Fusion Modules of Classical Lie Algebras and Image of Multilinear Polynomials on Nilpotent Lie Algebras

Abstract

Abstract This thesis explores two key facets of Lie algebras. In the initial segment, we delve into the fusion product modules associated with current Lie algebras of types A, B,C, and D. Fusion products of finite-dimensional cyclic modules defined in [11] form an important class of graded representations of current Lie algebras. For any finite-dimensional simple Lie algebra g of types A, B,C, or D, and given two dominant weights λ and µ of g, a g[t] module F λ ,µ is defined through generators and relations. In their work [4], Kus and Barth established the isomorphism between F λ ,µ and V (λ ) z 1 ∗ V (µ) z 2 , the fusion product of irreducible modules, specifically for Lie algebras of rank 2. In a distinct approach outlined in [17], the authors demonstrated the same isomorphism for sl 3 and furnished the graded character of such modules. Utilizing a series of short exact sequences, we derived a graded decomposition for these modules and confirmed the isomorphism F λ ,µ ∼ = V (λ ) z 1 ∗V (µ) z 2 for Lie algebras A n , B n ,C n , D n , where λ and µ are both multiples of either the first or the last fundamental weight. Additionally, we provided the graded character for this fusion module. In the second part of the thesis, our focus shifts to the image of a multilinear Lie polynomial of degree 2 on nilpotent Lie algebras. A non-zero multilinear Lie polynomial in two variables is essentially a non-zero scalar multiple of the Lie bracket of these variables. Denoting a nilpotent Lie algebra as L, we define the image set of multilinear Lie polynomials of degree 2 as w(L)-not necessarily a vector space. The smallest such nilpotent Lie algebra over a field k with the characteristic not equal to 2 is identified as L 6,21 (0). Our investigation involvesx imposing constraints on the spanning set of images, specifically focusing on the derived Lie subalgebra L ′ = [L, L].