Double Ore extensions

An Ore extension \(B = A[x; \sigma, \delta]\) adjoins a variable \(x\) to a ring \(A\) with a twisted multiplication \(x a = \sigma(a) x + \delta(a)\), for an automorphism \(\sigma\) and a \(\sigma\)-derivation \(\delta\). Repeating the construction builds higher-dimensional algebras while keeping good homological behaviour; the quantum (skew) polynomial ring is the simplest example.

A double Ore extension adjoins two variables at once, producing a rank-two extension \((k_Q[x_1, x_2])_P[y_1, y_2; \sigma]\): a quantum plane \(k_Q[x_1, x_2]\) extended by \(y_1, y_2\).

The 26 families A–Z

The four-dimensional AS-regular domains of the form \((k_Q[x_1, x_2])_P[y_1, y_2; \sigma]\) were classified up to isomorphism by Zhang–Zhang (Double extension regular algebras of type (14641)), who found 26 of them, labelled A–Z.

\(\Sigma\)-\(M\)-duality

The count of 26 comes from a \(\Sigma\)-\(M\)-duality. A double Ore extension \((k_Q[x_1, x_2])_P[y_1, y_2; \sigma]\) is built from the quantum-plane data \(\Sigma = (k_Q, P)\) and the twisting data \(M = \sigma\); swapping the two pairs of variables \(x_i \leftrightarrow y_i\) sends a presentation to a dual one with the roles of \(\Sigma\) and \(M\) interchanged. Up to this duality (and the underlying twist equivalence) there are only 26 algebras. The duality pairs up \((E, J), (F, I), (N, P), (T, U), (W, Z)\) — isomorphic via the exchange — while \(B, C, M, O, R, S\) are self-dual.

The 26 families:

• double Ore A
• double Ore B
• double Ore C
• double Ore D
• double Ore E
• double Ore F
• double Ore G
• double Ore H
• double Ore I
• double Ore J
• double Ore K
• double Ore L
• double Ore M
• double Ore N
• double Ore O
• double Ore P
• double Ore Q
• double Ore R
• double Ore S
• double Ore T
• double Ore U
• double Ore V
• double Ore W
• double Ore X
• double Ore Y
• double Ore Z

Their invariants: