Constructions

Every construction below is built in exact arithmetic and checked against every axiom with classify before it is stored. Commutative and cocommutative results are not stored (they are function and group algebras). The resulting entries are listed under New families.

Family	Range	Entries
Bismash products	$\|G\| \le 32$	0
Abelian extensions	$\|G\| \le 24$	0
Quantum linear spaces	$\Gamma = \mathbb{Z}/n$ , $n \le 8$ , $\dim \le 64$	0
Liftings	the same, and $u_{q}(\mathfrak{sl}_{2})$ at $\ell = 5$	0
Twists of $K[G]$	$\|G\| \le 63$ , by $\mathbb{Z}_{2}^{2}$ and $\mathbb{Z}_{3}^{2}$	0

Bismash products

An exact factorization $G = F \cdot H$ , $F \cap H = 1$ , gives $hf = (h \triangleright f)(h \triangleleft f)$ and the Hopf algebra $K^{F} \# K[H]$ (Takeuchi) on the basis $e_{f} \# h$ :

(e_{f} \# h)(e_{f'} \# h') = \delta_{f,\, h \triangleright f'}\; e_{f} \# hh', \qquad \cop(e_{f} \# h) = \sum_{f_{1} f_{2} = f} (e_{f_{1}} \# h) \otimes \bigl(e_{f_{2}} \# (h^{-1} \triangleleft f_{1})^{-1}\bigr).

It is commutative exactly when $\triangleright$ is trivial and $H$ abelian, cocommutative exactly when $\triangleleft$ is trivial and $F$ abelian; those are skipped before building. Products with equal invariants are stored as one entry listing every construction: — new entries in the last run, with — constructions merged into them.

Abelian extensions

$K^{F} \#_{\sigma,\tau} K[H]$ keeps the nonzero positions of the bismash product and multiplies each structure constant by a root of unity $\zeta_{m}^{x_{v}}$ . On that support every axiom equates two monomials, so the axioms are linear equations for $x$ modulo $m$ , solved over $\mathbb{Z}$ by Smith normal form; rescaling the basis changes $x$ by a coboundary. One class per orbit of the automorphisms of $G$ fixing $F$ and $H$ is built, over the smallest cyclotomic field containing a representative. The extensions of $D_{8}$ that are neither commutative nor cocommutative are Kac–Paljutkin $H_{8}$ .

Quantum linear spaces and liftings

For $\Gamma = \mathbb{Z}/n$ and pairs $(g_{i}, \chi_{i})$ with $q_{i} = \chi_{i}(g_{i}) \ne 1$ and $\chi_{i}(g_{j})\chi_{j}(g_{i}) = 1$ :

g x_{i} g^{-1} = \chi_{i}(g)\, x_{i}, \quad x_{i} x_{j} = \chi_{j}(g_{i})\, x_{j} x_{i}, \quad x_{i}^{N_{i}} = 0, \quad \cop(x_{i}) = x_{i} \otimes 1 + g_{i} \otimes x_{i}.

One representative per Andruskiewitsch–Schneider class, so different entries are different algebras. Liftings deform two relations,

x_{i} x_{j} - \chi_{j}(g_{i})\, x_{j} x_{i} = \lambda_{ij}\,(1 - g_{i} g_{j}), \qquad x_{i}^{N_{i}} = \mu_{i}\,(1 - g_{i}^{N_{i}}),

with every allowed parameter set to 1; the product is computed by rewriting to the PBW basis. With $\lambda_{12} = 1$ , $E = x_{1}$ , $FK = x_{2}$ this is $u_{q}(\mathfrak{sl}_{2})$ : dimension 27 at a cube root of unity, and two entries of dimension 125 at a fifth root ( $q^{2} = \zeta_{5}, \zeta_{5}^{2}$ ), each with 5 group-likes, one 1-dimensional representation and $S$ of order 10.

Twists of group algebras

For a subgroup $V \cong (\mathbb{Z}/p)^{2}$ of $G$ and a nontrivial bicharacter $\omega$ on its characters, $J = \sum \omega(\chi, \psi)\, e_{\chi} \otimes e_{\psi}$ and $\cop^{J}(h) = J\cop(h)J^{-1}$ . Then $\operatorname{Rep}(K[G]^{J}) \simeq \operatorname{Rep}(G)$ , and every twist has the indicator traces of $K[G]$ . A normal Klein subgroup gave a cocommutative result on every group tried; for $\mathbb{Z}_{3}^{2}$ only $C_{3} \times S_{3}$ (with $|G| \le 32$ ) gives a new algebra.

Isomorphism

Entries of one family are merged when they agree on: the Wedderburn blocks of $H$ and $H^{*}$ , the numbers of group-likes and of 1-dimensional representations, the order of $S$ , and the indicator traces

\operatorname{Tr}_{H}\bigl(\Lambda^{[n]}\bigr) = \sum_{V} \dim(V)\, \nu_{n}(V), \qquad n = 1, 2, \dots

(for $K[G]$ , the number of $g$ with $g^{n} = 1$ ; at $n = 4$ they separate $H_{8}$ from $K[D_{8}]$ ). Equal invariants are not a proof. hopf_isomorphism(A, B) searches for an exact isomorphism: it fixes isomorphisms of the group-likes and of the characters compatible with their pairing, imposes the linear conditions these give (unit, counit, antipode, integral, multiplication by group-likes, actions of characters, central idempotents and their traces), then solves the remaining quadratic conditions where they become linear or univariate, extending to a cyclotomic field when a root needs it. A found map is verified exactly; an exhausted search proves non-isomorphism. It is run on every merge and on entries of different families with equal invariants, and the result is recorded in the entries.