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We identify the spaces of homogeneous polynomials in two variables K[Y^k, XY^{k-1}, ..., X^k] among representations of the Lie ring sl(2,K). This amounts to constructing a compatible K-linear structure on some abstract sl(2,K)-modules, where sl(2,K) is viewed as a Lie ring.
We identify the representations $mathbb{K}[X^k, X^{k-1}Y, dots, Y^k]$ among abstract $mathbb{Z}[mathrm{SL}_2(mathbb{K})]$-modules. One result is on $mathbb{Q}[mathrm{SL}_2(mathbb{Z})]$-modules of short nilpotence length and generalises a classical qu
We give a process to construct non-split, three-dimensional simple Lie algebras from involutions of sl(2,k), where k is a field of characteristic not two. Up to equivalence, non-split three-dimensional simple Lie algebras obtained in this way are par
We determine the multiplicity of the irreducible representation V(n) of the simple Lie algebra sl(2,C) as a direct summand of its fourth exterior power $Lambda^4 V(n)$. The multiplicity is 1 (resp. 2) if and only if n = 4, 6 (resp. n = 8, 10). For th
This paper examines the relationship between certain non-commutative analogues of projective 3-space, $mathbb{P}^3$, and the quantized enveloping algebras $U_q(mathfrak{sl}_2)$. The relationship is mediated by certain non-commutative graded algebras
In this paper we explore the possibility of endowing simple infinite-dimensional ${mathfrak{sl}_2(mathbb{C})}$-modules by the structure of the graded module. The gradings on finite-dimensional simple module over simple Lie algebras has been studied in [arXiv:1308.6089] and [arXiv:1601.03008].