Geodesic
Quantum representation theory and Manin matrices I: The finite-dimensional case
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 83 (2022) no. 1, pp. 87-179.

Voir la notice de l'article provenant de la source Math-Net.Ru

We construct a theory that describes a quantum (non-commutative) analogue of representations within the framework of “non-commutative linear geometry” set out in the work of Manin [Quantum groups and noncommutative geometry, Univ. Montréal, Centre de Recherches Mathématiques, Montréal, QC, 1988]. For this purpose, the concept of an internal $\mathrm{hom}$-functor is generalized to the case of parameterized adjunctions, and we construct a general approach to representations of monoids for a symmetric monoidal category with a parameter subcategory. A quantum theory of representations is then obtained by applying this approach to the monoidal category of a certain class of graded algebras with the Manin product, where the parameterizing subcategory consists of connected finitely generated quadratic algebras. We formulate this theory in the language of Manin matrices. We also obtain quantum analogues of the direct sum and tensor product of representations. Finally, we give some examples of quantum representations. 
@article{MMO_2022_83_1_a5,
     author = {A. V. Silantyev},
     title = {Quantum representation theory and {Manin} matrices {I:} {The} finite-dimensional case},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {87--179},
     publisher = {mathdoc},
     volume = {83},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MMO_2022_83_1_a5/}
}
TY  - JOUR
AU  - A. V. Silantyev
TI  - Quantum representation theory and Manin matrices I: The finite-dimensional case
JO  - Trudy Moskovskogo matematičeskogo obŝestva
PY  - 2022
SP  - 87
EP  - 179
VL  - 83
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MMO_2022_83_1_a5/
LA  - ru
ID  - MMO_2022_83_1_a5
ER  - 
%0 Journal Article
%A A. V. Silantyev
%T Quantum representation theory and Manin matrices I: The finite-dimensional case
%J Trudy Moskovskogo matematičeskogo obŝestva
%D 2022
%P 87-179
%V 83
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MMO_2022_83_1_a5/
%G ru
%F MMO_2022_83_1_a5
A. V. Silantyev. Quantum representation theory and Manin matrices I: The finite-dimensional case. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 83 (2022) no. 1, pp. 87-179. http://geodesic.mathdoc.fr/item/MMO_2022_83_1_a5/

[1] Drinfeld V. G., “Kvantovye gruppy”, Zap. nauchn. sem. LOMI, 155, 1986, 18–49 | Zbl

[2] Kassel K., Kvantovye gruppy, B-ka matematika, 5, Fazis, M., 1999

[3] Maklein S., Kategorii dlya rabotayuschego matematika, Fizmatlit, M., 2004

[4] Manin Yu. I., Lektsii po algebraicheskoi geometrii, v. 1, Affinnye skhemy, MGU, M., 1970 ; Манин Ю. И., Введение в теорию схем и квантовые группы, МЦНМО, М., 2020, 8–109 | MR

[5] Molev A. I., Operatory Sugavary dlya klassicheskikh algebr Li, MTsNMO, M., 2018 (elektronnoe izdanie) https://www.mccme.ru/free-books/Molev-Sugavara.pdf

[6] Reshetikhin N. Yu., Takhtadzhyan L. A., Faddeev L. D., “Kvantovanie grupp Li i algebr Li”, Algebra i analiz, 1:1 (1989), 178–206

[7] Talalaev D. V., “Kvantovaya sistema Godena”, Funkts. analiz i ego pril., 40:1 (2006), 86–91, arXiv: hep-th/0404153 | DOI | MR | Zbl

[8] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981

[9] Arnaudon D., Molev A., Ragoucy E., “On the $R$-matrix realization of Yangians and their representations”, Ann. H. Poincaré, 7 (2006), 1269–1325, arXiv: math/0511481 | DOI | MR | Zbl

[10] Borceux F., Handbook of categorical algebra, v. 2, Encyclopedia of Mathematics and its Applications, 51, Categories and structures, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[11] Chervov A., Falqui G., “Manin matrices and Talalaev's formula”, J. Phys. A, 41:19 (2009), 194006, arXiv: 0711.2236 | DOI | MR

[12] Chervov A. V., Molev A. I., “On higher-order Sugawara operators”, IMRN, 2009:9 (2009), 1612–1635, arXiv: 0808.1947 | MR | Zbl

[13] “Chervov A., Falqui G., Rubtsov V.”, Adv. in Appl. Math., 43:3 (2009), 239–315, arXiv: 0901.0235 | DOI | MR | Zbl

[14] Chervov A., Falqui G., Rubtsov V., Silantyev A., “Algebraic properties of Manin matrices. II: $q$-analogues and integrable systems”, Adv. in Appl. Math., 60 (2014), 25–89, arXiv: 1210.3529 | DOI | MR | Zbl

[15] Deligne P., Morgan J. W., “Notes on supersymmetry (following Joseph Bernstein)”, Quantum fields and strings: a course for mathematicians, v. 1, 2, AMS, Princeton, NJ, 1996/1997, 41–97 | MR

[16] Eckmann B., Hilton P., “Structure maps in group theory”, Fund. Math., 50 (1961), 207–221 | DOI | MR | Zbl

[17] Foata D., Han G.-N., “A new proof of the Garoufalidis"– Lê"– Zeilberger quantum MacMahon master theorem”, J. Algebra, 307:1 (2007), 424–431, arXiv: math.CO/0603464 | DOI | MR | Zbl

[18] Foata D., Han G.-N., “A basis for the right quantum algebra and the "$1=q$" principle”, J. Algebraic Combin., 27:2 (2008), 163–172, arXiv: math.CO/0603463 | DOI | MR | Zbl

[19] Fulton W., Algebraic curves. An introduction to algebraic geometry, Addison-Wesley Publ. Company, Redwood City, CA, 1989 (first published in 1969; revised in 2008) http://www.math.lsa.umich.edu/w̃fulton/CurveBook.pdf | MR | Zbl

[20] Garoufalidis S., Lê T. T. Q., Zeilberger D., “The quantum MacMahon master theorem”, Proc. Nat. Acad. of Sci., 103:38 (2006), 13928–13931, arXiv: math.QA/0303319 | DOI | MR | Zbl

[21] Isaev A., Ogievetsky O., “Half-quantum linear algebra”, Symmetries and groups in contemporary physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., 11, World Sci. Publ., Hackensack, NJ, 2013, 479–486, arXiv: 1303.3991 | MR | Zbl

[22] Khoroshkin S. M., “Central extension of the Yangian double”, Algèbre non commutative, groupes quantiques et invariants (Reims, 1995), Sémin. Congr., 2, SMF, Paris, 1997, 119–135 | MR

[23] Lang S., Differential and Riemannian manifolds, Graduate Texts in Math., 160, Springer-Verlag, Berlin–New York, 1995 | DOI | MR | Zbl

[24] Manin Yu. I., “Some remarks on Koszul algebras and quantum groups”, Ann. Inst. Fourier (Grenoble), 37:4 (1987), 191–205 | DOI | MR | Zbl

[25] Manin Yu. I., Quantum groups and noncommutative geometry, Univ. Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1988 | MR

[26] Manin Yu. I., “Multiparametric quantum deformation of the general linear supergroup”, Comm. Math. Phys., 123:1 (1989), 163–175 | DOI | MR | Zbl

[27] Manin Yu. I., Topics in non-commutative geometry, Porter Lectures, 3, Princeton Univ. Press, Princeton, NJ, 1991 | MR

[28] Manin Yu. I., “Notes on quantum groups and quantum de Rham complexes”, TMF, 92:3 (1992), 425–450 | MR | Zbl

[29] Molev A. I., Ragoucy E., “The MacMahon master theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{g}l}_{m|n}$”, Mosc. Math. J., 14:1 (2014), 83–119, arXiv: 0911.3447 | DOI | MR | Zbl

[30] Pareigis B., “A non-commutative non-cocommutative Hopf algebra in “nature””, J. Algebra, 70:2 (1981), 356–374 | DOI | MR | Zbl

[31] Algebraic geometry: an introduction, Springer-Verlag, London, 2008 | MR | MR | Zbl

[32] Porst H.-E., “On categories of monoids, comonoids, and bimonoids”, Quaest. Math., 31:2 (2008), 127–139 | DOI | MR | Zbl

[33] Rubtsov V., Silantyev A., Talalaev D., “Manin matrices, quantum elliptic commutative families and characteristic polynomial of elliptic Gaudin model”, SIGMA, 5 (2009), 110, 1–22, arXiv: 0908.4064 | MR

[34] Silantyev A., “Manin matrices for quadratic algebras”, SIGMA, 17 (2021), 066, 1–81, arXiv: 2009.05993 | MR