Almost primitive elements of free nonassociative algebras of small ranks

Klimakov, A.; Mikhalev, A.
September 2012
Journal of Mathematical Sciences;Sep2012, Vol. 185 Issue 3, p430
Academic Journal
Let K be a field, X = { x ,..., x }, and let F( X) be the free nonassociative algebra over the field K with the set X of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free. A subset M of nonzero elements of the algebra F( X) is said to be primitive if there is a set Y of free generators of F( X), F( X) = F( Y ), such that M ⊆ Y (in this case we have | Y| = | X| = n). A nonzero element u of the free algebra F( X) is said to be almost primitive if u is not a primitive element of the algebra F( X), but u is a primitive element of any proper subalgebra of F( X) that contains it. In this article, for free nonassociative algebras of rank 1 and 2 criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed. New examples of almost primitive elements of free nonassociative algebras of rank 3 are constructed.


Related Articles

  • Indecomposable large sets of Steiner triple systems with indices 5, 6. Cheng, Mei; Tian, Zi // Acta Mathematica Sinica;Nov2012, Vol. 28 Issue 11, p2169 

    A family ( X, B), ( X, B), ..., ( X, B) of q STS( υ)s is a λ-fold large set of STS( υ) and denoted by LSTS( υ) if every 3-subset of X is contained in exactly λ STS( υ)s of the collection. It is indecomposable and denoted by IDLSTS( υ) if there does not exist an LSTS(...

  • Necessary and sufficient conditions for a variety of Leibniz algebras to have polynomial growth. Mishchenko, S.P.; Cherevatenko, O.I. // Journal of Mathematical Sciences;Aug2008, Vol. 152 Issue 2, p282 

    We study the behavior of the codimension sequence of polynomial identities of Leibniz algebras over a field of characteristic 0. We prove that a variety V has polynomial growth if and only if the condition holds, where N 2 A is the variety of Lie algebras defined by the identity $$ (x_1 x_2...

  • ON CONMUTATIVE LEFT-NILALGEBRAS OF INDEX 4. GUTIERREZ FERNANDEZ, JUAN C. // Proyecciones - Journal of Mathematics;2008, Vol. 27 Issue 1, p103 

    We first present a solution to a conjecture of (Correa, Hentzel, Labra, 2002) in the positive. We show that if A is a commutative nonassociative algebra over a field of characteristic ? 2, 3, satisfying the identity x(x(xx)) = 0, then Lat1 Lat2... Lats = 0 if t1 + t2 + ... + ts = 10, where a ? A.

  • Deformed Quantum Calogero-Moser Problems and Lie Superalgebras. Sergeev, A. N.; Veselov, A. P. // Communications in Mathematical Physics;Mar2004, Vol. 245 Issue 2, p249 

    The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized root systems suggested by V. Serganova. For the classical series a recurrent formula for the...

  • A SHORT NOTE ON LCBA--FUZZY LOGIC WITH A NON-ASSOCIATIVE CONJUNCTION. KOLAŘÍK, MIROSLAV // Discussiones Mathematicae: General Algebra & Applications;2016, Vol. 36 Issue 1, p113 

    We significantly simplify the axiomatic system LCBA for fuzzy logic with a non-associative conjunction.

  • Hartman--Mycielski functor of non-metrizable compacta. Radul, Taras; Repov�, Du�an // Proceedings of the Indian Academy of Sciences: Mathematical Scie;Aug2008, Vol. 118 Issue 3, p467 

    We investigate certain topological properties of the normal functor H, introduced by the first author, which is a certain functorial compactification of the Hartman-Mycielski construction HM. We prove that H is always open and we also find the condition when H X is an absolute retract,...

  • The decomposition of metric n-Lie algebras and the uniqueness. Zhiqi Chen; Ke Liang // Journal of Mathematical Physics;May2010, Vol. 51 Issue 5, p053507 

    In this paper, we show that the orthogonal decomposition of any metric n-Lie algebra into indecomposable nondegenerate ideals is unique up to an isometry. Moreover, the decomposition is unique up to the order of the ideals if the center of the n-Lie algebra is zero.

  • Formality of Cyclic Chains. Willwacher, Thomas // IMRN: International Mathematics Research Notices;Sep2011, Vol. 2011 Issue 17, p3939 

    We prove the cyclic formality conjecture for chains, raised by Tsygan [�Formality conjectures for chains.� Differential Topology, Infinite-dimensional Lie Algebras, and Applications 261�74. American Mathematical Society Translation Series 2, 194. Providence, RI: American...

  • The Semicontinuous Quasi-uniformity of a Frame. Ferreira, Maria João; Picado, Jorge // Kyungpook Mathematical Journal;2006, Vol. 46 Issue 2, p189 

    The semicontinuous quasi-uniformity is known to be one of the most important examples of transitive quasi-uniformities. The aim of this paper is to show that various facts in classical topology connected with the semicontinuous quasi-uniformity and semicontinuous real functions may be easily...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics