FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2000, VOLUME 6, NUMBER 3, PAGES 757-776
T. V. Dubrovina
N. I. Dubrovin
Abstract
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The equation $x^n=g$ has been solved in the universal covering
group $\mathbb G$ of the group $\mathop{\mathrm{SL}}(2)$ .
If $g$ is not a central
element, then the $n$ -th root of $g$ exists and is unique. In the case
when $g$ belongs to the center of the universal covering $\mathbb G$ ,
the set of all solutions may be empty or may form a two-dimensional submanifold of the manifold $\mathbb G$ . The following two
questions are considered. (A) How wide may be this submanifold
from the algebraic point of view? (B) How can we complete
the group $\mathbb G$ with absent roots?
Of the results close to the main theorem one can mention
the following: the semigroup $\mathop{\mathrm{SL}}(2)^+$ ,
consisting of all matrices
$A\in\mathop{\mathrm{SL}}(2)$ with non-negative coefficients,
is complete, that
is one can derive any root from any element.
All articles are published in Russian.
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Last modified: December 8, 2000