Noncommutative Birational Geometry, Representations and Combinatorics.

Intro -- Chapter 1. Preface -- The number of right ideals of given codimension over a finite field -- 1. Introduction -- 2. Catalan numbers and -analogues -- 3. Enumeration of right ideals -- 4. Prefix-free and prefix-closed sets -- 5. Weak prefix bases for right ideals -- 6. Broad ideals and prefix bases -- 7. Sequential prefix bases -- 8. Trees associated to maximal prefix-free sets -- 9. Proof of Theorem 1 -- 10. Proof of Theorem 2 -- 11. Right congruences of the free monoid -- References -- Cocycle twists and extensions of braided doubles -- 0. Introduction -- 1. Cocycles, central extensions, the Schur multiplier, twisted group algebras -- 2. The Schur multiplier of _{ } -- 3. Cocycle twists and cocycle extensions of group-graded algebras -- 4. Cocycle twists and the lazy cohomology of a monoidal category -- 5. Cocycle twists of braidings and Nichols algebras -- 6. Extensions and twists of Yetter-Drinfeld modules -- 7. Braided doubles -- 8. Braided Heisenberg doubles and braided Weyl algebras -- 9. Extensions and twists of other braided doubles -- 10. Covering Cherednik algebras and spin Cherednik algebras -- References -- Quantum Chevalley groups -- 1. Introduction -- 2. Preliminaries -- 3. -commutators and proof of Theorem 1.8 -- 4. Preprojective and preinjective factorization -- 5. Examples -- Appendix A. Coproduct -- References -- The Reciprocal of ∑_{ ≥0} ⁿ ⁿ for non-commuting and , Catalan numbers and non-commutative quadratic equations -- 1. Inverting ∑_{ ≥0} ⁿ ⁿ. -- 2. Inversion of 1- in the general case -- 3. The Quasideterminant of a Jacobi Matrix -- Acknowledgement -- References -- Semi-invariants for gentle algebras -- 1. Introduction -- 2. Preliminaries -- 3. Varieties of representations for colored algebras -- 4. Semi-invariants for gentle algebras -- 5. Combinatorics of the rings of semi-invariants.

6. Matching Semigroups -- 7. Degree Bounds -- 8. Example -- References -- Monodromy of the trigonometric Casimir connection for ₂ -- 1. Introduction -- 2. The trigonometric Casimir connection of ₂ -- 3. The trigonometric Casimir connection of ₂ -- 4. Affine braid groups -- 5. The trigonometric KZ equations -- 6. The dual pair ( _{ }, ₂) and trigonometric connections -- 7. Monodromy of the trigonometric KZ equations -- 8. Quantum loop algebras -- 9. Quantum Weyl groups -- 10. The dual pair ( _{\ℎ } _{ }, _{\ } _{ } ) -- 11. Affine braid group actions on quantum matrix space -- 12. Monodromy theorems -- Appendix A. Monodromy of the trigonometric KZ equations (after Etingof-Geer-Schiffmann) -- Appendix B. Proof of Proposition 9.10 -- References -- Rational series in the free group and the Connes operator -- 1. Introduction -- 2. The Free Field, Rational Series, and the Connes Operator -- 3. Rank of the Connes Operators -- 4. Applications to \ \ and &lt -- ( )&gt -- -- Appendix A. Using Euler's Identity -- Appendix B. Extensions &amp -- Open Problems -- Appendix C. Lyndon Words in the Magnus Ordering -- References -- On natural maps from strata of quiver Grassmannians to ordinary Grassmannians -- 1. Introduction -- 2. Noncommutative cluster variables -- 3. Quantum cluster variables of rank 2 -- 4. Proof of Theorem 2(2) -- 5. Proof of Theorem 2(1) -- References -- Properties of a Minimal Non-Koszul (Γ) -- 1. Introduction -- 2. The Algebra (Γ_{ }) -- References -- Double Poisson brackets on free associative algebras -- 1. Introduction -- 2. Quadratic double Poisson brackets -- 3. Compatible linear and quadratic double Poisson bracket -- 4. Conclusions and perspectives -- References -- Generalizing the notion of rank to noncommutative quadratic forms -- Introduction.

1. Noncommutative Quadratic Forms -- 2. Rank of quadratic forms on two generators -- 3. Rank of quadratic forms on three generators -- References.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.