Standard Lyndon bases of Lie algebras and enveloping algebras

Abstract

It is well known that the standard bracketings of Lyndon words in an alphabet<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"><mml:semantics><mml:mi>A</mml:mi><mml:annotation encoding="application/x-tex">A</mml:annotation></mml:semantics></mml:math></inline-formula>form a basis for the free Lie algebra<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Lie left-parenthesis upper A right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext>Lie</mml:mtext></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">{\text {Lie}}(A)</mml:annotation></mml:semantics></mml:math></inline-formula>generated by<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"><mml:semantics><mml:mi>A</mml:mi><mml:annotation encoding="application/x-tex">A</mml:annotation></mml:semantics></mml:math></inline-formula>. Suppose that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g approximately-equals Lie left-parenthesis upper A right-parenthesis slash upper J"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">g</mml:mi></mml:mrow><mml:mo>≅</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext>Lie</mml:mtext></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {g} \cong {\text {Lie}}(A)/J</mml:annotation></mml:semantics></mml:math></inline-formula>is a Lie algebra given by a generating set<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"><mml:semantics><mml:mi>A</mml:mi><mml:annotation encoding="application/x-tex">A</mml:annotation></mml:semantics></mml:math></inline-formula>and a Lie ideal<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"><mml:semantics><mml:mi>J</mml:mi><mml:annotation encoding="application/x-tex">J</mml:annotation></mml:semantics></mml:math></inline-formula>of relations. Using a Gröbner basis type approach we define a set of "standard" Lyndon words, a subset of the set Lyndon words, such that the standard bracketings of these words form a basis of the Lie algebra<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">g</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {g}</mml:annotation></mml:semantics></mml:math></inline-formula>. We show that a similar approach to the universal enveloping algebra<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">g</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {g}</mml:annotation></mml:semantics></mml:math></inline-formula>naturally leads to a Poincaré-Birkhoff-Witt type basis of the enveloping algebra of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">g</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {g}</mml:annotation></mml:semantics></mml:math></inline-formula>. We prove that the standard words satisfy the property that any factor of a standard word is again standard. Given root tables, this property is nearly sufficient to determine the standard Lyndon words for the complex finite-dimensional simple Lie algebras. We give an inductive procedure for computing the standard Lyndon words and give a complete list of the standard Lyndon words for the complex finite-dimensional simple Lie algebras. These results were announced in [LR].