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

2013
*
arXiv
*
pre-print

We identify two subclasses

arXiv:1211.4436v2
fatcat:tx336gwsuvaebp5vcl7s5cbwue
*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*...##
###
NOTTINGHAM LIE ALGEBRAS WITH DIAMONDS OF FINITE TYPE

2004
*
International journal of algebra and computation
*

We study a class

doi:10.1142/s0218196704001633
fatcat:2s7be6ne45hzfeq7qay4icpilq
*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*. ...##
###
Thin loop algebras of Albert–Zassenhaus algebras

2007
*
Journal of Algebra
*

Here we consider several classes

doi:10.1016/j.jalgebra.2007.03.001
fatcat:toe3k4wktbgdrho7sbxlnecoom
*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, ...##
###
The earliest diamond of finite type in Nottingham algebras
[article]

2021
*
arXiv
*
pre-print

Finally, we determine how many

arXiv:2106.14796v1
fatcat:vyenxbzig5aypcnqzrjm7qp7uq
*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. ...##
###
Diamond distances in Nottingham algebras
[article]

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 ...

##
###
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]

2008
*
arXiv
*
pre-print

Infinite-dimensional thin

arXiv:0812.1250v1
fatcat:tr3emthw4vbu7lof7scb5bcbzi
*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*. ...##
###
Diamonds of finite type in thin Lie algebras
[article]

2005
*
arXiv
*
pre-print

Specifically, we prove that, under certain technical assumptions, the degree

arXiv:math/0511256v1
fatcat:w6uzx5ufxnfedai4e2quybx6zi
*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 ...##
###
Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class

1999
*
Journal of the Australian Mathematical Society
*

A subclass

doi:10.1017/s1446788700001142
fatcat:lbcuhu4svvak5fhno55oql4qxq
*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*. ...##
###
A sandwich in thin lie algebras

2022
*
Proceedings of the Edinburgh Mathematical Society
*

A thin

doi:10.1017/s0013091521000845
fatcat:xavvczsj5zcv7fjrvmsj45i6ue
*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. ...##
###
Constituents of graded Lie algebras of maximal class and chain lengths of thin Lie algebras
[article]

2021
*
arXiv
*
pre-print

Hence L_1 is a

arXiv:2011.04110v2
fatcat:bkuqcc3fajbmblxib6ojsjbtfi
*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. ...##
###
Gradings of non-graded Hamiltonian Lie algebras

2005
*
Journal of the Australian Mathematical Society
*

2 /

doi:10.1017/s1446788700010983
fatcat:pubzlrbx2jbq3c4cdpyd6z2seu
*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] . ...##
###
Gradings of non-graded Hamiltonian Lie algebras
[article]

2005
*
arXiv
*
pre-print

We describe several cyclic grading

arXiv:math/0510431v1
fatcat:7h4obhvlvffrzh3hi5vyfz33iq
*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] . ...##
###
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

1996
*
Quarterly Journal of Mathematics
*

an

doi:10.1093/qmath/47.3.279
fatcat:usy3embb3neqhphzbo6g2o3zoa
*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. ...
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