Lyndon basis for Free Lie algebras is well known in the literature.

My question is,

what is the analogous combinatorial model for the case of free Lie superalgebras? what is the super analogous of Lyndon words?

Kindly share your thoughts.

Thank you.


1 Answer 1


Theorem 2.2 in Leonid A. Bokut, Seok-Jin Kang, Kyu-Hwan Lee, Peter Malcolmson, Gröbner–Shirshov Bases for Lie Superalgebras and Their Universal Enveloping Algebras, Journal of Algebra 217, Issue 2, 15 July 1999, pp. 461--495 claims a basis formed by "super-Lyndon-Shirshov monomials" whenever the base field has characteristic $\neq 2, 3$. (I suspect the requirements on the characteristic come from unclarity about what a Lie superalgebra in characteristic $2$ or $3$ is.) The proof is relegated to references.

There seems to be a basis of right-normed brackets too: E. S. Chibrikov, The Right-Normed Basis for a Free Lie Superalgebra and Lyndon–Shirshov Words, Algebra Logika 45 (2006), issue 4, pp. 458--483.

  • $\begingroup$ I have checked the first paper. Very interesting. Thank you :). $\endgroup$
    – GA316
    Feb 9, 2019 at 2:24

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