The minimal dimensions of faithful representations for Heisenberg Lie superalgebras

Abstract

This paper aims to determine the minimal dimensions and super-dimensions of faithful representations for Heisenberg Lie superalgebras over an algebraically closed field of characteristic zero.

Minimal faithful representations of the free 2-step nilpotent Lie algebra of the rank $r$

“…The situation is similar for polycyclic groups, and in particular for finitely generated torsion-free nilpotent groups (τ -groups): the works of Auslander [1], and Jennings for τ -groups [15], show that these groups can be embedded into some group of matrices over the integers, but as in the case of Lie algebras, it is difficult to provide embeddings of low dimension (compared to the Hirsch length of the given group). Obtaining algorithms towards this end is an important problem (see [13,18,21]) and, in fact, the interest in low dimensional faithful representations also applies to other type of groups and algebras (see for instance [2,17,26]). This problem for τ -groups is very closely related to that for nilpotent Lie algebras, mainly by the exp and log maps, and many ideas are borrowed from each other (see [13]).…”

On the minimal dimension of faithful representations for current Heisenberg Lie superalgebras

“…Note that h 0,n,p is isomorphic to h 0,n if deg(p) = 1 (see definitions in the introduction). Hence the following examples imply that Lemma 2.1 is invalid when m = 0, which also tells us that there is a gap in [LC15,Lemma 2.3 and Theorem 2.5] for the Heisenberg Lie superalgebra h 0,n .…”

“…The above conjecture holds for Heisenberg Lie superalgebras (i.e. deg(p) = 1) by[LC15, Theorem 2.5] and for current Heisenberg Lie algebras (i.e., h m,0,p ) by[CR09, Theorem 1.1].…”

“…For any filiform Lie algebra g of dimension no smaller than 10, a lower bound of µ(g) was found (see [B95]). In the super case, the value of µ is determined only for abelian Lie superalgebras (see [LW12,WL16]), filiform Lie superalgebras (see [WCL15]) and Heisenberg Lie superalgebras (see [LC15]).…”

The Matrix Representation of Quasi Centroids of Heisenberg Superalgebra

“…( [7]) Heisenberg Lie superalgebras can be divided into Heisenberg Lie superalgebras with even centers Hm,n and Heisenberg Lie superalgebras with odd centers Hn:…”