Conformal S1₂ enveloping algebras

Automorphisms of Generalized Down-Up Algebras

“…Other classes of algebras to which our study applies are Le Bruyn's conformal sl 2 enveloping algebras [18], occuring as L(bx 2 + x, r, s, γ), for b ∈ K and rs = 0, and some of Witten's seven parameter deformations of the enveloping algebra of sl 2 [26] (see also [8,Thm. 2.6] and [11,Ex.…”

“…Generalized down-up algebras include all down-up algebras, the algebras similar to the enveloping algebra of sl 2 defined by Smith [25], Le Bruyn's conformal sl 2 enveloping algebras [18] and Rueda's algebras similar to the enveloping algebra of sl 2 [24]. The reader is encouraged to consult [11] for further details and references.…”

CONFORMAL $\mathcal{s}$$\mathcal{l}$2 ENVELOPING ALGEBRAS AS AMBISKEW POLYNOMIAL RINGS

“…where α, β, γ, δ, , ζ, η ∈ C. As was observed in [13], not all of these algebras will have desirable ring-theoretic properties, such as being a domain, having finite global dimension or a PBW basis. In the classical case the associated graded ring of U(sl 2 ) is a commutative polynomial ring, and so the above properties follow.…”

“…However, for these algebras the associated graded ring is not necessarily commutative. In [13], Le Bruyn gave the definition of a conformal sl 2 enveloping algebra as being those seven-parameter algebras above whose associated graded ring is an Auslander regular algebra of global dimension three. This allows conformal sl 2 enveloping algebras to enjoy some of the same good ring-theoretic and homological properties as in the classical case.…”

Isomorphisms Between Quantum Generalized Weyl Algebras

“…Le Bruyn (cf. [12] and [13]) studied the algebras M(ǫ) whose associated graded algebra are Auslander-regular. He proved that they determine a 3-parameter family of Witten deformation algebras, with defining relations:…”