In this paper we introduce and study n-point Virasoro algebras,Ṽa, which are natural generalizations of the classical Virasoro algebra and have as quotients multipoint genus zero Krichever-Novikov type algebras. We determine necessary and sufficient conditions for the latter two such Lie algebras to be isomorphic. Moreover we determine their automorphisms, their derivation algebras, their universal central extensions, and some other properties. The list of automorphism groups that occur is Cn, Dn, A 4 , S 4 and A 5 . We also construct a large class of modules which we call modules of densities, and determine necessary and sufficient conditions for them to be irreducible.
In this paper we study composition series and embeddings of Verma modules induced from "nonstandard" Borel subalgebras. This article can be viewed as a generalization of Futorny's work on imaginary Verma modules for Λ\ ' where the center of the Kac-Moody algebra acts nontrivially.Introduction. Let A be an indecomposable symmetrizable generalized Cartan matrix, fl(A) = n_ © f) 0 n+ the triangular decomposition of the Kac-Moody algebra g(A) and W the Weyl group for g(A). The "standard" Borel subalgebra b+ and its opposite faare defined to be b± = f) 0 n±. For affine Kac-Moody algebras, H. Jakobsen and V. Kac and independently V. Futornyi have found an explicit description of a set of representatives of the conjugacy classes of Borel subalgebras (see 1.1) under the action of W x {±1}. We will call all Borel subalgebras not conjugate to b+ or 6_, "nonstandard" Borel subalgebras. In particular for each subset X of the set of simple roots Π for g(A), one can construct a nonstandard Borel subalgebra b+ and then for each λ 6ή* one can use induction to obtain what we will call a "nonstandard" Verma module M x (λ). For example if X = Π then b+ is the standard Borel subalgebra and M(λ) is the "standard" Verma module. At the other extreme X = 0 one obtains the "natural" Borel subalgebra and what one might call a "natural" Verma module. A striking difference between Verma modules induced from a standard Borel and those induced from b+ for X C Π, is that these new nonstandard Verma modules have infinite dimensional weight spaces. Consequently many of the classical techniques used in the study of the composition series of standard Verma modules do not seem to apply to this new setting.
Abstract. Let m ∈ N, P (t) ∈ C[t]. Then we have the Riemann surfaces (commutative algebras)The Lie algebras Rm(P ) = Der(Rm(P )) and Sm(P ) = Der(Sm(P )) are called the m-th superelliptic Lie algebras associated to P (t). In this paper we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras (the Riemann surfaces) by using polynomial Pell equations.
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.
Abstract. This work expands to the setting of sl n (C) the results of H. Jakobsen and V. Kac and independently D. Bernard and G. Felder on the realization of sl 2 (C), in terms of infinite sums of partial differential operators. We note in the paper that, in the generic case, these geometric constructions are just realizations of the imaginary Verma modules studied by V. Futorny. (2000): Primary: 17B67, 81R10. Mathematics Subject Classifications
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