Geodesic
Explicit Central Elements of $U_q(\mathfrak{gl}(N+1))$
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

By using Drinfeld's central element construction and fusion of $R$-matrices, we construct central elements of the quantum group $U_q(\mathfrak{gl}(N+1))$. These elements are explicitly written in terms of the generators.
Keywords: quantum groups, Harish-Chandra isomorphism, central elements. 
@article{SIGMA_2023_19_a35,
     author = {Jeffrey Kuan and Keke Zhang},
     title = {Explicit {Central} {Elements} of $U_q(\mathfrak{gl}(N+1))$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a35/}
}
TY  - JOUR
AU  - Jeffrey Kuan
AU  - Keke Zhang
TI  - Explicit Central Elements of $U_q(\mathfrak{gl}(N+1))$
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2023
VL  - 19
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a35/
LA  - en
ID  - SIGMA_2023_19_a35
ER  - 
%0 Journal Article
%A Jeffrey Kuan
%A Keke Zhang
%T Explicit Central Elements of $U_q(\mathfrak{gl}(N+1))$
%J Symmetry, integrability and geometry: methods and applications
%D 2023
%V 19
%U http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a35/
%G en
%F SIGMA_2023_19_a35
Jeffrey Kuan; Keke Zhang. Explicit Central Elements of $U_q(\mathfrak{gl}(N+1))$. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a35/

[1] Arnaudon D., Bauer M., “Polynomial relations in the centre of $\mathcal{U}_q(\mathfrak{sl}(N))$”, Lett. Math. Phys., 30 (1994), 251–257, arXiv: hep-th/9310030 | DOI | MR | Zbl

[2] Carinci G., Giardinà C., Redig F., Sasamoto T., “Asymmetric stochastic transport models with $\mathcal{U}_q(\mathfrak{su}(1,1))$ symmetry”, J. Stat. Phys., 163 (2016), 239–279, arXiv: 1507.01478 | DOI | MR | Zbl

[3] Carinci G., Giardinà C., Redig F., Sasamoto T., “A generalized asymmetric exclusion process with $\mathcal{U}_q(\mathfrak{sl}_2)$ stochastic duality”, Probab. Theory Related Fields, 166 (2016), 887–933, arXiv: 1407.3367 | DOI | MR | Zbl

[4] Carter R.W., Finite groups of Lie type: conjugacy classes and complex characters, Pure Appl. Math. (New York), John Wiley Sons, Inc., New York, 1985 | MR | Zbl

[5] Dai Y., “Explicit generators of the centre of the quantum group”, Commun. Math. Stat. (to appear) , arXiv: 1912.04021 | DOI

[6] Drinfeld V.G., “Almost cocommutative Hopf algebras”, Leningrad Math. J., 1 (1990), 321–342 | MR | Zbl

[7] Faddeev L.D., Reshetikhin N.Yu., Takhtajan L.A., “Quantization of Lie groups and Lie algebras”, Algebraic Analysis, v. I, Academic Press, Boston, MA, 1988, 129–139 | DOI | MR

[8] Gould M.D., Zhang R.B., Bracken A.J., “Generalized Gel'fand invariants and characteristic identities for quantum groups”, J. Math. Phys., 32 (1991), 2298–2303 | DOI | MR | Zbl

[9] Jimbo M., “A $q$-analogue of $U(\mathfrak{gl}(N+1))$, Hecke algebra, and the Yang–Baxter equation”, Lett. Math. Phys., 11 (1986), 247–252 | DOI | MR | Zbl

[10] Khoroshkin S.M., Tolstoy V.N., “Universal $R$-matrix for quantized (super)algebras”, Comm. Math. Phys., 141 (1991), 599–617 | DOI | MR | Zbl

[11] Kirillov A.N., Reshetikhin N., “$q$-Weyl group and a multiplicative formula for universal $R$-matrices”, Comm. Math. Phys., 134 (1990), 421–431 | DOI | MR | Zbl

[12] Kuan J., A (2+1)-dimensional Gaussian field as fluctuations of quantum random walks on quantum groups, arXiv: 1601.04402

[13] Kuan J., “Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two”, J. Phys. A, 49 (2016), 115002, 29 pp., arXiv: 1504.07173 | DOI | MR | Zbl

[14] Kuan J., “A multi-species ${\rm ASEP}(q,j)$ and $q$-TAZRP with stochastic duality”, Int. Math. Res. Not., 2018 (2018), 5378–5416, arXiv: 1605.00691 | DOI | MR | Zbl

[15] Kuniba A., Mangazeev V.V., Maruyama S., Okado M., “Stochastic $R$ matrix for $U_q\big(A_n^{(1)}\big)$”, Nuclear Phys. B, 913 (2016), 248–277, arXiv: 1604.08304 | DOI | MR | Zbl

[16] Li J., “The quantum Casimir operators of ${\rm U}_q(\mathfrak{gl}_n)$ and their eigenvalues”, J. Phys. A, 43 (2010), 345202, 9 pp., arXiv: 1003.3729 | DOI | MR | Zbl

[17] Li L., Xia L., Zhang Y., “On the centers of quantum groups of $A_n$-type”, Sci. China Math., 61 (2018), 287–294 | DOI | MR | Zbl

[18] Lusztig G., “Canonical bases arising from quantized enveloping algebras”, J. Amer. Math. Soc., 3 (1990), 447–498 | DOI | MR | Zbl

[19] Mudrov A., “Quantum conjugacy classes of simple matrix groups”, Comm. Math. Phys., 272 (2007), 635–660, arXiv: math.QA/0412538 | DOI | MR | Zbl

[20] Rosso M., “Représentations irréductibles de dimension finie du $q$-analogue de l'algèbre enveloppante d'une algèbre de Lie simple”, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 587–590 | MR | Zbl

[21] Tanisaki T., “Killing forms, Harish-Chandra isomorphisms, and universal $R$-matrices for quantum algebras”, Internat. J. Modern Phys. A, 7, supp. 01b (1992), 941–961 | DOI | MR | Zbl

[22] Xi N.H., “Root vectors in quantum groups”, Comment. Math. Helv., 69 (1994), 612–639 | DOI | MR | Zbl

[23] Zhang R.B., Gould M.D., Bracken A.J., “Generalized Gel'fand invariants of quantum groups”, J. Phys. A, 24 (1991), 937–943 | DOI | MR | Zbl

[24] Zhang R.B., Gould M.D., Bracken A.J., “Quantum group invariants and link polynomials”, Comm. Math. Phys., 137 (1991), 13–27 | DOI | MR | Zbl