• Prove that irreducible representations of $(\mathfrak{g}\otimes A)^T$ are restrictions of those of $\mathfrak{g}\otimes A$, using equivariance and character group analysis.

Experimental results

Research questions

• RQ1 How can finite-dimensional irreducible representations of $\mathfrak{g}\otimes A$ be classified when $\mathfrak{g}$ is a simple Cartan-type Lie superalgebra?
• RQ2 What is the role of the even subalgebra $\mathfrak{g}_0\otimes A$ in constructing irreducible representations of $\mathfrak{g}\otimes A$?
• RQ3 Under what conditions do all irreducible representations of the twisted algebra $(\mathfrak{g}\otimes A)^T$ arise as restrictions of representations of $\mathfrak{g}\otimes A$?
• RQ4 How does the action of a finite abelian group $T$ on $\operatorname{MaxSpec}(A)$ affect the classification of irreducible representations of $(\mathfrak{g}\otimes A)^T$?

Key findings

• All finite-dimensional irreducible representations of $\mathfrak{g}\otimes A$ are irreducible quotients of Generalized Kac Modules induced from $\mathfrak{g}_0\otimes A$.
• For $\mathfrak{g} = S(n), \tilde{S}(n), H(n)$, all finite-dimensional irreducible representations of $\mathfrak{g}\otimes A$ are tensor products of evaluation representations.
• For $\mathfrak{g} = W(n)$, irreducible finite-dimensional representations of $\mathfrak{g}\otimes A$ are parametrized by irreducible finite-dimensional representations of $\mathfrak{g}_0\otimes A$ with finite support.
• Every finite-dimensional irreducible representation of the twisted Cartan map Lie superalgebra $(\mathfrak{g}\otimes A)^T$ is isomorphic to the restriction of an irreducible representation of $\mathfrak{g}\otimes A$.
• When $T$ acts freely on $\operatorname{MaxSpec}(A)$, irreducible representations of $(\mathfrak{g}\otimes A)^T$ are parametrized by tuples of irreducible representations $\pi_i$ and distinct maximal ideals $\mathfrak{m}_i$ lying in distinct $T$-orbits.
• The isomorphism class of a representation $\varphi_{\mathfrak{m}_i^{n_i}}^T(\pi_i)$ is preserved under $T$-action, allowing reduction to representatives from distinct orbits.