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Fundamentalnaya i Prikladnaya Matematika, 2000, Volume 6, Issue 3, Pages 757–776
(Mi fpm501)
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This article is cited in 4 scientific papers (total in 4 papers)
Roots in the universal covering group of the unimodular $2\times2$-matrix group
T. V. Dubrovina, N. I. Dubrovin Vladimir State University
Abstract:
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.
Received: 01.04.2000
Citation:
T. V. Dubrovina, N. I. Dubrovin, “Roots in the universal covering group of the unimodular $2\times2$-matrix group”, Fundam. Prikl. Mat., 6:3 (2000), 757–776
Linking options:
https://www.mathnet.ru/eng/fpm501 https://www.mathnet.ru/eng/fpm/v6/i3/p757
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Abstract page: | 260 | Full-text PDF : | 113 | First page: | 2 |
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