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Hopf Algebras, Tensor Categories and Related Topics
| Hopf Algebras, Tensor Categories and Related Topics |
| Autore | Andruskiewitsch Nicolás |
| Edizione | [1st ed.] |
| Pubbl/distr/stampa | Providence : , : American Mathematical Society, , 2021 |
| Descrizione fisica | 1 online resource (372 pages) |
| Disciplina | 512/.55 |
| Altri autori (Persone) |
LiuGongxiang
MontgomerySusan |
| Collana | Contemporary Mathematics |
| Soggetto topico |
Hopf algebras
Tensor algebra Geometry, Algebraic Algebraic geometry -- Special varieties -- Supervarieties [See also 32C11, 58A50] Associative rings and algebras {For the commutative case, see 13-XX} -- Representation theory of rings and algebras -- Representations of quivers and partially ordered sets Associative rings and algebras {For the commutative case, see 13-XX} -- Rings and algebras arising under various constructions -- Smash products of general Hopf actions [See also 16T05] Associative rings and algebras {For the commutative case, see 13-XX} -- Hopf algebras, quantum groups and related topics -- Hopf algebras and their applications [See also 16S40, 57T05] Associative rings and algebras {For the commutative case, see 13-XX} -- Hopf algebras, quantum groups and related topics -- Yang-Baxter equations Nonassociative rings and algebras -- Lie algebras and Lie superalgebras {For Lie groups, see 22Exx} -- Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 8 Category theory; homological algebra {For commutative rings see 13Dxx, for associative rings 16Exx, for groups 20Jxx, for topological groups and related structures 57Txx; see also 55Nxx and 55Uxx for $K$-theory [See also 16E20, 18F25] -- Grothendieck groups and $K_0$ [See also 13D15, 18F30] -- Frobenius induction, Burnside and representation rings Group theory and generalizations -- Linear algebraic groups and related topics {For arithmetic theory, see 11E57, 11H56; for geometric theory, see 14Lxx, 22Exx; for other methods in representation the |
| ISBN |
9781470464271
1470464276 |
| Classificazione | 14M3016G2016S4016T0516T2517B3718D1018D2519A2220G42 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Cover -- Title page -- Contents -- Preface -- List of speakers and titles of talks -- BiHom Hopf algebras viewed as Hopf monoids -- Introduction -- 1. Some combinatorial background -- 2. Cosemigroups in ⁺-monoidal categories -- 3. Semigroups in ₊-monoidal categories -- 4. Bisemigroups in ⁺ ₊-duoidal categories -- 5. Comonoids in ⁺ ⁰-monoidal categories -- 6. Monoids in ₀ ₊-monoidal categories -- 7. Bimonoids in ⁺₀ ⁰₊-duoidal categories -- References -- Survey on Hopf algebras of GK-dimension 1 and 2 -- 1. Introduction -- 2. Hopf algebras of GK-dimension 1 -- 3. Hopf algebras of GK-dimension 2 -- 4. Further comments, projects and open questions -- Acknowledgments -- References -- On braided double-biproduct Hopf algebras -- Introduction -- 1. Preliminaries -- 2. Some braided isomorphisms -- 3. Double-biproduct Hopf algebras -- 4. 2-cocycle deformations on double-biproduct Hopf algebras -- 5. Some examples -- Acknowledgments -- References -- Module Eilenberg-Watts calculus -- 1. Introduction -- 2. More about the Eilenberg-Watts calculus -- 3. Balanced functors and twistings -- 4. Eilenberg-Watts calculus for module functors -- 5. Eilenberg-Watts calculus for balanced functors -- 6. Transporting (co)monads along Eilenberg-Watts -- References -- Coherent unit actions on braided operads and Hopf algebras -- 1. Introduction -- 2. Braided algebraic operads -- 3. Coherent unit actions and braided -Hopf algebras -- 4. Criteria for braided binary quadratic regular operads with coherent unit actions -- Acknowledgments -- References -- A novel quantum affine algebra of type ₁⁽¹⁾ and its PBW basis -- 1. Introduction -- 2. Admissible quantum affine algebras _{ }(̂ ₂) of type ₁⁽¹⁾ -- 3. New quantum affine algebra _{ }(̂ ₂) of type ₁⁽¹⁾ and its quantum Weyl group.
4. Definition of imaginary and real root vectors -- 5. A Poincaré-Birkhoff-Witt basis of _{ }(̂ ₂) -- Acknowledgments -- References -- Finite quasi-quantum groups over abelian groups -- 1. Introduction -- 2. Normalized cocycles on finite abelian groups and braided Gr-categories -- 3. Finite quasi-quantum groups of diagonal type: A brief summary -- 4. Finite quasi-quantum groups of nondiagonal type: A first attempt -- 5. Some further problems -- References -- On biproducts and extensions -- Introduction -- 1. Preliminaries -- 2. The biproduct in the first case -- 3. Representations -- 4. Hopf subalgebras of dimension 16 -- 5. Hopf subalgebras of dimension 2 -- 6. Hopf subalgebras of dimensions 4 and 8 -- 7. The biproduct in the second case -- References -- On the notion of exact sequence: From Hopf algebras to tensor categories -- 1. Introduction -- 2. Preliminaries on Hopf algebras and tensor categories -- 3. Exact sequences of Hopf algebras -- 4. Exact sequences of tensor categories -- 5. Exact sequences of finite tensor categories with respect to a module category -- 6. Examples and open questions -- References -- Algebraic supergroups and their representations -- 1. Introduction -- 2. Algebraic supergroups -- 3. Representations of algebraic supergroups -- 4. Quasireductive supergroups admitting maximal parabolic super-subgroups -- Acknowledgments -- References -- Hopf algebraic methods in finite tensor categories: Characters and integrals -- 1. Introduction -- 2. Representation theory and Yetter-Drinfeld modules -- 3. Preliminaries: Finite abelian categories -- 4. Induction to the Drinfeld center -- 5. Categorical class functions -- 6. Integral theory and its applications -- 7. Applications to fusion categories -- Acknowledgments -- References -- Indecomposability of weak Hopf algebras -- 1. Introduction. 2. Definitions and basic properties related with weak bialgebras -- 3. The direct sum construction of weak bialgebras and Hopf algebras -- 4. The direct sum construction of weak Hopf algebras and quantum doubles -- 5. Categorical interpretation of the direct sum construction and its generalizations -- Acknowledgments -- References -- Quasi-Hopf algebras of Cartan type and small quasi-quantum groups -- 1. Introduction -- 2. Preliminaries -- 3. Finite-dimensional Quasi-Hopf algebras -- 4. Genuine Quasi-Hopf algebras of Cartan type -- 5. Small quasi-quantum groups -- References -- Back Cover. |
| Record Nr. | UNINA-9910970687603321 |
Andruskiewitsch Nicolás
|
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| Providence : , : American Mathematical Society, , 2021 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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On Finite GK-Dimensional Nichols Algebras over Abelian Groups
| On Finite GK-Dimensional Nichols Algebras over Abelian Groups |
| Autore | Andruskiewitsch Nicolás |
| Edizione | [1st ed.] |
| Pubbl/distr/stampa | Providence : , : American Mathematical Society, , 2021 |
| Descrizione fisica | 1 online resource (142 pages) |
| Disciplina | 512/.55 |
| Altri autori (Persone) |
AngionoIván
HeckenbergerIstván |
| Collana | Memoirs of the American Mathematical Society |
| Soggetto topico |
Hopf algebras
Associative rings and algebras -- Hopf algebras, quantum groups and related topics -- Ring-theoretic aspects of quantum groups Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Quantum groups (quantized enveloping algebras) and related deformations |
| ISBN |
9781470466367
1470466368 |
| Classificazione | 16T2017B37 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Cover -- Title page -- List of Tables -- Chapter 1. Introduction -- 1.1. Antecedents -- 1.2. Points and blocks -- 1.3. The main result -- 1.3.1. The class of braided vector spaces -- 1.3.2. Diagonal type -- 1.3.3. Flourished graphs -- 1.3.4. Organization of the paper and scheme of the proof -- 1.3.5. About the proofs -- 1.3.6. The Poseidon Nichols algebras -- 1.4. Applications -- 1.4.1. Examples of Hopf algebras -- 1.4.2. Domains -- 1.4.3. Co-Frobenius Hopf algebras -- Chapter 2. Preliminaries -- 2.1. Conventions -- 2.2. Nichols algebras of diagonal type -- 2.3. On the Gelfand-Kirillov dimension -- 2.3.1. Basic facts -- 2.3.2. A criterium for infinite \GK -- Chapter 3. Yetter-Drinfeld modules of dimension 2 -- 3.1. Indecomposable modules and blocks -- 3.2. The Jordan plane -- 3.3. The super Jordan plane -- 3.4. Filtrations of Nichols algebras -- 3.5. Proof of Theorem 3.1.2 -- Chapter 4. Yetter-Drinfeld modules of dimension 3 -- 4.1. The setting -- 4.1.1. A block and a point -- 4.1.2. A pale block and a point -- 4.1.3. Indecomposable of dimension 3 -- 4.1.4. Notations -- 4.1.5. Strong interaction -- 4.2. Weak interaction -- 4.2.1. Preparations -- 4.2.2. Proof of Theorem 4.1.3 -- 4.2.3. Proof of Theorem 4.1.1, weak interaction -- 4.3. The Nichols algebras with finite \GK -- 4.3.1. The Nichols algebra \cB(\lstr(1,\ghost)) -- 4.3.2. The Nichols algebra \cB(\lstr(-1,\ghost)) -- 4.3.3. The Nichols algebra \cB(\lstr₋(1,\ghost)) -- 4.3.4. The Nichols algebra \cB(\lstr₋(-1,\ghost)) -- 4.3.5. The Nichols algebra \cB(\lstr( ,1)) -- 4.4. Mild interaction -- 4.4.1. The Nichols algebra \cB(\cyc₁) -- Chapter 5. One block and several points -- 5.1. The setting -- 5.2. Proof of Theorem 5.1.1 ( =1) -- 5.2.1. Weak interaction and the algebra -- 5.2.2. | |=2 -- 5.2.3. | |> -- 2 -- 5.3. The Nichols algebras with finite \GK, _{\diag} connected.
5.3.1. The Nichols algebra \cB(\lstr( (1|0)₁ -- )), ∈\G_{ }', ≥3 -- 5.3.2. The Nichols algebra \cB(\lstr( (1|0)₁ -- )), ∉\G_{∞} -- 5.3.3. The Nichols algebra \cB(\lstr( (1|0)₂ -- )) -- 5.3.4. The Nichols algebra \cB(\lstr( (1|0)₃ -- )) -- 5.3.5. The Nichols algebra \cB(\lstr( (2|0)₁ -- )) -- 5.3.6. The Nichols algebra \cB(\lstr( (2|1) -- )) -- 5.3.7. The Nichols algebra \cB(\lstr( ₂,2)) -- 5.3.8. The Nichols algebra \cB(\lstr( _{ -1})) -- 5.4. Proof of Theorem 5.1.2 ( =-1) -- 5.4.1. Connected components of _{\diag} -- 5.4.2. The Nichols algebra \cB(\cyc₂) -- 5.4.3. Several components -- 5.4.4. The Nichols algebras with finite \GK, several connected components in _{\diag} -- Chapter 6. Two blocks -- 6.1. The setting -- 6.2. ₁=1 -- 6.3. ₁= ₂=-1 -- Chapter 7. Several blocks, several points -- 7.1. Notations -- 7.2. Several blocks, one point -- 7.3. The Nichols algebras \pos(\bq,\ghost) -- 7.4. Several blocks, several points -- Chapter 8. Appendix -- 8.1. Nichols algebras over abelian groups -- 8.1.1. The context -- 8.1.2. A pale block and a point -- 8.1.3. The block has =1 -- 8.1.4. The block has =-1 -- 8.1.5. The block has = ∈\G₃' -- 8.2. Admissible flourished diagrams -- Bibliography -- Back Cover. |
| Record Nr. | UNINA-9910966336303321 |
|
Andruskiewitsch Nicolás
|
||
| Providence : , : American Mathematical Society, , 2021 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||