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Register a new userAnother class of simple graded Lie conformal algebras that cannot be embedded into general Lie conformal algebras
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In a previous paper by the authors, we obtain the first example of a finitely freely generated simple $mathbb Z$-graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying $mathbb Z$-graded simple Lie conformal algebras ${cal G}=oplus_{i=-1}^infty{cal G}_i$ satisfying the following, (1) ${cal G}_0cong{rm Vir}$, the Virasoro conformal algebra; (2) Each ${cal G}_i$ for $ige-1$ is a ${rm Vir}$-module of rank one. These algebras include some Lie conformal algebras of Block type.
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In this paper, we introduce the notion of completely non-trivial module of a Lie conformal algebra. By this notion, we classify all finite irreducible modules of a class of $mathbb{Z}^+$-graded Lie conformal algebras $mathcal{L}=bigoplus_{i=0}^{infty} mathbb{C}[partial]L_i$ satisfying $ [{L_0}_lambda L_0]=(partial+2lambda)L_0,$ and $[{L_1}_lambda L_i] eq 0$ for any $iin mathbb{Z}^+$. These Lie conformal algebras include Block type Lie conformal algebra $mathcal{B}(p)$ and map Virasoro Lie conformal algebra $mathcal{V}(mathbb{C}[T])=Virotimes mathbb{C}[T]$. As a result, we show that all non-trivial finite irreducible modules of these algebras are free of rank one as a $mathbb{C}[partial]$-module.
In the present paper, we prove that any finite non-trivial irreducible module over a rank two Lie conformal algebra $mathcal{H}$ is of rank one. We also describe the actions of $mathcal{H}$ on its finite irreducible modules explicitly. Moreover, we show that all finite non-trivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one.
Let $min N$, $P(t)in C[t]$. Then we have the Riemann surfaces (commutative algebras) $R_m(P)=C[t^{pm1},u | u^m=P(t)]$ and $S_m(P)=C[t , u| u^m=P(t)].$ The Lie algebras $mathcal{R}_m(P)=Der(R_m(P))$ and $mathcal{S}_m(P)=Der(S_m(P))$ 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 essential feature of a root-graded Lie algebra L is the existence of a split semisimple subalgebra g with respect to which L is an integrable module with weights in a possibly non-reduced root system S of the same rank as the root system R of g. Examples include map algebras (maps from an affine scheme to g, S = R), matrix algebras like sl_n(A) for a unital associative algebra A (S = R = A_{n-1}), finite-dimensional isotropic central-simple Lie algebras (S properly contains R in general), and some equivariant map algebras. In this paper we study the category of representations of a root-graded Lie algebra L which are integrable as representations of g and whose weights are bounded by some dominant weight of g. We link this category to the module category of an associative algebra, whose structure we determine for map algebras and sl_n(A). Our results unify previous work of Chari and her collaborators on map algebras and of Seligman on isotropic Lie algebras.
We prove that the tensor product of a simple and a finite dimensional $mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $mathfrak{q}(n)$-supermodules to that of simple $mathfrak{sl}_n$-modules. Rough structure of simple $mathfrak{q}(n)$-supermodules, considered as $mathfrak{sl}_n$-modules, is described in terms of the combinatorics of category $mathcal{O}$.
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