Abstract: All quasi-affine connected Generalized Dynkin Diagram with rank and are found. All quasi-affine Nichols (Lie braided) algebras with rank and are also found.
Submitted 9 April, 2024; originally announced June 2024.

Comments: 338 pages

MSC Class: 16W30; 16G10

arXiv:2403.19695 [pdf, ps, other] math.AG
Classification of quasi-affine Generalized Dynkin Diagrams with Rank

Authors: Zhengtang Tan, Shouchuan Zhang

Abstract: All quasi-affine connected Generalized Dynkin Diagrams with rank are found. All quasi-affine Nichols (Lie braided) algebras with rank are also found.
Submitted 17 March, 2024; originally announced March 2024.

Comments: 77 pages. arXiv admin note: substantial text overlap with arXiv:2202.09940

MSC Class: 16W30; 16G10

arXiv:2202.09940 [pdf, ps, other] math.AG
Classification of quasi-affine Generalized Dynkin Diagrams with Rank

Authors: Zhengtang Tan, Shouchuan Zhang

Abstract: All quasi-affine connected Generalized Dynkin Diagram with rank are found. All quasi-affine Nichols (Lie braided) algebras with rank are also found.
Submitted 7 May, 2024; v1 submitted 20 February, 2022; originally announced February 2022.

Comments: 79 pages

MSC Class: 16W30; 16G10

arXiv:2007.06538 [pdf, ps, other] math.QA math.RA
doi
10.1142/s0219498821501875
Nichols algebras over classical Weyl groups

Authors: Zhengtang Tan, Weicai Wu, Shouchuan Zhang

Abstract: We show that except in several cases conjugacy classes of classical Weyl groups and are of type {\rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({\rm YD} in short )modules over the classical Weyl groups are infinite dimensional.
Submitted 10 July, 2020; originally announced July 2020.

Comments: 12 pages. arXiv admin note: text overlap with arXiv:1307.8227

MSC Class: 16W30; 22E60; 11F23

arXiv:1810.05951 [pdf, ps, other] math.CO math.RA
The relationship between graphs and Nichols braided Lie algebras

Authors: Weicai Wu, Shouchuan Zhang, Zhengtang Tan

Abstract: In this paper we give the relationship between the connected components of pure generalized Dynkin graphs and Nichols braided Lie algebras.
Submitted 23 May, 2019; v1 submitted 13 October, 2018; originally announced October 2018.

Comments: 6 pages. arXiv admin note: substantial text overlap with arXiv:1704.06810

arXiv:1704.06810 [pdf, ps, other] math.QA
Structures of Nichols (braided) Lie algebras of diagonal type

Authors: Weicai Wu, Jing Wang, Shouchuan Zhang, Yao-Zhong Zhang

Abstract: Let be a braided vector space of diagonal type. Let , and be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over , respectively. We show that a monomial belongs to if and only if that this monomial is connected. We obtain the basis for of arithmetic root systems and the dimension… ▽ More
Submitted 16 October, 2017; v1 submitted 22 April, 2017; originally announced April 2017.

Comments: 23 pages. Version to appear in Journal of Lie Theory

MSC Class: 16W30; 22E60; 05L25

Journal ref: Journal of Lie Theory 28 (2018), 357-380

arXiv:1701.02406 [pdf, ps, other] math.QA
Basis of Nichols Braided Lie Algebras

Authors: Weicai Wu, Jing Wang, Shouchuan Zhang

Abstract: Assume that is a braided vector space with diagonal type. It is shown that a monomial belongs to Nichols braided Lie algebra if and only if this monomial is connected. A basis of Nichols braided Lie algebra and dimension of Nichols braided Lie algebra of finite Cartan type are obtained.
Submitted 12 April, 2020; v1 submitted 9 January, 2017; originally announced January 2017.

arXiv:1607.07955 [pdf, ps, other] math.QA
Finiteness of Nichols Algebras and Nichols (Braided) Lie Algebras

Authors: Weicai Wu, Shouchuan Zhang, Yao-Zhong Zhang

Abstract: It is shown that if is connected Nichols algebra of diagonal type with , then \dim (\mathfrak L^-(V)) = \infty (resp. \dim (\mathfrak L(V)) = \infty ) ( resp. \dim (\mathfrak B(V)) = \infty ) if and only if Δ(\mathfrak B(V)) is an arithmetic root system and the quantum numbers (i.e. the fixed parameters) of generalized Dynkin diagrams of V are of fi… ▽ More
Submitted 17 March, 2019; v1 submitted 27 July, 2016; originally announced July 2016.

Comments: 24 pages

MSC Class: 16W30; 22E60

arXiv:1409.3769 [pdf, ps, other] math.QA
doi
10.1142/S0129167X15500822
On Nichols (braided) Lie algebras

Authors: Weicai Wu, Shouchuan Zhang, Yao-Zhong Zhang

Abstract: We prove {\rm (i)} Nichols algebra \mathfrak B(V) of vector space V is finite-dimensional if and only if Nichols braided Lie algebra \mathfrak L(V) is finite-dimensional; {\rm (ii)} If the rank of connected V is 2 and \mathfrak B(V) is an arithmetic root system, then \mathfrak B(V) = F \oplus \mathfrak L(V); and {\rm (iii)} if Δ(\mathfrak B(V)) is an arithmetic root system and ther… ▽ More
Submitted 12 July, 2015; v1 submitted 12 September, 2014; originally announced September 2014.

Comments: 29 Pages; Substantially revised version; To appear in International Journal of Mathematics

MSC Class: 16W30; 16G10

Journal ref: Int. J. Math. 26, 1550082 (2015)

arXiv:1401.3795 [pdf, ps, other] math.QA
Relationship between Nichols braided Lie algebras and Nichols algebras

Authors: Weicai Wu, Shouchuan Zhang, Yao-Zhong Zhang

Abstract: We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra \mathfrak B(V) is finite-dimensional if and only if Nichols braided Lie algebra \mathfrak L(V) is finite-dimensional if there does not exist any m-infinity element in \mathfrak B(V); (ii) Nichols Lie algebra \mathfrak L^-(V) is infinite di… ▽ More
Submitted 21 May, 2014; v1 submitted 15 January, 2014; originally announced January 2014.

Comments: LeTex 18 pages, need JOLT-macros to compile. To appear in Journal of Lie Theory

Journal ref: Journal of Lie Theory 25 (2015), 45--63

arXiv:1401.2864 [pdf, ps, other] math.QA
Quantum Theory in Design

Authors: Zhengtang Tan, Shouchuan Zhang

Abstract: In this paper, we use braiding diagrams to present rules of shapes and designs. That is, we design colour, design size, design brightness, design codes by means of braiding.
Submitted 13 January, 2014; originally announced January 2014.

Comments: page 13. arXiv admin note: text overlap with arXiv:math/0511251

arXiv:1307.8227 [pdf, ps, other] math.QA
Nichols algebras over classical Weyl groups, Fomin-Kirillov algebras and Lyndon basis

Authors: Shouchuan Zhang, Weicai Wu, Zhengtang Tan, Yao-Zhong Zhang

Abstract: We show that except in several cases conjugacy classes of classical Weyl groups W(B_n) and W(D_n) are of type {\rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({\rm YD} in short )modules over the classical Weyl groups are infinite dimensional. We establish the relationship between Fomin-Kirillov algebra \mathcal E_n and Nichols algebra… ▽ More
Submitted 3 March, 2017; v1 submitted 31 July, 2013; originally announced July 2013.

Comments: 45pages

MSC Class: 16W30; 22E60; 11F23

arXiv:1307.6794 [pdf, ps, other] math.QA
Classification of quasi-affine Generalized Dynkin Diagrams with Rank 4

Authors: Zhengtang Tan, Shouchuan Zhang

Abstract: All quasi-affine connected Generalized Dynkin Diagram with rank = 4 are found. All quasi-affine Nichols (Lie braided) algebras with rank 4 are also found.
Submitted 10 January, 2025; v1 submitted 25 July, 2013; originally announced July 2013.

Comments: 109 pages

MSC Class: 16W30; 22E60

arXiv:1201.2064 [pdf, ps, other] math.NT
Finite dimensional Nichols algebras over finite cyclic groups

Authors: Weicai Wu, Shouchuan Zhang, Yao-Zhong Zhang

Abstract: All finite dimensional Nichols algebras with diagonal type of connected finite dimensional Yetter-Drinfeld modules over finite cyclic group \mathbb Z_n are found. It is proved that finite dimensional Nichols algebra over \mathbb Z_2 is a quantum linear space and Nichols algebra of connected Yetter-Drinfeld module V over \mathbb Z_n with \dim V >3 is infinite dimensional.
Submitted 10 September, 2013; v1 submitted 10 January, 2012; originally announced January 2012.

Comments: 22 pages, substantially shortened and reference added. Version to appear in Journal of Lie Theory

Journal ref: Journal of Lie Theory 24 (2014) 351-372

arXiv:1006.3700 [pdf, ps, other] math.QA
Universal Enveloping Algebras of Braided m-Lie Algebras

Authors: Lingwei Guo, Shouchuan Zhang, Jieqiong He

Abstract: Universal enveloping algebras of braided m-Lie algebras and PBW theorem are obtained by means of combinatorics on words.
Submitted 13 September, 2014; v1 submitted 18 June, 2010; originally announced June 2010.

Comments: 9pages

MSC Class: 16W30; 16G10

arXiv:0910.1280 [pdf, ps, other] math.QA
doi
10.1142/S0129167X12500668
Pointed Hopf Algebras with classical Weyl Groups

Authors: Shouchuan Zhang, Yao-Zhong Zhang

Abstract: We prove that Nichols algebras of irreducible Yetter-Drinfeld modules over classical Weyl groups A \rtimes \mathbb S_n supported by \mathbb S_n are infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter-Drinfeld modules over classical Weyl groups A \rtimes \mathbb S_n supported by A to be finite dimensional.
Submitted 7 January, 2012; v1 submitted 7 October, 2009; originally announced October 2009.

Comments: Combined with arXiv:0902.4748 plus substantial changes. To appear International Journal of Mathematics

MSC Class: 16W30; 16G10

Journal ref: Int. J. Math. 23 (2012) 1250066

arXiv:0909.0435 [pdf, ps, other] math.QA
The character tables of centralizers in Sporadic Simple Groups of {\rm McL}

Authors: Shouchuan Zhang, Jieqiong He, Guichao Wu

Abstract: To classify the finite dimensional pointed Hopf algebras with G= {\rm McL} we obtain the representatives of conjugacy classes of G and all character tables of centralizers of these representatives by means of software {\rm GAP}.
Submitted 2 September, 2009; originally announced September 2009.

Comments: 339pages

MSC Class: 16W30; 20D06

arXiv:0906.0410 [pdf, ps, other] math.GR math.QA
On Pointed Hopf Algebras with Sporadic Simple Groups {\rm HS} and {\rm Co3}

Authors: Shouchuan Zhang, Jing Cheng

Abstract: Every non quasi- -1-type Nichols algebra is infinite dimensional. All quasi- -1-type Nichols algebra over sporadic simple groups {\rm HS} and {\rm Co3} are found.
Submitted 12 September, 2009; v1 submitted 1 June, 2009; originally announced June 2009.

Comments: 9pages

MSC Class: 16W30; 16G10

arXiv:0906.0408 [pdf, ps, other] math.GR
The character tables of centralizers in Sporadic Simple Groups of {\rm HS} and {\rm Co_3}

Authors: Shouchuan Zhang, Jing Cheng, Jieqiong He

Abstract: To classify the finite dimensional pointed Hopf algebras with G= {\rm HS} or {\rm Co3} we obtain the representatives of conjugacy classes of G and all character tables of centralizers of these representatives by means of software {\rm GAP}.
Submitted 1 June, 2009; originally announced June 2009.

Comments: 409 pages

MSC Class: 16W30; 20D06

arXiv:0902.4748 [pdf, ps, other] math.QA
The radicals of (Braided) Lie algebras

Authors: Lingwei Guo, Shouchuan Zhang, Junqin Li

Abstract: The general theory of the radicals of Lie algebras are established. Baer radicals of untwisted affine Lie algebras are found.
Submitted 26 May, 2014; v1 submitted 26 February, 2009; originally announced February 2009.

Comments: 14pages

MSC Class: 17B65; 17B67

arXiv:0807.2406 [pdf, ps, other] math.QA
On pointed Hopf algebras associated with the symmetric groups

Authors: Nicolás Andruskiewitsch, Fernando Fantino, Shouchuan Zhang

Abstract: It is an important open problem whether the dimension of the Nichols algebra B(O,ρ) is finite when O is the class of the transpositions and ρis the sign representation, with m>= 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs (O,ρ) might give rise to finite-dimensional Nichols algebras.
Submitted 24 July, 2008; v1 submitted 15 July, 2008; originally announced July 2008.

Comments: 14 pages; typos added in abstract

MSC Class: 16W30; 17B37

Journal ref: Manuscripta Math., Vol. 128, N. 3 (March 2009). Pages 359-371

arXiv:0804.2602 [pdf, ps, other] math.QA
On Pointed Hopf Algebras with Weyl Groups of exceptional type

Authors: Shouchuan Zhang, Yao-Zhong Zhang, Peng Wang, Jing Cheng, Hui Yang

Abstract: All -1-type pointed Hopf algebras and central quantum linear spaces with Weyl groups of exceptional type are found. It is proved that every non -1-type pointed Hopf algebra with real G(H) is infinite dimensional and every central quantum linear space over finite group is finite dimensional. It is proved that except a few cases Nichols algebras of reducible Yetter-Drinfeld modules over Weyl g… ▽ More
Submitted 4 April, 2009; v1 submitted 16 April, 2008; originally announced April 2008.

Comments: 44pages

MSC Class: 16W30; 16G10

arXiv:0804.2005 [pdf, ps, other] math.QA
The character tables of centralizers in Weyl Group of E_8 V