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Construction of Finitely Presented Lie Algebras and Superalgebras

VLADIMIR P. GERDT, VLADIMIR V KORNYAK
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
We describe here an algorithm for constructing the basis of a finitely presented Lie (super)algebra and its commutator table, and its implementation in C.  ...  The finite presentations also indicate a way to q-quantize Lie (super)algebras.  ...  Ufnarovsky for fruitful discussions and useful remarks. This work was supported in part by the INTAS project No. 93-0893.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jsco.1996.0016">doi:10.1006/jsco.1996.0016</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gemyc4cdjzc4pdj3yn4y7zxbvu">fatcat:gemyc4cdjzc4pdj3yn4y7zxbvu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170927164630/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/567/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDc0NzcxNzE5NjkwMDE2NA%3D%3D.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/be/13/be13396373e31207dba2cd4703ae23082721ec5e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jsco.1996.0016"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

An algorithm for analysis of the structure of finitely presented Lie algebras

Vladimir P. Gerdt, Vladimir V. Kornyak
<span title="1997-01-01">1997</span> <i title="Centre pour la Communication Scientifique Directe (CCSD)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/aagtqr2vajamvduhte7kigeygi" style="color: black;">Discrete Mathematics &amp; Theoretical Computer Science</a> </i> &nbsp;
We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language.  ...  The finite presentations also indicate a way to q-quantize Lie algebras.  ...  Ufnarovsky for fruitful discussions and useful remarks. This work was supported in part by the INTAS project No. 93-893.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.46298/dmtcs.243">doi:10.46298/dmtcs.243</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4hboqy3axvbwllbinbowcelmla">fatcat:4hboqy3axvbwllbinbowcelmla</a> </span>
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An Implementation in C of an Algorithm for Construction of Finitely Presented Lie Superalgebras

V. Gerdt, V. Kornyak
<span title="">1996</span> <i title="Institute of Mathematics and Computer Science of the Academy of Sciences of Moldova"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/xdcxxn6dzrcflkahbh3lz5z5ja" style="color: black;">Computer Science Journal of Moldova</a> </i> &nbsp;
The program is based on an algorithm of constructing complete set of relations called also standard basis or Grobner basis of ideal of free Lie (super)algebra generated by the input set of relations.  ...  The purpose of this paper is to describe a C program FPLSA for investigating finitely presented Lie algebras and superalgebras.  ...  A finitely presented Lie (super)algebra L F with an empty set of defining relations is called a free Lie (super)algebra.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://doaj.org/article/4e3932f8ed1a4926b33f7f47d83539bd">doaj:4e3932f8ed1a4926b33f7f47d83539bd</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eekwzltiefcg5hj3s5gwgzh25i">fatcat:eekwzltiefcg5hj3s5gwgzh25i</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171202063855/http://www.math.md/nrofdownloads.php?file=/files/csjm/v4-n3/v4-n3-(pp399-427).pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/fd/07/fd07a2e122075699b321b126c00bf9c9ebd50358.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a>

Noncommutative Grobner Bases for Almost Commutative Algebras [article]

Huishi Li
<span title="2007-01-04">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Therefor, every quotient algebra of the enveloping algebra U(g) of a finite dimensional K-Lie algebra g is, as a noncommutative algebra of the form A=K< X> /I, defined by a finite Gröbner basis in K< X  ...  algebra G(A) is commutative, then I is generated by a finite Gröbner basis.  ...  Equivalently, for a finite dimensional K-Lie algebra g, every quotient algebra of the enveloping algebra U (g) of g, viewed as a quotient of some free K-algebra, is defined by a finite Gröbner basis.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0701120v1">arXiv:math/0701120v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xdte4flr4vaylinvqq7mkdkfki">fatcat:xdte4flr4vaylinvqq7mkdkfki</a> </span>
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Gröbner bases of ideals invariant under endomorphisms

Vesselin Drensky, Roberto La Scala
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
We calculate the Groebner S-bases of the ideal corresponding to the universal enveloping algebra of the free nilpotent of class 2 Lie algebra and of the T-ideal generated by the polynomial identity [x,  ...  In the latter case, if |X|>2, the ordinary Groebner basis is infinite and our Groebner S-basis is finite. We obtain also explicit minimal Groebner bases of these ideals.  ...  Acknowledgements The authors are very grateful to the anonymous referees for the numerous suggestions which led to the improvement of the exposition and the extension of the list of references.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jsc.2006.04.004">doi:10.1016/j.jsc.2006.04.004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/d2blbtk6hvgpljqei3zwmtb5ry">fatcat:d2blbtk6hvgpljqei3zwmtb5ry</a> </span>
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Gröbner bases in universal enveloping algebras of Leibniz algebras

Manuel A. Insua, Manuel Ladra
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
We give a proof of the Poincaré-Birkhoff-Witt theorem for universal enveloping algebras of finite dimensional Leibniz algebras using Gröbner bases in a free associative algebra.  ...  On the finiteness of Gröbner bases computation in quotients of the free algebra. Appl. Algebra Engrg. Comm. Comput. 11,.  ...  In the present paper we give a different proof of this theorem using Gröbner bases.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jsc.2007.07.020">doi:10.1016/j.jsc.2007.07.020</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fi7dlsj65nbrnabnokiwqqpwfy">fatcat:fi7dlsj65nbrnabnokiwqqpwfy</a> </span>
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Gröbner bases for p-group algebras [article]

David J. Green
<span title="2009-10-09">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
For the so-called Jennings word ordering, based on a special power-conjugate group presentation, the associated monomial algebra is a group invariant.  ...  Experiment shows that the reverse length-lexicographical word ordering consistently yields far smaller Gr\"obner bases for modular p-group algebras than the length-lexicographical ordering.  ...  I am very grateful to the Institute for the continuing use of their computing facilities. Gröbner bases First we recall the basics about Gröbner bases in free associative algebras.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0910.1699v1">arXiv:0910.1699v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wlnq5yo2vvhyzpkw3wi3oebfii">fatcat:wlnq5yo2vvhyzpkw3wi3oebfii</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-0910.1699/0910.1699.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a8/15/a81544b324268d7e709dce66aeb6bed6692bfd74.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0910.1699v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Page 1776 of Mathematical Reviews Vol. , Issue 2000c [page]

<span title="">2000</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
(D-BCHMM; Bochum) On infinite Grobner bases in free algebras. Indag. Math. (N.S.) 9 (1998), no. 4, 491-501.  ...  In the paper under review the author studies Groébner bases of ideals in the (noncommutative) free associative algebra K (S) gen- erated by a finite ordered set S over a field K.  ... 
<span class="external-identifiers"> </span>
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Page 3144 of Mathematical Reviews Vol. , Issue 2002E [page]

<span title="">2002</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Constructing bases of finitely presented Lie algebras using Grobner bases in free algebras.  ...  Using well-known techniques of standard bases of ideals of free non-associative algebras, the authors obtain an algorithm for com- puting a basis of a finitely presented finite-dimensional Lie algebra  ... 
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Page 143 of Mathematical Reviews Vol. , Issue 2002A [page]

<span title="">2002</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Mat. 36 (1972), 1173-1219; MR 48 #8588] Then, in Sections 3-6, the authors consider bases of free Lie algebras, combinatorics of Lyndon-Shirshov words, Lie algebras with one defining relation, algebraically  ...  Now as a result of the efforts of the first author and some other mathematicians, One starts to use more and more the notion of standard) Grobner-Shirshov bases instead of Grébner bases, not only in the  ... 
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A Presentation of The Frege Lie Algebra F/γ3 (F)'

Gülistan Kaya GÖK
<span title="2016-04-15">2016</span> <i title="Mathematical Sciences and Applications E-Notes"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/b7x4owk6nzg5lbka6smojvqxqy" style="color: black;">Mathematical sciences and applications e-notes</a> </i> &nbsp;
By using the technique of Gröbner-Shirshov bases we show that the Lie algebra F/γ 3 (F ) has the presentation x, y | ∆ , where ∆ is the minimal Gröbner basis of the algebra γ 3 (F ) .  ...  Let F be a free Lie algebra generated by the free generators x and y.  ...  The technique of Gröbner-Shirsov bases is very useful in the study of presentations of Lie algebras, associative algebras, groups, etc., by generators and defining relations (see [4] , [5] , [6] ,  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.36753/mathenot.421354">doi:10.36753/mathenot.421354</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rdevaaetzvaulnczscssai2ujy">fatcat:rdevaaetzvaulnczscssai2ujy</a> </span>
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An effective criterion for Nielsen-Schreier varieties [article]

Vladimir Dotsenko, Ualbai Umirbaev
<span title="2022-05-11">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Using methods of the operad theory, we propose an effective combinatorial criterion for that property in the case of algebras over a field of zero characteristic.  ...  All algebras of a certain type are said to form a Nielsen-Schreier variety if every subalgebra of every free algebra is free.  ...  Not only had he been interested in Nielsen-Schreier varieties of algebras throughout his mathematical life [7, 8, 9, 10] , but also his mathematical heritage is strongly connected with the operad theory  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2205.05364v1">arXiv:2205.05364v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ixmyc7n2znh5jeon466ox6uvee">fatcat:ixmyc7n2znh5jeon466ox6uvee</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220515221910/https://arxiv.org/pdf/2205.05364v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/10/cb/10cb2c95103b805c199888e7fcf68e7c4fdc0c12.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2205.05364v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Operadic approach to HNN-extensions of Leibniz algebras [article]

Georg Klein, Chia Zargeh
<span title="2022-03-11">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We construct HNN-extensions of Lie di-algebras in the variety of di-algebras and provide a presentation for the replicated HNN-extension of a Lie di-algebras.  ...  As an application of HNN-extensions, we prove that Lie di-algebras are embedded in their HNN-extension.  ...  The theory of Gröbner-Shirshov bases is parallel to the theory of Gröbner bases and was introduced for ideals of free (commutative, anti-commutative) non associative algebras, free Lie algebras and simplicity  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2203.05776v1">arXiv:2203.05776v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pdiy7o46e5dffcjjw33bhnuwoq">fatcat:pdiy7o46e5dffcjjw33bhnuwoq</a> </span>
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Some remarks on the Akivis algebras and the Pre-Lie algebras

Yuqun Chen, Yu Li
<span title="">2011</span> <i title="Institute of Mathematics, Czech Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z3svrlve6velboutarmbhdzqzy" style="color: black;">Czechoslovak Mathematical Journal</a> </i> &nbsp;
Based on it, he gave the algorithm to construct a Gröbner-Shirshov basis for any ideal of a free Lie algebra. The same algorithm is valid in the associative case.  ...  problem for any finitely presented homogeneous Lie algebra.  ...  Bokut for his kind guidance, useful discussions and enthusiastic encouragement.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10587-011-0041-y">doi:10.1007/s10587-011-0041-y</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/x6ubrx4qanfntlrmg6t5h433tu">fatcat:x6ubrx4qanfntlrmg6t5h433tu</a> </span>
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Plural, a Non–commutative Extension of Singular: Past, Present and Future [chapter]

Viktor Levandovskyy
<span title="">2006</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In the system, we provide rich functionality for symbolic computation within a wide class of noncommutative algebras.  ...  We describe the non-commutative extension of the computer algebra system Singular, called Plural.  ...  Acknowledgments I am grateful to Gert-Martin Greuel, Hans Schönemann and Gerhard Pfister for long and fruitful cooperation, and for their role in the development of Plural.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11832225_13">doi:10.1007/11832225_13</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/e6tjiwcaabgfrazaex427wz5ca">fatcat:e6tjiwcaabgfrazaex427wz5ca</a> </span>
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