HNN Extensions in Lie Superalgebras

• The paper introduces HNN extensions of Lie superalgebras that adjoin a stable homogeneous generator to extend a graded algebra while preserving key bracket relations.
• It employs Gröbner–Shirshov bases to derive a normal form and structural decomposition, ensuring faithful embedding and a PBW-type presentation.
• Applications include embedding countable-dimensional Lie superalgebras and analyzing finite generation of ideals via controlled free Lie-superalgebraic constructions.

Searching arXiv for the cited HNN-extension papers and related Gröbner–Shirshov basis work in Lie superalgebras. HNN extensions of Lie superalgebras are constructions in which a Lie superalgebra $L=L_{0}\oplus L_{1}$ is enlarged by adjoining a new homogeneous generator $t$ and imposing bracket relations that encode a prescribed homogeneous derivation $d$ on a graded subalgebra $A\subseteq L$. In the formulations developed in "HNN-extension of Lie superalgebras" (Ladra et al., 24 Jan 2026) and "HNN extensions of Lie superalgebras" (Kochloukova et al., 9 Sep 2025), the resulting object is the Lie superalgebra

$H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$

with $|t|=|d|$, over a ground field of characteristic $\ne 2,3$. The construction is accompanied by embedding theorems, Gröbner–Shirshov normal forms, an explicit structural decomposition, and applications to two-generator embeddings and finite generation problems.

1. Definition and algebraic setting

Let $L=L_{0}\oplus L_{1}$ be a Lie superalgebra, let $A\subseteq L$ be a graded subalgebra, and let $d:A\to L$ be a homogeneous derivation of degree $t$0. The derivation condition is

$t$1

The HNN extension of $t$2 relative to $t$3 is defined by adjoining a new homogeneous generator $t$4 of degree $t$5 and imposing the relations

$t$6

together with all original brackets in $t$7 (Ladra et al., 24 Jan 2026).

An equivalent presentation is given when $t$8 and $t$9 are described by homogeneous generators and relations. In that case,

$d$0

where $d$1 is any preimage of $d$2 (Kochloukova et al., 9 Sep 2025).

The standing hypothesis that the base field has characteristic $d$3 is used so that the super-Jacobi identities and PBW-type arguments work in the usual way. The data also require that $d$4 be graded and that the derivation $d$5 be graded, either even or odd (Ladra et al., 24 Jan 2026).

2. Presentations and Gröbner–Shirshov framework

A concrete construction proceeds by choosing a homogeneous basis $d$6 of $d$7 containing a basis $d$8 of the subalgebra $d$9, together with a total order

$A\subseteq L$0

The Lie bracket on $A\subseteq L$1 is written in terms of structure constants

$A\subseteq L$2

with

$A\subseteq L$3

and the Jacobi identities are encoded by the relations

$A\subseteq L$4

The derivation $A\subseteq L$5 is expressed on $A\subseteq L$6 by

$A\subseteq L$7

subject to the compatibility condition

$A\subseteq L$8

that restates the derivation law in coordinates (Ladra et al., 24 Jan 2026).

The HNN extension is then presented as a quotient of the free Lie superalgebra on $A\subseteq L$9 by homogeneous relations

$H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$0

$H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$1

$H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$2

The set

$H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$3

is shown, via the Composition–Diamond–Shirshov approach, to be a Gröbner–Shirshov basis: all compositions among the defining relations either vanish or reduce to lower-order relations (Ladra et al., 24 Jan 2026).

The later treatment makes the Gröbner–Shirshov machinery more explicit. It works in the free associative superalgebra $H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$4 on a $H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$5-graded set $H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$6, viewed as a Lie superalgebra by

$H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$7

A total order on $H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$8 induces a length-lex order on $H=\langle L,t:[t,a]=d(a)\;(a\in A)\rangle,$9. The relevant monomials are super–Lyndon–Shirshov words: either Lyndon–Shirshov words, or squares $|t|=|d|$0 of odd Lyndon–Shirshov words. Each such word $|t|=|d|$1 has a unique standard bracketing $|t|=|d|$2 whose leading associative word is $|t|=|d|$3. The normal-form theorem states that if $|t|=|d|$4 is generated by a Gröbner–Shirshov set $|t|=|d|$5, then the images of all super–Lyndon–Shirshov monomials $|t|=|d|$6 whose underlying word $|t|=|d|$7 contains no subword $|t|=|d|$8 with $|t|=|d|$9, form a $\ne 2,3$0-basis of $\ne 2,3$1 (Kochloukova et al., 9 Sep 2025).

3. Universal embedding theorem

A central theorem states that every Lie superalgebra $\ne 2,3$2 embeds into its HNN extension $\ne 2,3$3 (Ladra et al., 24 Jan 2026). The argument is combinatorial and depends on the Gröbner–Shirshov basis obtained from the defining relations.

The proof outline uses the Composition–Diamond lemma for Lie superalgebras, cited there with Bokut–Kang–Lee–Malcolmson, together with Shirshov’s lemma: if $\ne 2,3$4 lies in the ideal generated by $\ne 2,3$5, then its leading monomial contains the leading monomial of some $\ne 2,3$6 as a subword. Since none of the leading monomials of $\ne 2,3$7 lies entirely in words on the original alphabet $\ne 2,3$8, no nonzero element of the subalgebra generated by $\ne 2,3$9 is annihilated by the defining relations. Therefore the natural map

$L=L_{0}\oplus L_{1}$0

is injective (Ladra et al., 24 Jan 2026).

This embedding theorem is the Lie-superalgebraic analogue of the injectivity property classically associated with HNN constructions in combinatorial group theory. In the super setting, however, the proof runs through Gröbner–Shirshov reduction rather than through van Kampen diagrams or Britton’s lemma (Ladra et al., 24 Jan 2026).

4. Normal forms, bases, and direct-sum structure

The explicit basis theory developed in (Kochloukova et al., 9 Sep 2025) refines the basic embedding result by describing the internal structure of the HNN extension. If one chooses a homogeneous complement $L=L_{0}\oplus L_{1}$1 of $L=L_{0}\oplus L_{1}$2 in $L=L_{0}\oplus L_{1}$3, writes

$L=L_{0}\oplus L_{1}$4

and forms the associated set of relations $L=L_{0}\oplus L_{1}$5, then $L=L_{0}\oplus L_{1}$6 is closed under all Lie compositions. Hence the HNN extension admits a normal form in terms of super–Lyndon–Shirshov monomials whose underlying words avoid the leading subwords

$L=L_{0}\oplus L_{1}$7

These monomials form a linear basis of

$L=L_{0}\oplus L_{1}$8

(Kochloukova et al., 9 Sep 2025).

The same analysis yields a PBW-type description of $L=L_{0}\oplus L_{1}$9. The summary states that the PBW basis of $A\subseteq L$0 may be written in the form

$A\subseteq L$1

where each block $A\subseteq L$2 is a nonzero ordered monomial in the $A\subseteq L$3, each $A\subseteq L$4, and $A\subseteq L$5. The corresponding Lyndon–Shirshov bracketings give a basis of $A\subseteq L$6 (Kochloukova et al., 9 Sep 2025).

The structural theorem is sharper still. Define

$A\subseteq L$7

Then

$A\subseteq L$8

where $A\subseteq L$9 is the free Lie superalgebra on the set $d:A\to L$0 (Kochloukova et al., 9 Sep 2025). Equivalently, inside $d:A\to L$1 the subalgebra generated by the associative words $d:A\to L$2 is free on those words, and the Lyndon–Shirshov bracketings produce the free Lie subalgebra $d:A\to L$3.

This decomposition isolates the original Lie superalgebra $d:A\to L$4 and the new free part generated by the stable letter $d:A\to L$5 together with ordered words in the complementary generators. A plausible implication is that HNN extensions in this setting are not merely embedding devices but also explicit mechanisms for adjoining a controlled free Lie-superalgebraic component.

5. Applications to embedding and finite generation

One application is an embedding theorem for countable-dimensional Lie superalgebras. If $d:A\to L$6 has at most countable dimension, one may write a countable generating set $d:A\to L$7, form the free product

$d:A\to L$8

and inside $d:A\to L$9 consider the graded subalgebra $t$00 generated by the left-nested commutators

$t$01

By a standard lemma these $t$02 form a free basis of $t$03. Defining

$t$04

the freeness of $t$05 implies that $t$06 extends uniquely to a derivation of $t$07 into $t$08. The HNN extension

$t$09

is generated by $t$10, contains $t$11 and hence $t$12, and is therefore a two-generator Lie superalgebra containing $t$13. Injectivity of $t$14 follows from the universal embedding theorem (Ladra et al., 24 Jan 2026).

A second application concerns finite generation of ideals. The summary of (Kochloukova et al., 9 Sep 2025) states that if $t$15 is finitely presented and has an ideal $t$16 with

$t$17

then $t$18 is finitely generated as a Lie superalgebra, provided $t$19 does not itself contain a nonabelian free Lie subsuperalgebra. The proof uses a “shifting-to-HNN” argument to realize

$t$20

for suitable finitely generated $t$21. The structure theorem then forces $t$22, because otherwise the free part $t$23 would be nontrivial and would produce a nonabelian free Lie superalgebra inside $t$24 (Kochloukova et al., 9 Sep 2025).

Together, these applications show that HNN extensions serve both as embedding instruments and as tools for deriving structural finiteness consequences.

6. Relation to the classical group-theoretic HNN construction

The analogy with the classical HNN extension of group theory is explicit in the source material. In both settings one adjoins a stable letter $t$25 to an original object and imposes relations describing the action of $t$26 on a distinguished substructure. In both settings the construction is used to embed the original object into a larger one with prescribed additional relations (Ladra et al., 24 Jan 2026).

The differences are equally important. In groups, one starts from subgroup isomorphisms $t$27 and imposes relations of the form

$t$28

For Lie superalgebras, one starts instead from a derivation $t$29 on a graded subalgebra $t$30 and imposes bracket relations

$t$31

The group-theoretic proofs of injectivity typically use van Kampen diagrams or Britton’s lemma, whereas the Lie-superalgebraic proof uses Gröbner–Shirshov bases and the Composition–Diamond lemma. The $t$32-grading has no direct analogue in the classical group setting, although in the super context it affects the sign rules in the bracket identities (Ladra et al., 24 Jan 2026).

This comparison clarifies the conceptual role of the construction. The Lie-superalgebraic HNN extension mirrors the group-theoretic pattern at the level of adjoining a stable letter and enforcing a prescribed action, but the mechanism is super-derivational rather than conjugational, and its technical implementation is fundamentally reduction-theoretic.

7. Scope and conceptual significance

The papers (Ladra et al., 24 Jan 2026) and (Kochloukova et al., 9 Sep 2025) establish a coherent theory of HNN extensions for Lie superalgebras over fields of characteristic $t$33. The theory includes a precise definition, a Gröbner–Shirshov presentation, a normal-form theorem in terms of super–Lyndon–Shirshov monomials, the universal embedding theorem, and the structural decomposition

$t$34

Within this framework, the HNN extension is not only a formal enlargement of $t$35. It preserves $t$36 faithfully inside a larger Lie superalgebra, yields explicit bases and forbidden-subword descriptions, and supports embedding results such as the theorem that every Lie superalgebra of at most countable dimension embeds into a two-generator Lie superalgebra (Ladra et al., 24 Jan 2026). It also supports finiteness arguments for ideals in finitely presented Lie superalgebras under the absence of nonabelian free Lie subsuperalgebras (Kochloukova et al., 9 Sep 2025).

The combination of superalgebraic derivations, Composition–Diamond methods, and free Lie superalgebra structure places HNN extensions of Lie superalgebras within the broader program of using Gröbner–Shirshov techniques to obtain explicit and functorial embedding constructions.