LEADER 06437nam 22006013 450 001 9910966336303321 005 20231110232625.0 010 $a9781470466367 010 $a1470466368 035 $a(CKB)4940000000609983 035 $a(MiAaPQ)EBC6715033 035 $a(Au-PeEL)EBL6715033 035 $a(RPAM)22488292 035 $a(PPN)258258403 035 $a(OCoLC)1266906978 035 $a(EXLCZ)994940000000609983 100 $a20210901d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aOn Finite GK-Dimensional Nichols Algebras over Abelian Groups 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2021. 210 4$dİ2021. 215 $a1 online resource (142 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.271 311 08$a9781470448301 311 08$a1470448300 320 $aIncludes bibliographical references. 327 $aCover -- 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(\cyc?) -- 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. 327 $a5.3.1. The Nichols algebra \cB(\lstr( (1|0)? -- )), ?\G_{ }', ?3 -- 5.3.2. The Nichols algebra \cB(\lstr( (1|0)? -- )), ?\G_{?} -- 5.3.3. The Nichols algebra \cB(\lstr( (1|0)? -- )) -- 5.3.4. The Nichols algebra \cB(\lstr( (1|0)? -- )) -- 5.3.5. The Nichols algebra \cB(\lstr( (2|0)? -- )) -- 5.3.6. The Nichols algebra \cB(\lstr( (2|1) -- )) -- 5.3.7. The Nichols algebra \cB(\lstr( ?,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(\cyc?) -- 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 -- 6.3. ?= ?=-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 = ?\G?' -- 8.2. Admissible flourished diagrams -- Bibliography -- Back Cover. 330 $a"We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GKdim for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over Z with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite GKdim if and only if the size of the block is 2 and the eigenvalue is 1; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite GKdim if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite GKdim. Consequently we present several new examples of Nichols algebras with finite GKdim, including two not in the class alluded to above. We determine which among these Nichols algebras are domains"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aHopf algebras 606 $aAssociative rings and algebras -- Hopf algebras, quantum groups and related topics -- Ring-theoretic aspects of quantum groups$2msc 606 $aNonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Quantum groups (quantized enveloping algebras) and related deformations$2msc 615 0$aHopf algebras. 615 7$aAssociative rings and algebras -- Hopf algebras, quantum groups and related topics -- Ring-theoretic aspects of quantum groups. 615 7$aNonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Quantum groups (quantized enveloping algebras) and related deformations. 676 $a512/.55 686 $a16T20$a17B37$2msc 700 $aAndruskiewitsch$b Nicola?s$01714852 701 $aAngiono$b Iva?n$01800593 701 $aHeckenberger$b Istva?n$01800594 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910966336303321 996 $aOn Finite GK-Dimensional Nichols Algebras over Abelian Groups$94345447 997 $aUNINA