Realizations of the three-point Lie algebra sl(2,ℛ) ⊕ (Ω_ℛ∕dℛ)

Abstract

Abstract. We describe the universal central extension of the three point current algebra sl(2, R) where R = C[t, t −1 , u | u 2 = t 2 + 4t] and construct realizations of it in terms of sums of partial differential operators. IntroductionIt is well known from the work of C. Kassel and J.L. Loday (see [20], and [21]) that if R is a commutative algebra and g is a simple Lie algebra, both defined over the complex numbers, then the universal central extensionĝ of g ⊗ R is the vector space (g ⊗ R) ⊕ Ω 1 R /dR where Ω 1 R /dR is the space of Kähler differentials modulo exact forms (see [21]). The vector spaceĝ is made into a Lie algebra by defining . We find such a realization in the setting where g = sl(2, C) and R = C[t, t −1 , u|u 2 = t 2 + 4t] is the three point algebra. In Kazhdan and Luszig's explicit study of the tensor structure of modules for affine Lie algebras (see [22] and [23]) the ring of functions regular everywhere except at a finite number of points appears naturally. This algebra M. Bremner gave the name n-point algebra. In particular in the monograph [16, Ch. 12] algebras of the form ⊕ n i=1 g((t − x i )) ⊕ Cc appear in the description of the conformal blocks. These contain the n-point algebras g ⊗ C[(t − x 1 ) −1 , . . . , (t − x N ) −1 ] ⊕ Cc modulo part of the center Ω R /dR. M. Bremner explicitly described the universal central extension of such an algebra in [3].Consider now the Riemann sphere C ∪ {∞} with coordinate function s and fix three distinct points a 1 , a 2 , a 3 on this Riemann sphere. Let R denote the ring of rational functions with poles only in the set {a 1 , a 2 , a 3 }. It is known that the automorphism group P GL 2 (C) of C(s) is simply 3-transitive and R is a subring of C(s), so that R is isomorphic to the ring of rational functions with poles at {∞, 0, 1, a}. Motivated by this isomorphism one sets a = a 4 and here the 4-point, u] where u 2 = t 2 − 2bt + 1 with b a complex number not equal to ±1. Then M. Bremner has shown us that R a ∼ = S b . As the later, being Z 2 -graded, is a cousin to super Lie algebras, and is thus more immediately amendable to the theatrics of conformal field theory. Moreover Bremner has given an explicit description of the universal central extension of g ⊗ R, in terms of ultraspherical 1991 Mathematics Subject Classification. Primary 17B67, 81R10.

Representations for three-point Lie algebras of genus zero

“…The 2-cocycle of W p in Definition 3.2 is slightly different from one of 2-cocycles obtained by Cox and Jurisich[10].…”

“…Remark 2.3. Note that we have slightly modified the original relations in [10] by replacing the central elements ω 0 , ω 1 ofĝ p to −(k + + k − ), −k − , respectively. Remark 2.4.…”

The 3-point Virasoro algebra and its action on a Fock space

“…The following is a straightforward computation, with corrections to the version in [CJ14] (where some formulas for the 4-point algebra were inadvertently included).…”

“…Schlichenmaier has a somewhat different description of the three point algebra as having coordinate ring C[(z 2 −a 2 ) k , z(z 2 −a 2 ) k | k ∈ Z] where a = 0 (see [Sch03a]). In [CJ14] it was noted that R ∼ = C[t, t −1 , u | u 2 = t 2 + 4t], and thus the three point algebra resembles S b above. Besides Bremner's article mentioned above, other work on the universal central extension of 3-point algebras can be found in [BT07].…”

On self-similar Lie algebras and virtual endomorphisms

“…There is a vast literature about the structure, central extensions and representations of these algebras, cf. [13], [14] and [15] and references therein. These algebras are particular cases of Krichever-Novikov algebras L ⊗ X studied in [28], [29] in connection with the string theory in Minkowski space, where X is the algebra of meromorphic functions on a Riemann surface of any genus with a finite number of poles.…”