On the rank of an element of a free Lie algebra (1995)

Let <italic>L</italic> be a free Lie algebra over an arbitrary field <italic>K</italic>, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet x 1 comma ellipsis comma x Subscript n Baseline comma ellipsis EndSet comma n greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {x_1}, \ldots ,{x_n}, \ldots \} ,n \geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, be a free basis of <italic>L</italic>. We define the rank of an element <italic>u</italic> of <italic>L</italic> as the least number of free generators on which the image of <italic>u</italic> under an arbitrary automorphism of <italic>L</italic> can depend. We prove that for a homogeneous element <italic>u</italic> of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m \geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, to have rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is equivalent to another property which in the most interesting and important case when the algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L equals upper L Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">L = {L_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a finite rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> looks as follows: an arbitrary endomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ</mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an automorphism if and only if <italic>u</italic> belongs to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis phi left-parenthesis upper L Subscript n Baseline right-parenthesis right-parenthesis Superscript m"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{(\phi ({L_n}))^m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This yields in particular a simple algorithm for finding the rank of a homogeneous element and also for finding a particular automorphic image of this element realizing the rank.