Non-Abelian Extensions of Bol Algebras

• The paper establishes a cohomological classification of non-abelian extensions using (2,3)-cocycles and semidirect sum constructions.
• It unifies abelian and non-abelian extension theories by analyzing automorphism lifting, Wells exact sequences, and obstruction maps.
• The paper further explores quadratic Bol algebras with T*-extensions, emphasizing invariant bilinear forms and dual coadjoint representations.

Searching arXiv for the cited Bol algebra extension papers and closely related context. arXiv search query: (Issa, 18 Mar 2026) Non-abelian extensions of Bol algebras concern short exact sequences

$$0\longrightarrow V \stackrel{i}{\longrightarrow}\hat B\stackrel{p}{\longrightarrow}B\longrightarrow 0$$

in which both the quotient $B$ and the kernel $V$ are Bol algebras and the kernel is not assumed abelian. In the recent literature, this problem is treated cohomologically through non-abelian $(2,3)$-cocycles and structurally through obstruction theory for lifting automorphisms; a complementary but narrower construction arises in the quadratic setting, where coadjoint representations and $T^*$-extensions provide canonical quadratic extensions by the dual space $T^*$ rather than a full theory of arbitrary non-abelian kernels (Zhang et al., 2 Oct 2025, Issa, 18 Mar 2026).

1. Bol-algebraic framework

A Bol algebra is a vector space equipped with a bilinear product $*$ and a trilinear product $[\, ,\, ,\, ]$ satisfying skew-symmetry in the binary product, skew-symmetry in the first two variables of the ternary product, the cyclic identity

$[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$

and two compatibility identities linking the binary and ternary operations. In the left-Bol convention used for quadratic theory, the same structure is written as $(T,*,[\![\, ,\, ,\, ]\!])$, with identities $B$0–$B$1; in particular, $B$2 is a Lie triple system, and a Bol algebra is a Lie triple system plus a compatible binary product (Issa, 18 Mar 2026).

For extension theory, the key point is that Bol algebras are binary-ternary objects, so extension data must control both products simultaneously. This is already visible at the representation level: a module over a Bol algebra $B$3 is a quadruple $B$4, where $B$5 and $B$6, subject to compatibility identities such as

$B$7

together with commutator and mixed-action relations involving $B$8, $B$9, and $V$0. A fundamental structural proposition states that $V$1 is a representation of $V$2 if and only if the semidirect sum $V$3 becomes a Bol algebra under

$V$4

$V$5

This semidirect-sum criterion is the linear model from which both abelian and non-abelian extension theories develop (Zhang et al., 2 Oct 2025).

2. From abelian $V$6-cohomology to non-abelian extension data

The abelian starting point is Issa’s $V$7-cohomology for a Bol algebra $V$8 with coefficients in a representation $V$9. A $(2,3)$0-cochain is a pair

$(2,3)$1

where $(2,3)$2 is skew-symmetric and $(2,3)$3 is skew in the first two variables, and $(2,3)$4 also satisfies the cyclic identity

$(2,3)$5

Coboundaries are determined by a linear map $(2,3)$6 and a companion $(2,3)$7, producing explicit formulas for $(2,3)$8 and $(2,3)$9, and the resulting cohomology group is

$T^*$0

In the abelian case, this theory classifies extensions in the usual cohomological manner (Zhang et al., 2 Oct 2025).

The non-abelian theory retains the $T^*$1-shape but enlarges the data set. Given a non-abelian extension

$T^*$2

and a linear section $T^*$3 with $T^*$4, one extracts maps

$T^*$5

$T^*$6

by

$T^*$7

$T^*$8

$T^*$9

$T^*$0

The extension therefore determines a quintuple $T^*$1, and the theory shows that this quintuple is precisely the appropriate non-abelian $T^*$2-cocycle data (Zhang et al., 2 Oct 2025).

3. Non-abelian $T^*$3-cocycles and reconstructed extensions

A non-abelian $T^*$4-cocycle on $T^*$5 with values in $T^*$6 is a quintuple

$T^*$7

satisfying basic skew-symmetry identities

$T^*$8

$T^*$9

$*$0

together with further compatibility identities labeled $*$1–$*$2, which encode the interaction between the Bol structures on $*$3 and $*$4 and the twisting data $*$5 (Zhang et al., 2 Oct 2025).

From such a cocycle one defines a Bol-algebra structure on the direct sum $*$6 by

$*$7

$*$8

The decisive proposition is an if-and-only-if statement: $*$9 is a Bol algebra exactly when $[\, ,\, ,\, ]$0 is a non-abelian $[\, ,\, ,\, ]$1-cocycle. This identifies non-abelian extension theory as a deformation problem for the split object $[\, ,\, ,\, ]$2, with $[\, ,\, ,\, ]$3 and $[\, ,\, ,\, ]$4 measuring the failure of the section to preserve the two Bol operations and $[\, ,\, ,\, ]$5 recording the induced action of $[\, ,\, ,\, ]$6 on the kernel (Zhang et al., 2 Oct 2025).

The same formalism clarifies the distinction between genuinely non-abelian and merely abelian extension theory. When the kernel algebra $[\, ,\, ,\, ]$7 is non-abelian, the reconstruction formulas retain the intrinsic products $[\, ,\, ,\, ]$8 and $[\, ,\, ,\, ]$9; consequently, the extension is not a linear perturbation of a module extension but a full coupling of two Bol algebra structures.

4. Equivalence and classification by non-abelian cohomology

Two non-abelian extensions

$[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$0

are equivalent when there exists a Bol algebra homomorphism $[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$1 restricting to the identity on $[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$2 and inducing the identity on $[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$3. The set of equivalence classes is denoted

$[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$4

On the cocycle side, two cocycles

$[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$5

are equivalent if there exists a linear map $[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$6 satisfying the five transformation identities

$[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$7

$[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$8

together with

$[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,$9

expressed by the corresponding formulas involving $(T,*,[\![\, ,\, ,\, ]\!])$0, $(T,*,[\![\, ,\, ,\, ]\!])$1, and $(T,*,[\![\, ,\, ,\, ]\!])$2. The resulting set of equivalence classes is denoted

$(T,*,[\![\, ,\, ,\, ]\!])$3

This is a non-abelian cohomology set rather than, in general, a cohomology group (Zhang et al., 2 Oct 2025).

The classification theorem states

$(T,*,[\![\, ,\, ,\, ]\!])$4

Concretely, if $(T,*,[\![\, ,\, ,\, ]\!])$5 is an extension and $(T,*,[\![\, ,\, ,\, ]\!])$6 a section, then

$(T,*,[\![\, ,\, ,\, ]\!])$7

is well-defined; equivalent extensions yield equivalent cocycles, equivalent cocycles yield isomorphic split models

$(T,*,[\![\, ,\, ,\, ]\!])$8

by maps of the form $(T,*,[\![\, ,\, ,\, ]\!])$9, and every cohomology class is realized by some extension. The result is the Bol-algebra analogue of classical extension classification in group and Lie-theoretic settings, but adapted to a binary-ternary algebraic category (Zhang et al., 2 Oct 2025).

5. Automorphism lifting, inducibility, and Wells exact sequences

The automorphism problem asks when a pair

$B$00

extends to an automorphism of the middle term $B$01. Such a pair is called inducible if there exists

$B$02

such that

$B$03

Equivalently, the extension diagram commutes with $B$04 on the quotient, $B$05 on the kernel, and $B$06 on the total algebra (Zhang et al., 2 Oct 2025).

Let $B$07 be the cocycle induced by a section $B$08. Then $B$09 is inducible if and only if there exists a linear map $B$10 satisfying five explicit identities: $B$11

$B$12

and the corresponding compatibility conditions for $B$13, $B$14, and $B$15. These are exactly the relations needed for the candidate lift

$B$16

to be a Bol algebra automorphism of $B$17 compatible with $B$18 (Zhang et al., 2 Oct 2025).

The cohomological formulation is cleaner. For each pair $B$19, the theory defines a transformed cocycle

$B$20

by pullback along $B$21 and pushforward along $B$22. The pair $B$23 is inducible if and only if $B$24 and $B$25 are equivalent. This yields the Wells obstruction map

$B$26

$B$27

which is independent of the choice of section. The group homomorphism

$B$28

fits into the exact sequence

$B$29

and, using $B$30, into the refined sequence

$B$31

The interpretation is explicit: $B$32 and $B$33, so inducibility is exactly the vanishing of the Wells obstruction (Zhang et al., 2 Oct 2025).

6. Quadratic Bol algebras, coadjoint representations, and $B$34-extensions

A distinct but closely related line of work considers quadratic left Bol algebras. A quadratic Bol algebra is a pair $B$35 in which $B$36 is a left Bol algebra and $B$37 is a nondegenerate symmetric bilinear form satisfying binary invariance

$B$38

and ternary invariance in either of the equivalent forms

$B$39

For Lie triple systems, these two ternary invariance conditions are equivalent, and this equivalence underlies the definition of quadratic left Bol algebra. The quadratic paper also proves a restrictive low-dimensional result: there does not exist any quadratic structure on a real two-dimensional Bol algebra with nonzero binary operation, so the only two-dimensional quadratic real Bol algebras are quadratic Lie triple systems (Issa, 18 Mar 2026).

The quadratic setting is relevant to extension theory because it supplies canonical dual and coadjoint representations. If $B$40 is quadratic, the adjoint representation

$B$41

has a dual representation

$B$42

called the coadjoint representation, and the map

$B$43

is an isomorphism from the adjoint representation to the coadjoint representation. Outside the quadratic setting, the naïve dual triple $B$44 is not automatically a representation; it exists if and only if the additional conditions

$B$45

$B$46

are satisfied. This sharply separates the quadratic adjoint case from the general nonquadratic case (Issa, 18 Mar 2026).

Starting from a $B$47-cocycle $B$48 with respect to the coadjoint representation, one defines operations on $B$49 by

$B$50

$B$51

Then

$B$52

is a Bol algebra. The canonical bilinear form $B$53 on $B$54, defined by

$B$55

is symmetric and nondegenerate, and it is invariant precisely when

$B$56

The quadratic Bol algebra $B$57 is called the $B$58-extension of $B$59 by $B$60; when $B$61, the construction gives the trivial extension. If the binary product vanishes and $B$62, this recovers the $B$63-extension of a Lie triple system. The paper explicitly notes that this framework does not develop a full theory of non-abelian extensions in the sense of the later Wells-sequence work, but it does provide key ingredients for extension theory: coadjoint representations, cocycles, and a canonical extension by the dual space (Issa, 18 Mar 2026).

7. Scope, special cases, and conceptual boundaries

The non-abelian extension theory and the quadratic $B$64-extension theory address related but nonidentical problems. The former classifies all short exact sequences with arbitrary Bol-algebra kernel $B$65 through the non-abelian cohomology set $B$66 and analyzes the extensibility of automorphisms through the Wells obstruction map and exact sequence. The latter constructs a specific extension

$B$67

attached to a quadratic Bol algebra and its coadjoint representation, with explicit invariance conditions ensuring that the resulting algebra remains quadratic (Zhang et al., 2 Oct 2025, Issa, 18 Mar 2026).

A common source of confusion is the role of the cocycles $B$68. In both settings they twist the binary and ternary operations, but their ambient meanings differ. In the non-abelian theory they are part of a quintuple $B$69 encoding a genuinely non-abelian kernel and its action data; in the quadratic $B$70-construction they are cocycles with values in the coadjoint representation on $B$71. The quadratic paper further states that the resulting extension is not central in general; rather, it is a semidirect-type or cocycle-twisted extension determined by the coadjoint action (Issa, 18 Mar 2026).

The abelian case appears as a limiting regime of the non-abelian theory. When $B$72 is abelian, the extension theory reduces to the earlier abelian classification by $B$73, and the Wells sequence simplifies to

$B$74

where $B$75 is the subgroup of compatible pairs $B$76 satisfying

$B$77

This establishes the non-abelian framework as a genuine generalization rather than a reformulation of the abelian theory (Zhang et al., 2 Oct 2025).

Taken together, these developments place non-abelian extensions of Bol algebras within a coherent cohomological and automorphism-theoretic framework. The general theory provides classification by $B$78 and obstruction theory via Wells exact sequences, while the quadratic theory isolates a canonical class of dual-space extensions whose existence depends on invariant bilinear forms and coadjoint representations.