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Nottingham Lie algebras with diamonds of finite and infinite type [article]

Marina Avitabile, Sandro Mattarei
2013 arXiv   pre-print
We identify two subclasses of Nottingham Lie algebras as loop algebras of finite-dimensional simple Lie algebras of Hamiltonian Cartan type.  ...  We consider a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series.  ...  Nottingham algebras with diamonds of finite and infinite type In this section we will construct Nottingham Lie algebras with p s − 1 diamonds of infinite type separated by single occurrences of a diamond  ... 
arXiv:1211.4436v2 fatcat:tx336gwsuvaebp5vcl7s5cbwue

NOTTINGHAM LIE ALGEBRAS WITH DIAMONDS OF FINITE TYPE

A. CARANTI, S. MATTAREI
2004 International journal of algebra and computation  
We study a class of positively graded Lie algebras with a pattern of homogeneous components similar to that of the graded Lie algebra associated to the Nottingham group with respect to its lower central  ...  These algebras we call Nottingham Lie algebras. We are naturally led to discuss some related features of thin Lie algebras with the second diamond in weight q.  ...  In this paper we concentrate on those algebras where all diamonds are of finite type.  ... 
doi:10.1142/s0218196704001633 fatcat:2s7be6ne45hzfeq7qay4icpilq

Thin loop algebras of Albert–Zassenhaus algebras

Marina Avitabile, Sandro Mattarei
2007 Journal of Algebra  
Here we consider several classes of thin Lie algebras with second diamond in degree q.  ...  In particular, we identify the Lie algebras in one of these classes with suitable loop algebras of certain Albert-Zassenhaus Lie algebras.  ...  Nottingham Lie algebras with diamonds of finite types In this section we produce a Nottingham Lie algebra with diamonds of finite types and not all in the prime field, as the loop algebra of H (2; (1,  ... 
doi:10.1016/j.jalgebra.2007.03.001 fatcat:toe3k4wktbgdrho7sbxlnecoom

The earliest diamond of finite type in Nottingham algebras [article]

Marina Avitabile, Sandro Mattarei
2021 arXiv   pre-print
Finally, we determine how many diamonds of type ∞ may precede the earliest diamond of finite type in an arbitrary Nottingham algebra.  ...  We prove several structural results on Nottingham algebras, a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with  ...  Nottingham algebras with diamonds of finite and infinite type Let L be an infinite-dimensional Nottingham algebra with second diamond L q and standard generators x and y.  ... 
arXiv:2106.14796v1 fatcat:vyenxbzig5aypcnqzrjm7qp7uq

Diamond distances in Nottingham algebras [article]

Marina Avitabile, Sandro Mattarei
2021 arXiv   pre-print
Nottingham algebras are a class of just-infinite-dimensional, modular, ℕ-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central  ...  As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type ∞.  ...  In particular, another such situation arises where L m is a diamond of type 1 and L m+q−1 is a diamond with a type different from 1, as in the Nottingham algebras with all diamonds of finite type, see  ... 
arXiv:2011.05491v2 fatcat:l3muz4hu4jejxnieqarp7dtrku

Page 3295 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
Mattarei [“Nottingham Lie al- gebras with diamonds of finite type”, submitted] have shown that if such an algebra has third diamond of finite type “3 and not equal to —1 (and v4 #0 when y3 = 0), then the  ...  Of interest here is the case when p> 5 and the second diamond lies in dimension g. Such algebras are referred to as Nottingham Lie algebras.  ... 

The structure of thin Lie algebras up to the second diamond [article]

Marina Avitabile, Giuseppe Jurman, Sandro Mattarei
2008 arXiv   pre-print
Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of  ...  classical or of Cartan type depending on the location of the second diamond.  ...  In Section 3 we explain how graded Lie algebras of maximal class can sometimes be interpreted as thin Lie algebras of Nottingham type with fake second diamond.  ... 
arXiv:0812.1250v1 fatcat:tr3emthw4vbu7lof7scb5bcbzi

Diamonds of finite type in thin Lie algebras [article]

Sandro Mattarei, Marina Avitabile
2005 arXiv   pre-print
Specifically, we prove that, under certain technical assumptions, the degree of the earliest diamond of finite type in such a Lie algebra can only have a certain form, which does occur in explicit examples  ...  In one of the two main subclasses of thin Lie algebras the earliest diamond after that in degree one occurs in degree 2q-1, where q is a power of the characteristic.  ...  Thin Lie algebras with the second diamond in degree q have been called of Nottingham type in [Car97, Car99, Car98, CM04, AM], because the simplest example is the graded Lie algebra associated with the  ... 
arXiv:math/0511256v1 fatcat:w6uzx5ufxnfedai4e2quybx6zi

Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class

A. Caranti, S. Mattarei
1999 Journal of the Australian Mathematical Society  
A subclass of these algebras can be obtained via a twisted loop algebra construction from certain finite-dimensional, simple Lie algebras of Albert-Frank type.  ...  Another subclass of these algebras is strictly related to certain graded Lie algebras of maximal class, and exhibits a wide range of behaviours. 1991 Mathematics subject classification (Amer. Math.  ...  Then there is uniquely determined, infinite-dimensional (-l)-algebra T with second diamond in weight 2q -1, and second diamond of finite type. In T all diamonds are of finite type.  ... 
doi:10.1017/s1446788700001142 fatcat:lbcuhu4svvak5fhno55oql4qxq

A sandwich in thin lie algebras

Sandro Mattarei
2022 Proceedings of the Edinburgh Mathematical Society  
A thin Lie algebra is a Lie algebra $L$ , graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$ , and such that each non-zero ideal of $L$ lies  ...  We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.  ...  For the sake of this Introduction, we outline the example of the thin Lie algebra L associated with the Nottingham group, which has second diamond in degree p.  ... 
doi:10.1017/s0013091521000845 fatcat:xavvczsj5zcv7fjrvmsj45i6ue

Constituents of graded Lie algebras of maximal class and chain lengths of thin Lie algebras [article]

Sandro Mattarei
2021 arXiv   pre-print
Hence L_1 is a diamond, and if there are no other diamonds then L is a graded Lie algebra of maximal class.  ...  We present simpler proofs of some fundamental facts on graded Lie algebras of maximal class, and on thin Lie algebras, based on a uniform method, with emphasis on a polynomial interpretation.  ...  Thin Lie algebras with k = q have been named Nottingham algebras because a special case is related to the Nottingham group, see [Car97, AM07, AM] for a variety of examples and structural results.  ... 
arXiv:2011.04110v2 fatcat:bkuqcc3fajbmblxib6ojsjbtfi

Gradings of non-graded Hamiltonian Lie algebras

A. Caranti, S. Mattarei
2005 Journal of the Australian Mathematical Society  
2 / with a Block algebra.  ...  A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H .2 : n; !  ...  We remark that thin Lie algebras with diamonds of both finite and infinite types are constructed in [5] .  ... 
doi:10.1017/s1446788700010983 fatcat:pubzlrbx2jbq3c4cdpyd6z2seu

Gradings of non-graded Hamiltonian Lie algebras [article]

Andrea Caranti, Sandro Mattarei
2005 arXiv   pre-print
We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2;ω_2) (of dimension one less than a power of p) from which we construct infinite-dimensional thin Lie algebras.  ...  In the process we provide an explicit identification of H(2;ω_2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).  ...  We remark that thin Lie algebras with diamonds of both finite and infinite types are constructed in [AM05] .  ... 
arXiv:math/0510431v1 fatcat:7h4obhvlvffrzh3hi5vyfz33iq

Page 7715 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
(I-TRNT; Trento) Nottingham Lie algebras with diamonds of finite type. (English summary) Internat. J. Algebra Comput. 14 (2004), no. 1, 35-67.  ...  7715 17B Lie algebras and Lie superalgebras U,(g). In the authors’ previous work [J.  ... 

THIN GROUPS OF PRIME-POWER ORDER AND THIN LIE ALGEBRAS

A. CARANTI, S. MATTAREI, M. F. NEWMAN, C. M. SCOPPOLA
1996 Quarterly Journal of Mathematics  
an algebraic closure of the p-adic numbers, the Lie algebra ST is simple of type A 2 .  ...  For p = 5 this is where the Lie algebra associated with the Nottingham group comes in. By expanding + 2[tiyxy] -[uyyx] = -[w,yy] one gets [w,yy\ = 0.  ... 
doi:10.1093/qmath/47.3.279 fatcat:usy3embb3neqhphzbo6g2o3zoa
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