Realizations of the four point affine Lie algebra<b>sl</b>(2, R) ⊕(Ω<sub>R</sub>⁄dR)

Cox, Ben

doi:10.2140/pjm.2008.234.261

Cited by 18 publications

(14 citation statements)

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“…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.…”

Section: Introductionmentioning

confidence: 99%

On self-similar Lie algebras and virtual endomorphisms

2018

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We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type F Pn, which are variants of lamp-lighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra sln(k), where char(k) does not divide n affords non-trivial faithful self-similarity.

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Section: Introductionmentioning

confidence: 99%

On self-similar Lie algebras and virtual endomorphisms

2018

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“…In our previous work (see [Cox08,CJ14,BCF09,CF11]), the authors we used such detailed information to obtain certain free-field realizations of the three point, four point, elliptic affine and DJKM algebras depending on a parameter r = 0, 1 that correspond to two different normal orderings. These later realizations are analogues of Wakimoto type realizations which have been used by Schechtman and Varchenko and various other authors in the affine setting to pin down integral solutions to the Knizhnik-Zamolodchikov differential equations (see for example [ATY91], [Kur91], [EFK98], [SV90]).…”

Section: Introductionmentioning

confidence: 99%

On the universal central extension of hyperelliptic current algebras

Cox

2016

Proc. Amer. Math. Soc.

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“…Moreover Bremner has given an explicit description of the universal central extension of g ⊗ R, in terms of ultraspherical (Gegenbauer) polynomials where R is the four point algebra (see [5]). In [7] the first author gave a realization for the four point algebra where the center acts nontrivially.…”

Section: Introductionmentioning

confidence: 99%

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

Cox

Jurisich

2014

Pacific J. Math.

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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.

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Realizations of the four point affine Lie algebrasl(2, R) ⊕(Ω_R⁄dR)

Abstract: We construct free field realizations of the four point algebra sl(2, R)

Cited by 18 publications

References 17 publications

On self-similar Lie algebras and virtual endomorphisms

On self-similar Lie algebras and virtual endomorphisms

On the universal central extension of hyperelliptic current algebras

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

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Realizations of the four point affine Lie algebrasl(2, R) ⊕(ΩR⁄dR)

Abstract: We construct free field realizations of the four point algebra sl(2, R)

Cited by 18 publications

References 17 publications

On self-similar Lie algebras and virtual endomorphisms

On self-similar Lie algebras and virtual endomorphisms

On the universal central extension of hyperelliptic current algebras

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

Contact Info

Product

Resources

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Realizations of the four point affine Lie algebrasl(2, R) ⊕(Ω_R⁄dR)

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