[Paper Review] Representations of tame quivers and affine canonical bases

Zongzhu Lin, Jie Xiao+1 more|ArXiv.org|Jun 11, 2007

Algebraic structures and combinatorial models15 references3 citations

TL;DR

This paper constructs an integral PBW basis for the positive part of the quantized enveloping algebra of affine type A₁⁽¹⁾ using representations of the Kronecker quiver and geometric orderings from orbit and extension varieties. By ordering basis elements via dimension of orbit varieties and showing the transition matrix to a monomial basis is triangular with 1s on the diagonal, it realizes a bar-invariant basis, leading to an algebraic construction of the canonical basis for all symmetric affine Kac-Moody algebras.

ABSTRACT

An integral PBW-basis of type $A_1^{(1)}$ has been constructed by Zhang [Z] and Chen [C] using the Auslander-Reiten quiver of the Kronecker quiver. We associate a geometric order to elements in this basis following an idea of Lusztig [L1] in the case of finite type. This leads to an algebraic realization of a bar-invariant basis of $\uq2$. For any affine symmetric type, we obtain an integral PBW-basis of the generic composition algebra, by using an algebraic construction of the integral basis for a tube in [DDX], an embedding of the module category of the Kronecker quiver into the module category of the tame quiver, and a list of the root vectors of indecomposable modules according to the preprojective, regular, and preinjective components of the Auslander-Reiten quiver of the tame quiver. When the basis elements are ordered to be compatible with the geometric order given by the dimensions of the orbit varieties and the extension varieties, we can show that the transition matrix between the PBW-basis and a monomial basis is triangular with diagonal entries equal to 1. Therefore we obtain a bar-invariant basis. By a orthogonalization for the PBW-basis with the inner product, we finally give an algebraic way to realize the canonical bases of the quantized enveloping algebras of all symmetric affine Kac-Moody Lie algebras.

Motivation & Objective

• To provide an algebraic, representation-theoretic construction of the canonical basis for symmetric affine Kac-Moody algebras, avoiding geometric methods like perverse sheaves.
• To extend Lusztig’s finite-type approach—using geometric order and triangular transition matrices—to affine types via tame quiver representations.
• To resolve the lack of a longest Weyl group element in affine types by constructing a PBW basis compatible with geometric invariants such as orbit and extension variety dimensions.
• To establish a bar-invariant basis via orthogonalization of the PBW basis under a suitable inner product, yielding the canonical basis.

Proposed method

• Construct an integral PBW basis for type A₁⁽¹⁾ using the Auslander-Reiten quiver of the Kronecker quiver and Hall algebra techniques.
• Define a geometric order on basis elements by the dimension of orbit varieties associated to representations.
• Use the embedding of the Kronecker quiver’s module category into that of a general tame quiver to extend the construction to all affine symmetric types.
• Order the PBW basis elements so that the transition matrix to a monomial basis is upper triangular with diagonal entries equal to 1.
• Apply orthogonalization with respect to a canonical inner product to obtain a bar-invariant basis.
• Prove that the resulting basis satisfies the three defining properties of the canonical basis: invariance under bar involution, image under projection to v⁻¹ℚ[[v⁻¹]], and orthonormality modulo v⁻¹ℚ[[v⁻¹]] ∩ ℚ(v).

Experimental results

Research questions

• RQ1 Can a canonical basis for the positive part of a quantized affine enveloping algebra be constructed algebraically using quiver representations, without relying on intersection cohomology?
• RQ2 How can a PBW-type basis be ordered in affine type to ensure the transition matrix to a monomial basis is triangular with 1s on the diagonal?
• RQ3 Is it possible to realize the canonical basis via orthogonalization of a PBW basis in the context of tame quivers and Hall algebras?
• RQ4 Does the geometric order defined by orbit variety dimensions yield a compatible basis structure for constructing the canonical basis in affine type?
• RQ5 Can the construction be generalized to all symmetric affine Kac-Moody algebras using the tube components of the Auslander-Reiten quiver?

Key findings

• An integral PBW basis is constructed for the generic composition algebra of affine type A₁⁽¹⁾ using the Kronecker quiver and Hall algebra methods.
• The PBW basis elements are ordered by the dimension of orbit varieties, ensuring the transition matrix to the monomial basis is upper triangular with diagonal entries equal to 1.
• The orthogonalization of the PBW basis under the canonical inner product yields a bar-invariant basis, satisfying the three defining properties of the canonical basis.
• The resulting basis is shown to be the signed canonical basis in the sense of Lusztig, and thus the canonical basis of the quantized enveloping algebra.
• The construction provides a complete algebraic realization of the canonical basis for all symmetric affine Kac-Moody Lie algebras.
• The method resolves a question posed by Nakajima regarding the existence of such a canonical basis construction via algebraic means.