The Word Problem for Solvable Groups and Lie Algebras

1992 ◽  
pp. 61-67 ◽  
Author(s):  
O. Kharlampovich
Keyword(s):  
Lie Algebras ◽  
Word Problem ◽  
Solvable Groups

Varieties of lie algebras with solvable word problem

1993 ◽  
Vol 21 (10) ◽  
pp. 3571-3609
Author(s):  
O. Kharlampovich ◽  
D. Gildenhuys
Keyword(s):  
Lie Algebras ◽  
Word Problem ◽  
Solvable Word Problem ◽  
Solvable Word

Solvability of Equations in Classes of Solvable Groups and Lie Algebras

2017 ◽  
Vol 56 (3) ◽  
pp. 251-255
Author(s):  
V. A. Roman’kov
Keyword(s):  
Lie Algebras ◽  
Solvable Groups

The role of solvable groups in quantization of Lie algebras

1997 ◽  
Vol 83 (1) ◽  
pp. 106-112
Author(s):  
V. D. Lyakhovskii
Keyword(s):  
Lie Algebras ◽  
Solvable Groups

scholarly journals Gröbner–Shirshov Bases for Replicated Algebras

2017 ◽  
Vol 24 (04) ◽  
pp. 563-576 ◽  
Author(s):  
P.S. Kolesnikov
Keyword(s):  
Lie Algebras ◽  
Word Problem ◽  
Membership Problem ◽  
Leibniz Algebras ◽  
Universal Approach ◽  
Ideal Membership ◽  
Ideal Membership Problem ◽  
The Ideal

We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Gröbner–Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincaré–Birkhoff–Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD!-algebra).

CHARACTERIZATION OF SOLVABLE GROUPS AND SOLVABLE RADICAL

2013 ◽  
Vol 23 (05) ◽  
pp. 1011-1062 ◽  
Author(s):  
FRITZ GRUNEWALD ◽  
BORIS KUNYAVSKII ◽  
EUGENE PLOTKIN
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Solvable Groups ◽  
Infinite Groups ◽  
Open Problems ◽  
Solvable Radical ◽  
The Past ◽  
Arbitrary Finite Group

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.

THE WORD PROBLEM FOR SOME VARIETIES OF SOLVABLE LIE ALGEBRAS

1994 ◽  
Vol 04 (03) ◽  
pp. 481-491
Author(s):  
O. KHARLAMPOVICH ◽  
D. GILDENHUYS
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Word Problem ◽  
Characteristic Zero ◽  
Nilpotent Lie Algebras ◽  
Solvable Lie Algebras ◽  
Finitely Presented

The word problem is said to be solvable in a variety of Lie algebras if it is solvable in every algebra, finitely presented in this variety. Let [Formula: see text] denote the variety of (2-step nilpotent)-by-abelian Lie algebra and [Formula: see text] the variety of abelian-by-(2-step nilpotent) Lie algebras. It is proved that the word problem is unsolvable in the “interval” of varieties containing the variety [Formula: see text] (of centre-by-[Formula: see text] Lie algebras over a field of characteristic zero), and contained in the variety [Formula: see text].

Representations of Solvable Groups

1993 ◽  
Author(s):  
Olaf Manz ◽  
Thomas R. Wolf
Keyword(s):  
Solvable Groups
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