The Structures And Representations Of A Block Type Lie Algebra And Its Lie Conformal Algebra

Let L be the Block type Lie algebra over C with basis{Lα,i|(α,i)∈Z×Z} and Lie bracket [Lα,i,Lβ,j]=(β(i+1)-α(j+1))Lα+β,i+j. In this article, we are going to give a classification of all quasi-finite representations of the universal central extension algebra of L. We will also find the related conformal algebra of L and give a classification of its free intermediate series modules.We will first use some known results to find the universal central extension algebra of L. Then we shall give a classification of the quasi-finite representations for this cen-tral extension algebra. Similar algebras, like the W1+∞algebra, have been studied by many authors under different suppositions. The problem with this type algebras lies in that although they admit natural Z-gradings, each grade is still an infinite dimensional space. The modules of these algebras are usually very large in nature, which makes it hard to classify or study in detail. Therefore, we are about to discuss a class of modules with a certain finite properties, that is, the quasi-finite modules. However, for the Lie algebra L discussed in this paper, we are going to face an extra difficulty:Each grade here has even more basis elements than those algebras we discussed before. A typical realization via the polynomials is no longer possible. We need to base the realization of L on Laurent polynomials. In our work, first we shall give a rough classification of the quasi-finite modules, then we shall prove part of them are trivial, and finally we shall classify the rest of the modules in detail, which in turn will give us a thorough classification of our object.On the other hand, from L, we are able to construct a related Lie conformal al-gebra B using the method of formal distribution Lie algebras. Note that this is an example of infinite dimensional Lie conformal algebras, which are not well studied yet. In this article, we are mainly interested in classifying the free intermediate series modules of B. The classified result turns out to be quite an analogue to that of the Virasoro algebra. The classification is obtained via calculation. First we analyze the structure coefficients related to bases, then we build up a PDE to get a general form of these coefficients, finally we shall use this form to discuss the detail of the coefficients, fix them and finish the classification.