| Summary: | It is shown that for any integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≥</mo><mn>2</mn><mi>k</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mi>q</mi><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula>, there exists a real solvable Lie algebra of the first rank with a maximal torus of derivations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">t</mi></semantics></math></inline-formula> possessing the eigenvalue spectrum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>spec</mi><mrow><mo>(</mo><mi mathvariant="fraktur">t</mi><mo>)</mo></mrow><mo>=</mo><mfenced separators="" open="(" close=")"><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>k</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>⋯</mo><mo>,</mo><mi>N</mi></mfenced></mrow></semantics></math></inline-formula>, a nilradical of the nilpotence index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>−</mo><mi>k</mi></mrow></semantics></math></inline-formula> and a characteristic sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mi>k</mi><mo>,</mo><msup><mn>1</mn><mi>k</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula>.
|
|---|