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Register a new userVeranderungen uber einen Satz von Timmesfeld - II. Symmetric powers of Nat sl(2,K)
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Authors
Adrien Deloro
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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.
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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 quadratic theorem by Smith and Timmesfeld. Another one is on extending the linear structure on the module from the prime field to $mathbb{K}$. All proofs are by computation in the group ring using the Steinberg relations.
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 parametrised by a subgroup of the Brauer group of k and are characterised by the fact that their Killing form represents -2. Over local and global fields we re-express this condition in terms of Hilbert and Legendre Symbols and give examples of three-dimensional simple Lie algebras which can and cannot be obtained by this construction over the field of rationals.
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 these n we determine the multilinear polynomial identities of degree $le 7$ satisfied by the sl(2,C)-invariant alternating quaternary algebra structures obtained from the projections $Lambda^4 V(n) to V(n)$. We represent the polynomial identities as the nullspace of a large integer matrix and use computational linear algebra to find the canonical basis of the nullspace.
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 $S$, one for each $q in mathbb{C}^times$, having a degree-two central element $c$ such that $S[c^{-1}]_0 cong U_q(mathfrak{sl}_2)$. The non-commutative analogues of $mathbb{P}^3$ are the spaces $operatorname{Proj}_{nc}(S)$. We show how the points, fat points, lines, and quadrics, in $operatorname{Proj}_{nc}(S)$, and their incidence relations, correspond to finite dimensional irreducible representations of $U_q(mathfrak{sl}_2)$, Verma modules, annihilators of Verma modules, and homomorphisms between them.
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].
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