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A quantum cluster algebra structure on the semi-derived Hall algebra

Alessandro Contu

Abstract

Using Hernandez-Leclerc's isomorphism between the derived Hall algebra of a representation-finite quiver $Q$ and the quantum Grothendieck ring of the quantum loop algebra of the Dynkin type of $Q$, we lift the (quantum) cluster algebra structure of the quantum Grothendieck ring to the semi-derived Hall algebra, introduced by Gorsky, of the category of bounded complexes of projective modules over the path algebra of $Q$. We also construct a braid group action on the semi-derived Hall algebra, lifting Kashiwara-Kim-Oh-Park's braid group action on the quantum Grothendieck ring.

A quantum cluster algebra structure on the semi-derived Hall algebra

Abstract

Using Hernandez-Leclerc's isomorphism between the derived Hall algebra of a representation-finite quiver and the quantum Grothendieck ring of the quantum loop algebra of the Dynkin type of , we lift the (quantum) cluster algebra structure of the quantum Grothendieck ring to the semi-derived Hall algebra, introduced by Gorsky, of the category of bounded complexes of projective modules over the path algebra of . We also construct a braid group action on the semi-derived Hall algebra, lifting Kashiwara-Kim-Oh-Park's braid group action on the quantum Grothendieck ring.

Paper Structure

This paper contains 19 sections, 33 theorems, 230 equations, 3 figures.

Table of Contents

  1. introduction
  2. Semi-derived Hall algebras
  3. A quantum cluster algebra structure on the semi-derived Hall algebra
  4. Braid group action on the semi-derived Hall algebra
  5. Quantum cluster algebra strcutures on quantum Grothendieck rings
  6. The categories $\mathscr{C}_{\mathfrak{g}}^{[a,b],\mathfrak{s}}$
  7. Quantum Grothendieck rings
  8. Combinatorics of $i$-boxes and quantum seeds
  9. Derived and semi-derived Hall algebras
  10. Semi-derived Hall algebras of categories of complexes
  11. On the grading of the (semi-)derived Hall algebra
  12. The Hernandez--Leclerc isomorphism
  13. A lift of the exchange relations to the semi-derived Hall algebra
  14. A quantum cluster algebra structure on the semiderived Hall algebra
  15. A braid group action on the semi-derived Hall algebra
  16. ...and 4 more sections

Key Result

Theorem 1.1

There is an isomorphism of $\mathcal{K}(C^b(\mathcal{P}))$-graded $\mathbb{Q}(v^{1/2})$-algebras which makes the diagram (intro_commutative_square_[a,b]) commute.

Figures (3)

  • Figure 1: The quiver $\widetilde{Q}^{[-\infty,3],\mathfrak{s}}$ for $\mathfrak{g}$ of type $A_3$
  • Figure 2: The quiver ${Q}(\mathfrak{C})$ for $\mathfrak{g}$ of type $A_4$
  • Figure 3: The quiver $\widetilde{Q}(\mathfrak{C})$ for $\mathfrak{g}$ of type $A_4$

Theorems & Definitions (72)

  • Theorem 1.1: Theorem \ref{['teo_quantum_clust_semiderived_[a,b]']}
  • Theorem 1.2: Corollary \ref{['cor_clust_semiderived']}
  • Theorem 1.3: Theorem \ref{['teo_braid']}
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4: Hernandez2004_alg_approach
  • Proposition 2.5: Hernandez2004_alg_approach
  • Proposition 2.6
  • Proposition 2.7
  • ...and 62 more