Finite Irreducible Conformal Modules Research Articles

Extensions of finite irreducible conformal modules over Lie conformal algebras 𝒲(a,b,r)

In this article, we study extensions of conformal modules over Lie conformal algebras with three parameters a, b, , which contain the Virasoro Lie conformal algebra Vir, and as subalgebras. We obtain the existence conditions of extensions of nontrivial conformal modules over and give a complete classification of extensions of finite irreducible conformal modules of .

Representations of the associated Lie conformal algebra of the [formula omitted] algebra and beyond

We introduce the so-called infinite contact Lie conformal superalgebras KN(p) from finite Lie conformal superalgebras of Cartan type K. The annihilation algebra of the first member L in degenerate cases is exactly the associated graded Lie algebra of the filtered W-infinity algebra W1+∞. Using the conformal version of Lie's theorem, we first classify finite conformal modules over KN(p) and its derived subalgebra. Then we classify finite irreducible conformal modules over positive subalgebras of L and its derived subalgebra, which are shown to be weight modules, parameterized by polynomials, in the sense of D'Andrea and Kac [13]. Finally, we conceptually classify extensions between these conformal modules.

Representations of super deformation of Heisenberg-Virasoro type Lie conformal algebras

We introduce a class of Lie conformal superalgebra S(a, b), where a, b are complex parameters. These Lie conformal superalgebras can be viewed as the super deformation of Heisenberg-Virasoro type Lie conformal algebras. We classify the free conformal modules of rank over S(a, b) for all . Using a Cheng-Kac’s lemma, we also classify the finite irreducible conformal modules over S(a, b).

Classification of finite irreducible conformal modules for K4′

We classify finite irreducible modules over the conformal superalgebra K4′ by their correspondence with finite conformal modules over the associated annihilation superalgebra A(K4′). This is achieved by a complete classification of singular vectors in generalized Verma modules for A(K4′). We also show that morphisms between generalized Verma modules can be arranged in infinitely many bilateral complexes.

Classification of Finite Irreducible Conformal Modules over N = 2 Lie Conformal Superalgebras of Block Type

We introduce the N = 2 Lie conformal superalgebras \({\frak {K}}(p)\) of Block type, and classify their finite irreducible conformal modules for any nonzero parameter p. In particular, we show that such a conformal module admits a nontrivial extension of a finite conformal module M over K2 if p = − 1 and M has rank (2 + 2), where K2 is an N = 2 conformal subalgebra of \({\frak {K}}(p)\). As a byproduct, we obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal superalgebras \({\frak k}(n)\) for n ≥ 1. Composition factors of all the involved reducible conformal modules are also determined.

Representations of twisted infinite Lie conformal superalgebras

Following the twisting rule from Ramond Lie superalgebra to Neveu-Schwarz Lie superalgebra, we introduce the twisted version T(p) of a class of Z-graded Lie conformal superalgebras, where p is a nonzero parameter. We completely classify their small rank conformal modules and finite irreducible conformal modules, which turn out to be much richer than those over the original version. As applications, we also develop the representation theory of many new finite Lie conformal superalgebras.

Representations of Lie conformal algebras related to Galilean conformal algebras

We study the representation theory of Lie conformal analogues r(a, b) of certain Galilean conformal algebras, where are complex numbers. We first classify their finite irreducible conformal modules, and then solve the extension problem. In particular, we obtain several classes of new indecomposable conformal modules over r(a, b).

Finite irreducible conformal modules over the Lie conformal superalgebra [formula omitted

In the present paper, we introduce a class of infinite Lie conformal superalgebras S(p), which are closely related to Lie conformal algebras of extended Block type defined in [6]. Then all finite non-trivial irreducible conformal modules over S(p) for p∈C⁎ are completely classified. As an application, we also present the classifications of finite non-trivial irreducible conformal modules over finite quotient algebras s(n) for n≥1 and sh which is isomorphic to a subalgebra of Lie conformal algebra of N=2 superconformal algebra. Moreover, as a generalized version of S(p), the infinite Lie conformal superalgebras GS(p) are constructed, which have a subalgebra isomorphic to the finite Lie conformal algebra of N=2 superconformal algebra.

Extensions of finite irreducible modules of Lie conformal algebras and some Schrödinger-Virasoro type Lie conformal algebras

Lie conformal algebras are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this article, we give a complete classification of extensions of finite irreducible conformal modules of With a similar method, we characterize all extensions of finite irreducible conformal modules of Schrödinger-Virasoro type Lie conformal algebras TSV(a, b) and TSV(c).

Irreducible Representations of Finite Lie Conformal Algebras of Planar Galilean Type

It is well known that Galilean conformal algebras play important roles in the nonrelativistic anti-de Sitter/conformal field theory correspondence. The finite Lie conformal algebras P G ( a , b ) of planar Galilean type can be viewed as Lie conformal analogues of certain planar Galilean conformal algebras. In this paper, we classify finite irreducible conformal modules over P G ( a , b ) for all complex numbers a and b.

Extensions of conformal modules over Lie conformal algebras of Block type

We classify extensions between finite irreducible conformal modules over a class of infinite Lie conformal algebras B(p) of Block type, where p is a nonzero complex number. We find that although certain finite irreducible conformal modules over B(p) are simply conformal modules over its Virasoro conformal subalgebra Vir, there exist more nontrivial extensions between these conformal B(p)-modules. For extensions between other conformal modules, the situation becomes rather different. As an application, we also solve the extension problem for a series of finite Lie conformal algebras b(n) for n≥1.

Extensions of modules over a class of Lie conformal algebras 𝒲(b)

Let [Formula: see text] be a class of free Lie conformal algebras of rank two with [Formula: see text]-basis [Formula: see text] and relations [Formula: see text] where [Formula: see text] is a nonzero complex number. In this paper, we classify extensions between two finite irreducible conformal modules over the Lie conformal algebras [Formula: see text].

Finite irreducible modules of Lie conformal algebras 𝒲(a,b) and some Schrödinger–Virasoro type Lie conformal algebras

Lie conformal algebras [Formula: see text] are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we first give a complete classification of all finite nontrivial irreducible conformal modules of [Formula: see text]. It is shown that all such modules are of rank one. Moreover, with a similar method, all finite nontrivial irreducible conformal modules of Schrödinger–Virasoro type Lie conformal algebras [Formula: see text] and [Formula: see text] are characterized.

Classification of finite irreducible conformal modules over Lie conformal superalgebras of Block type

We introduce a class of infinite Lie conformal superalgebras S(p) of Block type, and classify their finite irreducible conformal modules for any nonzero parameter p. In particular, we show that such a conformal module admits a nontrivial extension of a finite conformal module over NS if p=−1, where NS is a Neveu-Schwarz conformal subalgebra of S(p). As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal superalgebras s(n) for n≥1.

Extensions of Schrödinger–Virasoro conformal modules

In this paper, we study extensions between two finite irreducible conformal modules over the Schrödinger–Virasoro conformal algebra and the extended Schrödinger–Virasoro conformal algebra. Also, we classify all finite nontrivial irreducible conformal modules over the extended Schrödinger–Virasoro conformal algebra. As a byproduct, we obtain a classification of extensions of Heisenberg–Virasoro conformal modules.

Classification of finite irreducible conformal modules over a class of Lie conformal algebras of Block type

We classify finite irreducible conformal modules over a class of infinite Lie conformal algebras B(p) of Block type, where p is a nonzero complex number. In particular, we obtain that a finite irreducible conformal module over B(p) may be a nontrivial extension of a finite conformal module over Vir if p=−1, where Vir is a Virasoro conformal subalgebra of B(p). As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal algebras b(n) for n≥1.

Classification of finite irreducible conformal modules over some Lie conformal algebras related to the Virasoro conformal algebra

In this paper, we classify all finite irreducible conformal modules over a class of Lie conformal algebras W(b) with b∈ℂ related to the Virasoro conformal algebra. Explicitly, any finite irreducible conformal module over W(b) is proved to be isomorphic to MΔ,α,β with Δ≠0 or β≠0 if b = 0, or MΔ,α with Δ≠0 if b≠0. As a byproduct, all finite irreducible conformal modules over the Heisenberg-Virasoro conformal algebra and the W(2, 2) Lie conformal algebra are classified. Finally, the same thing is done for the Schrödinger-Virasoro conformal algebra.

Finite irreducible representations of map Lie conformal algebras

In this paper, we study the finite representation theory of the map Lie conformal algebra [Formula: see text], where G is a finite simple Lie conformal algebra and A is a commutative associative algebra with unity over [Formula: see text]. In particular, we give a complete classification of nontrivial finite irreducible conformal modules of [Formula: see text] provided A is finite-dimensional.

Finite conformal modules over N=2,3,4 superconformal algebras

In this paper we continue the study of representation theory of formal distribution Lie superalgebras initiated by Cheng and Kac [Asian J. Math. 1, 181–193 (1997); 2, 153–156 (1998) (erratum)]. We study finite Verma-type conformal modules over the N=2, N=3 and the two N=4 superconformal algebras and also find explicitly all singular vectors in these modules. From our analysis of these modules we obtain a complete list of finite irreducible conformal modules over the N=2, N=3 and the two N=4 superconformal algebras.