B-Extensions: Unifying Extension Conditions

• B-extensions are diverse constructions across distinct fields, characterized by extension conditions that reveal hidden structures.
• They serve as diagnostic tests in complex analysis, enable homological transfer in finite-dimensional algebras, characterize normality in field theory, and resolve degeneracy in braided fusion categories.
• Applications include meromorphic extension tests, finitistic dimension estimates, Galois-type correspondences, and minimal modular extensions in braided fusion categories.

Searching arXiv for papers on “B-extensions” to ground the terminology and scope. B-extension is not a single standardized mathematical notion. In current arXiv usage, the term denotes several distinct constructions: a polyanalytic representation arising from meromorphic extension from circle families in one and several complex variables, a quotient bifinite extension $B\subseteq A$ in the homological theory of finite-dimensional algebras, a finite field extension $L/K$ whose endomorphism algebra is generated by differential operators and automorphisms, and a minimal nondegenerate extension of a braided fusion category (Globevnik, 2010, MacQuarrie et al., 2023, Bavula, 1 Sep 2025, Johnson-Freyd et al., 2021). Across these settings, the term consistently marks an extension problem supplemented by a strong structural criterion.

1. Terminological distribution

The principal usages represented in the literature are summarized below.

Setting	Definition or normal form	Main consequence
Complex analysis	$f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$	Test family for polyanalyticity and holomorphic extendibility (Globevnik, 2010)
Finite-dimensional algebras	$B\subseteq A$ with $\operatorname{pd}_{B^e}(A/B)<\infty$	Finitistic and global dimension transfer from $A$ to $B$ (MacQuarrie et al., 2023)
Field theory	$E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$	Equivalent to normality of $L/K$ (Bavula, 1 Sep 2025)
Braided fusion categories	Minimal nondegenerate extension of $\mathcal B$	Existence for every slightly degenerate braided fusion category (Johnson-Freyd et al., 2021)

These usages are mathematically unrelated in definition, ambient category, and proof method. What they share is the role of an extension condition as a diagnostic of hidden structure: analytic regularity, homological control, maximal symmetry, or modular completion.

2. The polyanalytic “B-extension form” in complex analysis

In one complex variable, the paper on meromorphic extension from small families of circles isolates a specific normal form, explicitly described as the B-extension form,

$L/K$0

with holomorphic coefficients on $L/K$1 (Globevnik, 2010). Here $L/K$2 is the open unit disc, and for $L/K$3,

$L/K$4

is the family of circles with hyperbolic center $L/K$5. For distinct $L/K$6, $L/K$7, and $L/K$8, the central equivalence states that for every circle $L/K$9, the function $f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$0 extends holomorphically through the disc bounded by $f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$1 if and only if

$f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$2

where each $f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$3 is holomorphic on $f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$4. Conversely, every such function admits meromorphic extension from every circle in $f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$5, with the only possible pole at the center $f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$6, of degree at most $f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$7.

This identifies $f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$8 as a test family for polyanalyticity of order $f(z)=g_0(z)+g_1(z)\bar z+\cdots+g_n(z)\bar z^n$9 among continuous functions on $B\subseteq A$0. The summary further states that the result is sharp and that continuity is essential. In this usage, “B-extension” does not mean an algebraic extension; it names a representation theorem for boundary data constrained by meromorphic extension from two Möbius-invariant circle families.

The same paper derives a several-complex-variables consequence for the unit ball $B\subseteq A$1. If $B\subseteq A$2 do not lie on a complex line, then the union of the families of complex lines through $B\subseteq A$3, $B\subseteq A$4, and $B\subseteq A$5 is a test family for holomorphic extendibility of continuous functions on the sphere $B\subseteq A$6. More precisely, if a continuous function on $B\subseteq A$7 extends holomorphically into $B\subseteq A$8 along every complex line in $B\subseteq A$9, then it extends holomorphically through the whole ball. The proof is described as reducing, via the Fourier–Bohr decomposition in the fiber variable, to the one-variable polyanalytic characterization. On $\operatorname{pd}_{B^e}(A/B)<\infty$0, a function admits an expansion

$\operatorname{pd}_{B^e}(A/B)<\infty$1

and each coefficient $\operatorname{pd}_{B^e}(A/B)<\infty$2 inherits a one-variable structure analogous to polyanalyticity of order $\operatorname{pd}_{B^e}(A/B)<\infty$3.

3. Quotient bifinite extensions in the homological theory of algebras

In the representation theory of finite-dimensional algebras, a quotient bifinite extension, or simply a B-extension, is an inclusion $\operatorname{pd}_{B^e}(A/B)<\infty$4 of finite-dimensional $\operatorname{pd}_{B^e}(A/B)<\infty$5-algebras such that the $\operatorname{pd}_{B^e}(A/B)<\infty$6-bimodule $\operatorname{pd}_{B^e}(A/B)<\infty$7 has finite projective dimension, equivalently $\operatorname{pd}_{B^e}(A/B)<\infty$8 (MacQuarrie et al., 2023). This is presented as weaker than several hypotheses previously used in the literature.

The main theorem is a downward transfer statement for the finitistic dimension conjecture. If $\operatorname{pd}_{B^e}(A/B)<\infty$9 is a quotient bifinite extension and $A$0, then $A$1. The proof strategy is to start from the exact sequence

$A$2

and then control projective dimensions of syzygies of a finitely generated $A$3-module $A$4 with finite projective dimension. The argument yields an explicit uniform estimate

$A$5

where $A$6.

A parallel result holds for global dimension: under the same quotient bifinite hypothesis, $A$7 implies $A$8. The converse is stated to be false in general. The summary also emphasizes that earlier results of Xi and others required stronger assumptions such as projectivity of $A$9 as a $B$0-bimodule or finite projective dimension of $B$1 as a $B$2-module, whereas here finite bimodule projective dimension of $B$3 is the decisive condition.

The examples are designed to show that the algebra $B$4 can be substantially more complicated than $B$5. In the cited constructions, $B$6 is a monomial algebra with finite finitistic dimension, while $B$7 is a subalgebra with additional non-monomial relations, including relations that are linear combinations of monomials. The projective resolution of $B$8 as a $B$9-bimodule is given explicitly. This rules out a simplistic interpretation in which quotient bifinite extensions merely reduce homological questions to “nicer” subalgebras.

4. B-extensions in field theory and the analogue of Galois theory

A different meaning appears in field theory. For a finite field extension $E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$0, let

$E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$1

The skew group algebra generated in $E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$2 by $E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$3 and $E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$4 is

$E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$5

The extension $E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$6 is called a B-extension if

$E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$7

(Bavula, 1 Sep 2025).

This definition is motivated as a characteristic-free unifier of the two extremal classes in classical Galois theory: Galois extensions and purely inseparable extensions. In the same framework, G-extensions satisfy $E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$8, while D-extensions satisfy $E(L/K)=\mathcal D(L/K)\rtimes G(L/K)$9. The central theorem asserts an exact equivalence: the class of B-extensions coincides with the class of normal finite field extensions. The summary lists several equivalent conditions, including

$L/K$0

and

$L/K$1

Consequently, all finite Galois extensions and all finite purely inseparable extensions are B-extensions, and so are normal extensions that are neither Galois nor purely inseparable.

The structural decomposition of the endomorphism algebra is

$L/K$2

where $L/K$3 is the maximal purely inseparable subfield over $L/K$4 in $L/K$5, and $L/K$6 is the maximal separable subfield. This decomposition gives a ring-theoretic factorization of normality into purely inseparable and Galois components.

The method is explicitly ring-theoretic and uses central simple algebras and the Double Centralizer Theorem. In this setting, subalgebras of $L/K$7 containing $L/K$8 correspond to intermediate subfields, and for normal extensions one has

$L/K$9

for each intermediate field $\mathcal B$0. The resulting correspondences are presented as analogues of the Galois correspondences for subfields and normal subfields. In this usage, the letter “B” is said to stand for “Bi,” referring to the two generating structures: automorphisms and differential operators.

5. Minimal nondegenerate extensions of braided fusion categories

In braided fusion category theory, the term B-extension is used for what the paper also calls a minimal nondegenerate extension or minimal modular extension (Johnson-Freyd et al., 2021). For a braided fusion category $\mathcal B$1, a minimal nondegenerate extension is a nondegenerate braided fusion category $\mathcal B$2 together with a fully faithful braided embedding $\mathcal B$3 such that the centralizer $\mathcal B$4 coincides with the Müger center $\mathcal B$5 of $\mathcal B$6.

The main theorem states that every slightly degenerate braided fusion category admits a minimal nondegenerate extension. Since a slightly degenerate braided fusion category is one whose Müger center is equivalent to $\mathcal B$7, the result implies that every pseudo-unitary super modular tensor category admits a minimal modular extension. The paper presents this as completing the program of characterizing minimal nondegenerate extensions of braided fusion categories.

The proof relies on fusion $\mathcal B$8-categories. The decisive object is the Drinfel'd centre $\mathcal B$9 of the fusion $L/K$00-category of module categories of $L/K$01. Minimal nondegenerate extensions of $L/K$02 correspond to certain trivializations of $L/K$03. In the slightly degenerate case, these trivializations are obstructed by a class in

$L/K$04

The obstruction is analyzed through a numerical invariant obtained by evaluating a certain two-dimensional topological field theory on a Klein bottle, and the paper proves that this obstruction always vanishes.

The same work develops an $L/K$05-matrix pairing for braided fusion $L/K$06-categories and proves that it is nondegenerate for $L/K$07. As a corollary, components of $L/K$08 are identified with blocks in the annular category of $L/K$09 and with homomorphisms from the Grothendieck ring of the Müger centre of $L/K$10 to the ground field. In this usage, a B-extension is therefore not an algebraic inclusion in the ring-theoretic sense, but a braided enlargement that removes degeneracy in the minimal possible way.

6. Comparative perspective and neighboring extension terminologies

The four usages above show that “B-extension” is field-specific rather than universal. In one complex variable it denotes a normal form for continuous functions detected by meromorphic extension from two Möbius-invariant circle families. In finite-dimensional algebra it denotes a subalgebra extension controlled by finite $L/K$11-bimodule projective dimension of $L/K$12. In field theory it denotes maximal symmetry of a finite extension, encoded by the equality $L/K$13. In braided fusion category theory it denotes a minimal nondegenerate completion.

Other arXiv uses of extension terminology involving the letter $L/K$14 are adjacent but distinct. The paper on extensions of $L/K$15-algebras studies essential extensions

$L/K$16

classified by $L/K$17 (Gabe et al., 2023). The paper on bosonic extensions introduces automorphisms on the bosonic extension of arbitrary type and shows that they satisfy the braid relations (Kashiwara et al., 2024). The thesis on the BMS group investigates generalized BMS and $L/K$18-BMS asymptotic symmetry groups (Ruzziconi, 2020). Particle-physics papers classify non-abelian gauge extensions of the Standard Model in view of $L/K$19-decay anomalies and analyze flavorful $L/K$20 extensions for rare $L/K$21-decays (Boucenna et al., 2016, Bause et al., 2021). This suggests terminological overlap rather than a shared technical definition.

A plausible implication is that “B-extension” functions best as a local term whose meaning is determined entirely by its ambient theory. In the analytic literature it is tied to polyanalyticity and test families; in homological algebra it is a transfer mechanism for finiteness properties; in field theory it reformulates normality through endomorphism algebras; and in fusion-category theory it resolves degeneracy by passage to a minimal nondegenerate ambient category.