Geodesic
Commutative Subalgebras of Quantum Algebras
Matematičeskie zametki, Tome 75 (2004) no. 2, pp. 208-221.

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In the present paper, a general assertion is proved, claiming that, for every associative algebra $\mathscr A$ without zero divisors which admits a valuation and a seminorm concordant with the valuation, the transcendence degree of an arbitrary commutative subalgebra does not exceed the maximal number of independent pairwise pseudocommuting elements of some basis of the algebra $\mathscr A$. The author shows that for such a algebra $\mathscr A$ one can take an arbitrary algebra of quantum Laurent polynomials, quantum analogs of the Weyl algebra, and also some universal coacting algebras. In the case of the algebra $\mathscr L$ of quantum Laurent polynomials, it is proved that the transcendence degree of a maximal commutative subalgebra of $\mathscr L$ coincides with the maximal number of independent pairwise commuting elements of the monomial basis of the algebra $\mathscr L$. 
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S. A. Zelenova. Commutative Subalgebras of Quantum Algebras. Matematičeskie zametki, Tome 75 (2004) no. 2, pp. 208-221. http://geodesic.mathdoc.fr/item/MZM_2004_75_2_a4/

[1] Artamonov V. A., Cohn P. M., “The skew field of rational functions on the quantum plane”, J. Math. Sci., 93:6 (1999), 824–829 | DOI | MR | Zbl

[2] Cohn P. M., “Centralisateur dans les corps libres”, Ecole de Printemps d'Informatique Théorique, “Séries formelles en variables non commutatives et applications”, ed. J. Berstel, Vieux-Boucau les Bains (Landes), 1978, 45–54 | MR | Zbl

[3] Artamonov V. A., “Telo kvantovykh ratsionalnykh funktsii”, UMN, 54:4 (1999), 151–152 | MR | Zbl

[4] Zelenova S. A., “Kommutativnye podalgebry koltsa kvantovykh mnogochlenov i tela kvantovykh loranovskikh ryadov”, Matem. sb., 192:3 (2001), 55–64 | MR | Zbl

[5] Alev J., Dumas F., “Sur le corps des fractions de certaines algèbres quantiques”, J. of Algebra, 170:1 (1994), 229–265 | DOI | MR | Zbl

[6] Demidov E. E., “O nekotorykh aspektakh teorii kvantovykh grupp”, UMN, 48:6 (1993), 39–74 | MR | Zbl

[7] Brown K. A., Goodearl K. R., Lecture on algebraic quantum groups, Birkhäuser, Basel–Boston, 2002 | MR

[8] Brookes C. J. B., “Crossed products and finitely presented groups”, J. Group Theory, 3 (2000), 433–444 | DOI | MR | Zbl