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Nagatoshi Sasano

Publications and source records attributed to Nagatoshi Sasano.

5 recordsLinked to original sources
arXiv2022

A class of Lie algebras who contains a class of Kac-Moody algebras

The theory of standard pentads is the theory aims to construct a graded Lie algebra whose local part consists of a given Lie algebra and its representation. In other words, using standard pentads, we can embed given Lie algebra and its representation into a larger graded Lie algebra. As special cases of Lie algebras associated with standard pentads, we have the notion of PC Lie algebras. Our aim of this paper is to show that the class of PC Lie algebras contains the class of Kac-Moody algebras, that is, to show that the notion of PC Lie algebras is an extension of Kac-Moody algebras.

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arXiv2017

Contragredient Lie algebras and Lie algebras associated with a standard pentad

We will construct standard pentads which are analogues of Cartan subalgebras, and moreover, we will study graded Lie algebras corresponding to these standard pentads. We call such pentads pentads of Cartan type and describe them by two positive integers and three matrices. The structures of their corresponding Lie algebras are related with contragredient Lie algebras.

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arXiv2017

Reduced contragredient Lie algebras and PC Lie algebras

The first aim of this paper is to show that any finite-dimensional reductive Lie algebra and its finite-dimensional completely reducible representation can be embedded into some PC Lie algebra. The second aim is to find the structure of a PC Lie algebra.

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arXiv2016

Lie algebras constructed with Lie modules and their positively and negatively graded modules

In this paper, we shall give a way to construct a graded Lie algebra $L(\mathfrak{g},ρ,V,{\cal V},B_0)$ from a standard pentad $(\mathfrak{g},ρ,V,{\cal V},B_0)$ which consists of a Lie algebra $\mathfrak{g}$ which has a non-degenerate invariant bilinear form $B_0$ and $\mathfrak{g}$-modules $(ρ, V)$ and ${\cal V}\subset \mathrm {Hom }(V,\mathfrak{k})$ all defined over a field $\mathfrak{k}$. In general, we do not assume that these objects are finite-dimensional. We can embed the objects $\mathfrak{g},ρ,V,{\cal V}$ into $L(\mathfrak{g},ρ,V,{\cal V},B_0)$. Moreover, we construct specific positively and negatively graded modules of $L(\mathfrak{g},ρ,V,{\cal V},B_0)$. Finally, we give a chain rule on the embedding rules of standard pentads.

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