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

By Philipp Lampe. 2010

48 citations recorded by the source corpus.

The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part U_q(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-Schroeer attached to M a category C_M of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra A(C_M). The dual semicanonical basis contains all cluster monomials. By construction, the cluster algebra A(C_M) 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 U_q(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(C_M) 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/290b10efa1134f09f9a34ea27817dcd43cdfdb37

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In Discrete Mathematics and Combinatorics

1. Bernard Leclerc
2. C. Geiss
3. J. Schröer
4. B. Keller
5. Pavlo Pylyavskyy
6. Osamu Iyama
7. Dylan Rupel
8. I. Reiten

In Geometry and Topology

1. Bernard Leclerc
2. C. Geiss
3. J. Schröer
4. B. Keller
5. Osamu Iyama
6. I. Reiten
7. Pavlo Pylyavskyy
8. Sergey Fomin

In Mathematical Physics

1. Bernard Leclerc
2. C. Geiss
3. J. Schröer
4. B. Keller
5. I. Reiten
6. Osamu Iyama
7. Pavlo Pylyavskyy
8. Sergey Fomin

Citations in the corpus

• Cluster structures on quantum coordinate rings (2011, result): “e γa,c(q)∈q−1Q[q−1]for every c 6= a; (b) Let β(a):=∑1≤k≤r akβk. Then σ(b(a))=qN(β(a))b(a). Proof — This is a restatement in our setup of [Ki, Theorem 4.26]. The same result was previously obtained in [La1] in a particular case. ✷ It follows from (10.27) that all quantum cluster monomials satisfy con”
• Linear recurrence relations for cluster variables of affine quivers (2010, unknown): “R T ] 1 6 Ju n 20 10”
• Tensor diagrams and cluster algebras (2012, background): “Lampe [33], for the cluster type at hand the appropriate element of the dual canonical basis is given by the quadratic Chebyshev polynomial of the second kind.”
• On a Quantum Analog of the Caldero–Chapoton Formula (2010, background): “,2): X−n = X(n+2,−n−1) + P p+r≤n n−r p q n+1−p r q (1.3) X(2r−n,2p−n−1); Xn+3 = X(−n−1,n+2) + P p+r≤n n−r p q n+1−p r q (1.4) X(2p−n−1,2r−n). This result was proved independently in a recent preprint [15], we present our proof for the convenience of the reader. Setting q = 1 we recover the Caldero-C”
• Framed BPS states (2010, unknown): “First, some of the formulae in [80] are closely related to laminations and line operators in SU(2) gauge theories.”