[Paper Review] Finite-dimensional algebras and quivers

Alistair Savage|ArXiv.org|May 5, 2005

Algebraic structures and combinatorial models12 references3 citations

TL;DR

This paper provides a comprehensive overview of finite-dimensional algebras and quivers, focusing on path algebras, Ringel-Hall algebras, and quiver varieties of Lusztig and Nakajima. It establishes a geometric realization of irreducible highest weight representations of Kac-Moody algebras via constructible functions on quiver varieties, showing that irreducible components of these varieties form a crystal basis under a geometrically defined crystal structure.

ABSTRACT

This is an overview article on finite-dimensional algebras and quivers, written for the Encyclopedia of Mathematical Physics. We cover path algebras, Ringel-Hall algebras and the quiver varieties of Lusztig and Nakajima.

Motivation & Objective

• To provide a unified overview of finite-dimensional algebras, quivers, and their representation theory for mathematical physics.
• To establish a connection between quiver varieties and the representation theory of Kac-Moody and quantum affine Lie algebras.
• To demonstrate how constructible functions on quiver varieties yield canonical bases and crystal graphs.
• To clarify the role of Ringel-Hall algebras and Nakajima's quiver varieties in realizing integrable representations.
• To present a geometric construction of canonical bases using irreducible components of quiver varieties.

Proposed method

• Construct the path algebra $kQ$ from a quiver $Q$, with basis given by paths and multiplication via concatenation.
• Define representations of quivers as assignments of vector spaces to vertices and linear maps to arrows, forming a category equivalent to $kQ$-modules.
• Introduce quiver varieties $\mathcal{L}(\mathbf{v},\mathbf{w})$ as moduli spaces of stable representations, with a $G_{\mathbf{V}}$-action and a $G_{\mathbf{V}}$-invariant Lagrangian structure.
• Define the space $M(\mathcal{L}(\mathbf{v},\mathbf{w}))$ of constructible functions on $\mathcal{L}(\mathbf{v},\mathbf{w})$, equipped with operators $e_i$, $f_i$, $h_i$ via proper pushforwards and pullbacks.
• Construct the irreducible highest weight representation $L(\mathbf{w})$ as the $G_{\mathbf{V}}$-invariant subspace generated by the constant function $\varphi$ on $\mathcal{L}(\mathbf{0},\mathbf{w})$.
• Establish an isomorphism $\Phi: L(\mathbf{v},\mathbf{w}) \to \mathbb{C}^{\operatorname{Irr}\mathcal{L}(\mathbf{v},\mathbf{w})}$, showing that the functions $g_Z$ associated to irreducible components $Z$ form a basis of $L(\mathbf{v},\mathbf{w})$.

Experimental results

Research questions

• RQ1 How can quiver varieties be used to geometrically realize integrable highest weight representations of Kac-Moody algebras?
• RQ2 What is the role of constructible functions and their operators $e_i$, $f_i$, $h_i$ in constructing canonical bases?
• RQ3 How do irreducible components of quiver varieties relate to crystal graphs and the structure of representations?
• RQ4 In what way do Nakajima's quiver varieties provide a geometric construction of canonical bases?
• RQ5 What is the precise relationship between the geometry of $\mathcal{L}(\mathbf{v},\mathbf{w})$ and the representation theory of quantum affine Lie algebras?

Key findings

• The space $L(\mathbf{w})$ of functions generated by acting on the constant function $\varphi$ with $f_i$ operators carries the structure of an irreducible highest weight integrable representation of $\mathfrak{g}$ with highest weight $\sum_{i\in Q_0} \mathbf{w}_i \omega_i$.
• The weight space decomposition of $L(\mathbf{w})$ is given by $L(\mathbf{v},\mathbf{w})$, with weight $\sum_{i\in Q_0} \mathbf{w}_i \omega_i - \mathbf{v}_i \alpha_i$.
• The map $\Phi: L(\mathbf{v},\mathbf{w}) \to \mathbb{C}^{\operatorname{Irr}\mathcal{L}(\mathbf{v},\mathbf{w})}$ is an isomorphism, showing that the functions $g_Z$ associated to irreducible components $Z$ form a basis of $L(\mathbf{v},\mathbf{w})$.
• Each basis function $g_Z$ is constant 1 on a dense open subset of $Z$ and vanishes outside $Z \cup K$ for some closed $G_{\mathbf{V}}$-invariant subset $K$ of lower dimension.
• The operators $e_i$, $f_i$, $h_i$ on $L(\mathbf{w})$ satisfy the defining relations of the Kac-Moody algebra $\mathfrak{g}$, endowing $L(\mathbf{w})$ with a $\mathfrak{g}$-module structure.
• The irreducible components of $\mathcal{L}(\mathbf{v},\mathbf{w})$ naturally carry the structure of a crystal graph, with $e_i$ and $f_i$ acting via geometric operations on the components.