Multiplet classification of highest weight modules over quantum universal enveloping algebras: the Uq(SL(3,C)) example in Groups St Andrews 1989 vol 1 LMS LNM #159

Representations of Quantum Groups, Symmetries in Science V (Lochau 1990), 93–135, Plenum Press, NY, 1991.

A chapter from Lecture Notes in Physics, 1990, Volume 370, here

Dobrev-Truini Irregular Uq(sl(3)) representations at roots of unity via Gel’fand–(Weyl)–Zetlin basis

Dobrev-Truini Polynomial realization of the Uq(sl(3)) Gel’fand–(Weyl)–Zetlin basis

MR1182163, MR1191199,

...and many others.

also there is a paper by Abdesselam, Arnaudon, Chakrabarti and a discussion of dimensions by Pereira here

• Dobrev, V. K., Multiplet classification of highest weight modules over quantum universal enveloping algebras: The $U_ q(sl(3,\mathbb{C}))$ example, Groups, Vol. 1, Proc. Int. Conf., St. Andrews/UK 1989, Lond. Math. Soc. Lect. Note Ser. 159, 87-104 (1991). Zbl 0758.17008.

• Dobrev, V. K., Representations of Quantum Groups, Gruber, B. (ed.) et al., Symmetries in Science V. Springer, Boston, MA (1991).

• Dobrev, V. K., Classification and characters of $U_q(sl(3,\mathbb{C}))$ representations, Quantum groups, Proc. 8th Int. Workshop Math. Phys., Clausthal/Germ. 1989, Lect. Notes Phys. 370, 107-117 (1990). Zbl 0727.17004.

• Dobrev, V. K.; Truini, P., Irregular $U_q(\mathrm{sl}(3))$ representations at roots of unity via Gel'fand–(Weyl)–Zetlin basis, J. Math. Phys. 38, No. 5, 2631-2651 (1997). Zbl 0965.17009.

• Dobrev, V. K.; Truini, P., Polynomial realization of the $\mathrm{U}_ q(\mathrm{sl}(3))$ Gel'fand–(Weyl)–Zetlin basis, J. Math. Phys. 38, No. 7, 3750-3767 (1997). Zbl 0882.17005.

• Dobrev, V. K., Characters of the $U_ q(sl(3,\mathbf{C}))$ highest weight modules, Eguchi, T. (ed.) et al., Common Trends in Mathematics and Quantum Field Theories. 1990 Yukawa international seminar school: Kansai Seminar House, Kyoto, Japan, May 10-16, 1990. Workshop: RIMS, Kyoto University, Japan, May 17-19, 1990. Tokyo: Yukawa Institute for Theoretical Physics, Prog. Theor. Phys., Suppl. 102, 137-158 (1990). Zbl 0784.17018. MR1182163.

• Dobrev, V. K., Representations of quantum groups for roots of $1$, Domokos, G. (ed.) et al., Nonperturbative Methods in Low Dimensional Quantum Field Theories (Debrecen, 1990), 69–105, World Sci. Publ., River Edge, NJ, 1991. MR1191199.

…and many others.

Also there is a paper by Abdesselam, Arnaudon, Chakrabarti:

• Abdesselam, B.; Arnaudon, D.; Chakrabarti, A., Representations of ${\mathcal U}_q(sl(N))$ at roots of unity, J. Phys. A, Math. Gen. 28, No. 19, 5495-5507 (1995). Zbl 0864.17016.

and a discussion of dimensions by Mariana Pereira here:

• Dimensions of simple $u_q(\mathfrak{sl}_3)$-modules.

I have tracked down some results on explicit classifications of simple modules for $u_q(\mathfrak{sl}_3)$. The general picture is that the simple modules are bigraded by the root lattice and look like towers of concentric hexagons.

For the benefit of anyone else interested, there is a long series of papers by Dobrev:

Multiplet classification of highest weight modules over quantum universal enveloping algebras: the Uq(SL(3,C)) example in Groups St Andrews 1989 vol 1 LMS LNM #159

Representations of Quantum Groups, Symmetries in Science V (Lochau 1990), 93–135, Plenum Press, NY, 1991.

A chapter from Lecture Notes in Physics, 1990, Volume 370, here

Dobrev-Truini Irregular Uq(sl(3)) representations at roots of unity via Gel’fand–(Weyl)–Zetlin basis

Dobrev-Truini Polynomial realization of the Uq(sl(3)) Gel’fand–(Weyl)–Zetlin basis

MR1182163, MR1191199,

...and many others.

also there is a paper by Abdesselam, Arnaudon, Chakrabarti and a discussion of dimensions by Pereira here

Some of the relevant material is hard to find and/or requires paying large sums of money to publishing corporations.

I have tracked down some results on explicit classifications of simple modules for $u_q(\mathfrak{sl}_3)$. The general picture is that the simple modules are bigraded by the root lattice and look like towers of concentric hexagons.

For the benefit of anyone else interested, there is a long series of papers by Dobrev:

Multiplet classification of highest weight modules over quantum universal enveloping algebras: the Uq(SL(3,C)) example in Groups St Andrews 1989 vol 1 LMS LNM #159

Representations of Quantum Groups, Symmetries in Science V (Lochau 1990), 93–135, Plenum Press, NY, 1991.

A chapter from Lecture Notes in Physics, 1990, Volume 370, here

Dobrev-Truini Irregular Uq(sl(3)) representations at roots of unity via Gel’fand–(Weyl)–Zetlin basis

Dobrev-Truini Polynomial realization of the Uq(sl(3)) Gel’fand–(Weyl)–Zetlin basis

MR1182163, MR1191199,

...and many others.

also there is a paper by Abdesselam, Arnaudon, Chakrabarti and a discussion of dimensions by Pereira here

Some of the relevant material is hard to find and/or requires paying large sums of money to publishing corporations.