### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 110, 10 pages      arXiv:1409.0274      https://doi.org/10.3842/SIGMA.2014.110
Contribution to the Special Issue on New Directions in Lie Theory

### Demazure Modules, Chari-Venkatesh Modules and Fusion Products

Bhimarthi Ravinder
The Institute of Mathematical Sciences, CIT campus, Taramani, Chennai 600113, India

Received September 11, 2014, in final form December 01, 2014; Published online December 12, 2014

Abstract
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $\theta$. Given two non-negative integers $m$, $n$, we prove that the fusion product of $m$ copies of the level one Demazure module $D(1,\theta)$ with $n$ copies of the adjoint representation ${\rm ev}_0 V(\theta)$ is independent of the parameters and we give explicit defining relations. As a consequence, for $\mathfrak{g}$ simply laced, we show that the fusion product of a special family of Chari-Venkatesh modules is again a Chari-Venkatesh module. We also get a description of the truncated Weyl module associated to a multiple of $\theta$.

Key words: current algebra; Demazure module; Chari-Venkatesh module; truncated Weyl module; fusion product.

pdf (354 kb)   tex (15 kb)

References

1. Chari V., Fourier G., Khandai T., A categorical approach to Weyl modules, Transform. Groups 15 (2010), 517-549, arXiv:0906.2014.
2. Chari V., Pressley A., Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191-223, math.QA/0004174.
3. Chari V., Shereen P., Venkatesh R., Wand J., A Steinberg type decomposition theorem for higher level Demazure modules, arXiv:1408.4090.
4. Chari V., Venkatesh R., Demazure modules, fusion products, and $Q$-systems, Comm. Math. Phys., to appear, arXiv:1305.2523.
5. Feigin B., Loktev S., On generalized Kostka polynomials and the quantum Verlinde rule, in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, Vol. 194, Amer. Math. Soc., Providence, RI, 1999, 61-79, math.QA/9812093.
6. Feigin E., The PBW filtration, Demazure modules and toroidal current algebras, SIGMA 4 (2008), 070, 21 pages, arXiv:0806.4851.
7. Fourier G., Littelmann P., Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), 566-593, math.RT/0509276.
8. Garland H., The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480-551.
9. Naoi K., Weyl modules, Demazure modules and finite crystals for non-simply laced type, Adv. Math. 229 (2012), 875-934, arXiv:1012.5480.
10. Venkatesh R., Fusion product structure of Demazure modules, Algebr. Represent. Theory, to appear, arXiv:1311.2224.