A quantum cluster algebra of Kronecker type and the dual canonical basis

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.Comment: 32 page

Quantum cluster algebras of type A and the dual canonical basis

The article concerns the subalgebra U_v^+(w) of the quantized universal enveloping algebra of the complex Lie algebra sl_{n+1} associated with a particular Weyl group element of length 2n. We verify that U_v^+(w) can be endowed with the structure of a quantum cluster algebra of type A_n. The quantum cluster algebra is a deformation of the ordinary cluster algebra Geiss-Leclerc-Schroeer attached to w using the representation theory of the preprojective algebra. Furthermore, we prove that the quantum cluster variables are, up to a power of v, elements in the dual of Lusztig's canonical basis under Kashiwara's bilinear form.Comment: 48 page

A microscopic Interpretation of the SM Higgs Mechanism

A model is presented where the Higgs mechanism of the Standard Model is deduced from the alignment of a strongly correlated fermion system in an internal space with $A_4$ symmetry. The ground state is constructed and its energy calculated. Finally, it is claimed that the model may be derived from a field theory in 6+1 dimensions.Comment: 24 pages, 3 figure

New Interactions in Top Quark Production and Decay at the Tevatron Upgrade

New interactions in top-quark production and decay are studied under the conditions of the Tevatron upgrade. Studying the process q qbar --> t tbar --> b mu nu tbar, it is shown how the lepton rapidity and transverse energy distribution are modified by nonstandard modifications of the g-t-tbar and the t-b-W vertex.Comment: Latex File and 3 eps Figure

Explaining nineteenth-century bilateralism: economic and political determinants of the Cobden-Chevalier network

This study investigates the empirical determinants of the treaty network of the 1860s and 1870s. It makes use of three central theories about the determinants of PTA formation, considering economic fundamentals from neoclassical and ‘new’ trade theory, political-economy variables, and international interaction due to trade diversion fears (dependence of later PTAs on former). These possible determinants are operationalized using a newly constructed dataset for bilateral cooperation and non-cooperation among 13 European Countries and the US. The results of logistic regression analysis show that the treaty network can be explained by a combination of ‘pure’ welfare-oriented economic theory with political economy and international interaction models.Preferential trade agreements, Anglo-French treaty of Commerce, Bilateralism, Political economy, Qualitative choice models

Explaining Nineteenth-Century Bilateralism: Economic and Political Determinants of the Cobden-Chevalier Network

This study investigates the empirical determinants of the treaty network of the 1860s and 1870s. It makes use of three central theories about the determinants of PTA formation, considering economic fundamentals from neoclassical and ‘new’ trade theory, political-economy variables, and international interaction due to trade diversion fears (dependence of later PTAs on former). These possible determinants are operationalized using a newly constructed dataset for bilateral cooperation and non-cooperation among 13 European Countries and the US. The results of logistic regression analysis show that the treaty network can be explained by a combination of ‘pure’ welfare-oriented economic theory with political economy and international interaction models.Cobden-Chevalier Network, Bilateralism