A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis

Abstract

The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part Uq(n) of the universal enveloping algebra of a Kac-Moody Lie algebra of type A(1) 1 . The examined subalgebra is associated with a terminal module M over the path algebra of the Kronecker quiver via an Weyl group element w of length four.Geiß-Leclerc-Schröer attached to M a category CM of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra A(CM ). The dual semicanonical basis contains all cluster monomials. By construction, the cluster algebra A(CM ) is a subalgebra of the graded dual of the (non-quantized) universal enveloping algebra U (n).We transfer to the quantized setup. Following Lusztig we attach to w a subalgebra U + q (w) of Uq(n). The subalgebra is generated by four elements that satisfy straightening relations; it degenerates to a commutative algebra in the classical limit q = 1. The algebra U + q (w) possesses four bases, a PBW basis, a canonical basis, and their duals. We prove recursions for dual canonical basis elements. The recursions imply that every cluster variable in A(CM ) is the specialization of the dual of an appropriate canonical basis element. Therefore, U + q (w) is a quantum cluster algebra in the sense of Berenstein-Zelevinsky. Furthermore, we give explicit formulae for the quantized cluster variables and for expansions of products of dual canonical basis elements.

Quantum unipotent subgroup and dual canonical basis

“…Our main result is to give a set up of a quantum analogue of Geiß-Leclerc-Schröer's results: In [32], S is the specialization of the dual canonical basis, while Σ is the dual semicanonical basis thanks to [22].…”

“…Some parts of Conjecture 1.1 were shown for A 2 , A 3 , A 4 cases with w = w 0 in [3] and [18, §12] and A (1) 1 with w = c 2 in [32]. The definition of the quantum cluster algebra A q (Γ w , Λ w ) will not be explained.…”

Atomic bases of cluster algebras of types A and Ã

“…It was recently proved in [9] that cluster monomials in A Q are always linearly independent over Z. Moreover, cluster monomials play a prominent role in the construction of Z-linear bases of A Q which are of interest with respect to the study of dual canonical bases of quantum groups; see, for instance, [20,22,24,26,27,33].…”

Monoidal categorification of cluster algebras

“…It is proved for g = A 2 , A 3 , A 4 and A q (n(w)) = A q (n) in [2] and [7, § 12]. When g = A (1) 1 , A n and w is a square of a Coxeter element, it is shown in [26] and [27] that the cluster variables belong to the upper global basis. When g is symmetric and w is a square of a Coxeter element, the conjecture is proved in [24].…”