## Osaka Journal of Mathematics

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

Nagatoshi Sasano

#### Abstract

In this paper, we shall give a way to construct a graded Lie algebra $L(\mathfrak{g},\rho,V,{\cal V},B_0)$ from a standard pentad $(\mathfrak{g},\rho,V,{\cal V},B_0)$ which consists of a Lie algebra $\mathfrak{g}$ which has a non-degenerate invariant bilinear form $B_0$ and $\mathfrak{g}$-modules $(\rho, V)$ and ${\cal V}\subset \mathrm {Hom }(V,F)$ all defined over a field $F$ with characteristic $0$. In general, we do not assume that these objects are finite-dimensional. We can embed the objects $\mathfrak{g},\rho,V,{\cal V}$ into $L(\mathfrak{g},\rho,V,{\cal V},B_0)$. Moreover, we construct specific positively and negatively graded modules of $L(\mathfrak{g},\rho,V,{\cal V},B_0)$. Finally, we give a chain rule on the embedding rules of standard pentads.

#### Article information

Source
Osaka J. Math., Volume 54, Number 3 (2017), 533-568.

Dates
First available in Project Euclid: 7 August 2017

https://projecteuclid.org/euclid.ojm/1502092827

Mathematical Reviews number (MathSciNet)
MR3685591

Zentralblatt MATH identifier
06775421

Subjects

#### Citation

Sasano, Nagatoshi. Lie algebras constructed with Lie modules and their positively and negatively graded modules. Osaka J. Math. 54 (2017), no. 3, 533--568. https://projecteuclid.org/euclid.ojm/1502092827

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