Skip to main content
DA / EN

Gæsteforelæsning Seminar

28.10.2026   at 15:00 - 16:00

QM Research Seminar: Hall algebras and complete realisations of quantum algebras

<strong>Speaker: Pallav Goyal&nbsp;</strong>(​University of Southern Denmark).<br><br><strong>Abstract:&nbsp;<br></strong>Ringel used the Hall algebra of a quiver to construct the positive half of the quantised enveloping algebra of the corresponding Kac–Moody algebra, and Lusztig’s geometric version of this construction produced canonical bases of these positive halves having remarkable positivity properties. A limitation of this approach is that it only yields the positive half of the algebra. <br><br>​​​​​​​This phenomenon is ubiquitous in the field: ‘halves’ of many non-commutative algebras (quantum groups, toroidal algebras, DAHAs, etc.) have been obtained as Hall algebras, but examples of ‘entire’ algebras being realised intrinsically are rare. I will survey some previous work on this problem and then describe joint work with Peter Samuelson giving complete Hall-theoretic realisations of several classes of quantum algebras, including shifted quantum affine algebras and quantum coordinate rings of matrices.

Speaker: Pallav Goyal (​University of Southern Denmark).

Abstract: 
Ringel used the Hall algebra of a quiver to construct the positive half of the quantised enveloping algebra of the corresponding Kac–Moody algebra, and Lusztig’s geometric version of this construction produced canonical bases of these positive halves having remarkable positivity properties. A limitation of this approach is that it only yields the positive half of the algebra.

​​​​​​​This phenomenon is ubiquitous in the field: ‘halves’ of many non-commutative algebras (quantum groups, toroidal algebras, DAHAs, etc.) have been obtained as Hall algebras, but examples of ‘entire’ algebras being realised intrinsically are rare. I will survey some previous work on this problem and then describe joint work with Peter Samuelson giving complete Hall-theoretic realisations of several classes of quantum algebras, including shifted quantum affine algebras and quantum coordinate rings of matrices.
28 Oct