Noncommutative Double Bruhat cells and their factorizations

Quadratic-linar algebras associated with factorizations of noncommutative polynomials and noncommutative differential polynomials

This is the first of series of talks presented at a permanent Rutgers workshop on noncommutative algebra and geometry. We study here quadratic and quadratic-linear algebras defined by factorizations of noncommutative polynomials and differential polynomials. Such algebras posses a natural derivation and give us a new understanding of a nature of noncommutative symmetric functions.

Factorizations of Polynomials over Noncommutative Algebras and Sufficient Sets of Edges in Directed Graphs

To directed graphs with unique sink and source we associate a noncommutative associative alsgebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when coefficients of the polynomial can be rationally expressed via elements of a given set of pseudo-roots (edges). Our results are based on a new theorem for directed graphs also proved in this paper.

The Berenstein-Zelevinsky quantum cluster algebra conjecture

We prove the Berenstein-Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite dimensional simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [4]. We furthermore prove that the corresponding upper quantum cluster algebras coincide with the constructed quantum cluster algebras and exhibit a large number of explicit quantum seeds. Along the way a detailed study of the properties of quantum double Bruhat cells from the viewpoint of noncommutative UFDs is carried out and a quantum analog of the Fomin-Zelev…

弯曲DG模与非交换二次超曲面的矩阵分解

Curved DG Modules and Matrix Factorizations from Noncommutative Quadric Hypersurfaces

本文研究非交换二次超曲面范畴的对偶性，构造从分次模到弯曲DG模同伦范畴的忠实函子，并在一定条件下将其限制到矩阵分解稳定范畴，证明偶数克利福德代数与PBW形变同构。

On the classification and properties of noncommutative duplicates

We give an explicit description of the set of all factorization structures, or twisting maps, existing between the algebras k^2 and k^2, and classify the resulting algebras up to isomorphism. In the process we relate several different approaches formerly taken to deal with this problem, filling a gap that appeared in a recent paper by Cibils. We also provide a counterexample to a result concerning the Hochschild (co)homology appeared in a paper by J.A. Guccione and J.J. Guccione.

Decomplexification of the Capelli identities and holomorphic factorization

The Capelli identities claim $det(A)det(B) = det(AB+correction)$ for certain matrices with noncommutative entries. They have applications in representation theory and integrable systems. We propose new examples of these identities, constructed according to the following principle. For several known identities for $n$ by $n$ matrices we construct new identity for $2n$ by $2n$ matrices where each element $z$ of the original matrix is substituted by 2x2 matrix of the form $[real(z) ~imag(z); ~ -imag(z) ~ real(z)]$, i.e. we view the original identity as complex valued, while the new identity is it…