{"ID":23029916,"CreatedAt":"2026-09-17T05:12:50.116335254Z","UpdatedAt":"2026-09-17T05:12:50.116335254Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.19086","arxiv_id":"2609.19086","title":"On the number of modular pairs in finite dimensional Lie algebras on finite fields","abstract":"Given a finite dimensional Lie algebra $L$ on a finite field $\\mathbb{F}_{p^n}$ of prime power order $p^n$ (with $n$ positive integer and $p$ prime), we consider the number of modular pairs $(A,B)$ in the lattice of all subalgebras $\\mathcal{L}(L)$ and introduce the notion of ``subalgebra commutativity degree'' of $L$. This represents the probability to find that two randomly chosen subalgebras $A$ and $B$ of $L$ are permutable. We investigate the subalgebra commutativity degree of $L$ in connection with recent techniques of algebraic combinatorics and number theory, providing upper and lower bounds which may influence the structure of $L$. A specific study for the subalgebra commutativity degree of Heisenberg algebras is executed.","short_abstract":"Given a finite dimensional Lie algebra $L$ on a finite field $\\mathbb{F}_{p^n}$ of prime power order $p^n$ (with $n$ positive integer and $p$ prime), we consider the number of modular pairs $(A,B)$ in the lattice of all subalgebras $\\mathcal{L}(L)$ and introduce the notion of ``subalgebra commutativity degree'' of $L$....","url_abs":"https://arxiv.org/abs/2609.19086","url_pdf":"https://arxiv.org/pdf/2609.19086v1","authors":"[\"Seid Kassaw Muhie\",\"Daniele Ettore Otera\",\"Francesco G. Russo\"]","published":"2026-09-16T17:25:39Z","proceeding":"math.RA","tasks":"[\"math.RA\",\"math.CO\",\"math.GR\"]","methods":"[]","has_code":false}