The Composition Algebra of a Cyclic Quiver

Eisenstein series and quantum affine algebras

“…These results can be viewed as "continual" analogs of results of Ringel [38][39][40] and Lusztig [31][32][33] on realizing quantum Kac-Moody algebras by means of Hall algebras associated with the category of representations of a quiver, instead of the category of coherent sheaves on a curve. The two types of categories have very similar properties; in particular, they have homological dimension 1.…”

“…It is instructive to compare Theorem (5.2.1) with the results of Rintgel [38][39][40] and Lusztig [31][32][33] on representations of quivers. Namely, let F be a quiver, i.e., a finite oriented graph without edgesloops.…”

Generic extensions and composition monoids of cyclic quivers

“…We now recall from [20,22] the definition of (generic) Ringel-Hall and composition algebras of △ n and make a comparison of the monoid algebra ZM and its subalgebra ZM c with the degenerate Ringel-Hall algebra H 0 and the degenerate composition algebra C 0 of △ n . We also deduce some relations in the degenerate composition algebra C 0 .…”

“…It is known from [20,22] that the u i = u [S i ] satisfy the so-called fundamental relations u i u j = u j u i if i, j ∈ I and j = i ± 1,…”

Uhlenbeck Spaces for $\mathbb A^2$ and Affine Lie Algebra $\widehat{\mathfrak{sl}}_n$

“…We begin with some notation. For On the other hand, Ringel's explicit description of the order κ ≤ κ in [22], 4.7 implies that we must have K(κ) ≥ K(κ ) whenever κ ≤ κ . The Lemma follows.…”