Groups, Rings, Group Rings, and Hopf Algebras.
Bergen, Jeffrey.
Groups, Rings, Group Rings, and Hopf Algebras. - 1st ed. - 1 online resource (294 pages) - Contemporary Mathematics ; v.688 . - Contemporary Mathematics .
Cover -- Title page -- Contents -- Preface -- The Dixmier-Moeglin equivalence for extensions of scalars and Ore extensions -- 1. Introduction -- 2. The Dixmier-Moeglin equivalence under base change -- 3. Linear operators on rings -- 4. Proof of Theorem 1.6 -- Acknowledgments -- References -- Nagata-Higman and rings with involution -- References -- On left symmetric color algebras -- 1. Introduction -- 2. Left symmetric color algebras and nondegenerate color symmetric 2-cocycles -- 3. Lifting the derivations of into the derivations of * -- Acknowledgements -- References -- On the automorphism group of rational group algebras of finite groups -- 1. Introduction -- 2. Preliminaries -- 3. Group algebras of simple groups -- 4. Non-simple groups -- References -- Graded simple modules and loop modules -- 1. Introduction -- 2. Graded simple modules -- 3. Loop modules -- 4. The groupoids \frM( ) and \frN( ) -- 5. Graded simple modules with finite-dimensional centralizers -- 6. Graded simple modules with simple centralizers -- 7. Finite-dimensional graded simple modules in characteristic zero -- References -- Symmetric groups and fixed points on modules: An application of group theory to topology -- 1. Introduction -- 2. Theorem 2.1 -- 3. The norm map -- References -- Free unit groups in group rings and division rings: My collaboration with Don Passman -- 1. Introduction -- 2. Our first work -- 3. The following years -- 4. The proof of Theorem 3.2 -- 5. Involutions in group rings -- 6. Our exploits in division rings -- References -- Group rings and Jordan decomposition -- 1. Introduction -- 2. Matrix Rings -- 3. Group Rings -- 4. Future Work -- References -- On the Toeplitz-Jacobson algebra and direct finiteness -- 1. Introduction -- 2. The results -- 3. The direct finiteness conjecture and other outstanding problems -- Acknowledgment -- References. Frobenius divisibility for Hopf algebras -- Introduction -- 1. Symmetric Algebras -- 2. Hopf Algebras -- Acknowledgement -- References -- Generalized nil-Coxeter algebras, cocommutative algebras, and the PBW property -- 1. Introduction -- 2. Cocommutative algebras, smash products, and the PBW theorem -- 3. Characterization via deformation theory -- 4. The case of bialgebras and Hopf algebras -- 5. Generalized nil-Coxeter algebras and grouplike algebras -- 6. Deformations over cocommutative algebras with nilpotent maximal ideals -- Acknowledgments -- References -- -subgroups of units in ℤ -- 1. Introduction -- 2. Known results -- 3. Frobenius Groups -- 4. Crucial examples for simple linear groups -- 5. Conjugacy in larger group rings -- References -- On the classification of finite-dimensional semisimple Hopf algebras -- 0. Introduction -- 1. Abelian extensions -- 2. Structure of ²_(\myk _,\myk^,\tl) -- 3. The Isomorphism Theorems -- 4. Almost Abelian Hopf Algebras of Dimension \le ⁴ -- 5. Some old classification results revisited -- 6. Appendices -- References -- Zero divisors in group rings of wreath products of groups -- 1. Introduction. -- 2. Preliminaries. -- 3. The Proof of Theorem I. -- 4. Proofs of Theorems II and III. -- 5. Embedding in simple Artinian rings and division rings. -- 6. Concluding remarks. -- References -- The lattice of submodules of a multiplicity-free module -- 1. Introduction. -- 2. Distributive Lattices. -- 3. Applications and Examples. -- References -- Star group identities on units of group algebras -- 1. Introduction -- 2. Basic facts -- 3. SLC groups -- 4. Symmetric units -- 5. Star group identities -- References -- A note on group algebras of locally compact groups -- 1. Preliminaries -- 2. Automorphism-invariant group algebras of locally compact groups. 3. Generalized group algebras of locally compact groups -- 4. Open Problems -- References -- Elementary construction of Lusztig's canonical basis -- 1. Introduction -- 2. Notation -- 3. Braid group action and PBW bases -- 4. Equality mod and piecewise linear bijections -- 5. Triangularity of bar involution and the canonical basis -- 6. Properties of the canonical basis -- 7. Example: Crystal operators from piecewise linear bijections -- Acknowledgements -- References -- Back Cover.
This volume contains the proceedings of the International Conference on Groups, Rings, Group Rings, and Hopf Algebras, held October 2-4, 2015 at Loyola University, Chicago, IL, and the AMS Special Session on Groups, Rings, Group Rings, and Hopf Algebras, held October 3-4, 2015, at Loyola University, Chicago, IL. Both conferences were held in honor of Donald S. Passman's 75th Birthday. Centered in the area of group rings and algebras, this volume contains a mixture of cutting edge research topics in group theory, ring theory, algebras and their representations, Hopf algebras and quantum groups.
9781470440428
Group algebras--Congresses.
Electronic books.
QA174.G768 2017
512.4
Groups, Rings, Group Rings, and Hopf Algebras. - 1st ed. - 1 online resource (294 pages) - Contemporary Mathematics ; v.688 . - Contemporary Mathematics .
Cover -- Title page -- Contents -- Preface -- The Dixmier-Moeglin equivalence for extensions of scalars and Ore extensions -- 1. Introduction -- 2. The Dixmier-Moeglin equivalence under base change -- 3. Linear operators on rings -- 4. Proof of Theorem 1.6 -- Acknowledgments -- References -- Nagata-Higman and rings with involution -- References -- On left symmetric color algebras -- 1. Introduction -- 2. Left symmetric color algebras and nondegenerate color symmetric 2-cocycles -- 3. Lifting the derivations of into the derivations of * -- Acknowledgements -- References -- On the automorphism group of rational group algebras of finite groups -- 1. Introduction -- 2. Preliminaries -- 3. Group algebras of simple groups -- 4. Non-simple groups -- References -- Graded simple modules and loop modules -- 1. Introduction -- 2. Graded simple modules -- 3. Loop modules -- 4. The groupoids \frM( ) and \frN( ) -- 5. Graded simple modules with finite-dimensional centralizers -- 6. Graded simple modules with simple centralizers -- 7. Finite-dimensional graded simple modules in characteristic zero -- References -- Symmetric groups and fixed points on modules: An application of group theory to topology -- 1. Introduction -- 2. Theorem 2.1 -- 3. The norm map -- References -- Free unit groups in group rings and division rings: My collaboration with Don Passman -- 1. Introduction -- 2. Our first work -- 3. The following years -- 4. The proof of Theorem 3.2 -- 5. Involutions in group rings -- 6. Our exploits in division rings -- References -- Group rings and Jordan decomposition -- 1. Introduction -- 2. Matrix Rings -- 3. Group Rings -- 4. Future Work -- References -- On the Toeplitz-Jacobson algebra and direct finiteness -- 1. Introduction -- 2. The results -- 3. The direct finiteness conjecture and other outstanding problems -- Acknowledgment -- References. Frobenius divisibility for Hopf algebras -- Introduction -- 1. Symmetric Algebras -- 2. Hopf Algebras -- Acknowledgement -- References -- Generalized nil-Coxeter algebras, cocommutative algebras, and the PBW property -- 1. Introduction -- 2. Cocommutative algebras, smash products, and the PBW theorem -- 3. Characterization via deformation theory -- 4. The case of bialgebras and Hopf algebras -- 5. Generalized nil-Coxeter algebras and grouplike algebras -- 6. Deformations over cocommutative algebras with nilpotent maximal ideals -- Acknowledgments -- References -- -subgroups of units in ℤ -- 1. Introduction -- 2. Known results -- 3. Frobenius Groups -- 4. Crucial examples for simple linear groups -- 5. Conjugacy in larger group rings -- References -- On the classification of finite-dimensional semisimple Hopf algebras -- 0. Introduction -- 1. Abelian extensions -- 2. Structure of ²_(\myk _,\myk^,\tl) -- 3. The Isomorphism Theorems -- 4. Almost Abelian Hopf Algebras of Dimension \le ⁴ -- 5. Some old classification results revisited -- 6. Appendices -- References -- Zero divisors in group rings of wreath products of groups -- 1. Introduction. -- 2. Preliminaries. -- 3. The Proof of Theorem I. -- 4. Proofs of Theorems II and III. -- 5. Embedding in simple Artinian rings and division rings. -- 6. Concluding remarks. -- References -- The lattice of submodules of a multiplicity-free module -- 1. Introduction. -- 2. Distributive Lattices. -- 3. Applications and Examples. -- References -- Star group identities on units of group algebras -- 1. Introduction -- 2. Basic facts -- 3. SLC groups -- 4. Symmetric units -- 5. Star group identities -- References -- A note on group algebras of locally compact groups -- 1. Preliminaries -- 2. Automorphism-invariant group algebras of locally compact groups. 3. Generalized group algebras of locally compact groups -- 4. Open Problems -- References -- Elementary construction of Lusztig's canonical basis -- 1. Introduction -- 2. Notation -- 3. Braid group action and PBW bases -- 4. Equality mod and piecewise linear bijections -- 5. Triangularity of bar involution and the canonical basis -- 6. Properties of the canonical basis -- 7. Example: Crystal operators from piecewise linear bijections -- Acknowledgements -- References -- Back Cover.
This volume contains the proceedings of the International Conference on Groups, Rings, Group Rings, and Hopf Algebras, held October 2-4, 2015 at Loyola University, Chicago, IL, and the AMS Special Session on Groups, Rings, Group Rings, and Hopf Algebras, held October 3-4, 2015, at Loyola University, Chicago, IL. Both conferences were held in honor of Donald S. Passman's 75th Birthday. Centered in the area of group rings and algebras, this volume contains a mixture of cutting edge research topics in group theory, ring theory, algebras and their representations, Hopf algebras and quantum groups.
9781470440428
Group algebras--Congresses.
Electronic books.
QA174.G768 2017
512.4