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Data publikacji: 13 lipca 2024

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1

Zhang, Yang, i Xiangui Zhao. "Gelfand–Kirillov dimension of differential difference algebras". LMS Journal of Computation and Mathematics 17, nr 1 (2014): 485–95. http://dx.doi.org/10.1112/s1461157014000102.

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AbstractDifferential difference algebras, introduced by Mansfield and Szanto, arose naturally from differential difference equations. In this paper, we investigate the Gelfand–Kirillov dimension of differential difference algebras. We give a lower bound of the Gelfand–Kirillov dimension of a differential difference algebra and a sufficient condition under which the lower bound is reached; we also find an upper bound of this Gelfand–Kirillov dimension under some specific conditions and construct an example to show that this upper bound cannot be sharpened any further.

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2

BERGEN, JEFFREY, i PIOTR GRZESZCZUK. "GK DIMENSION AND LOCALLY NILPOTENT SKEW DERIVATIONS". Glasgow Mathematical Journal 57, nr 3 (18.12.2014): 555–67. http://dx.doi.org/10.1017/s0017089514000482.

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AbstractLet A be a domain over an algebraically closed field with Gelfand–Kirillov dimension in the interval [2,3). We prove that if A has two locally nilpotent skew derivations satisfying some natural conditions, then A must be one of five algebras. All five algebras are Noetherian, finitely generated, and have Gelfand–Kirillov dimension equal to 2. We also obtain some results comparing the Gelfand–Kirillov dimension of an algebra to its subring of invariants under a locally nilpotent skew derivation.

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3

Lezama, Oswaldo, i Helbert Venegas. "Gelfand–Kirillov dimension for rings". São Paulo Journal of Mathematical Sciences 14, nr 1 (24.04.2020): 207–22. http://dx.doi.org/10.1007/s40863-020-00166-4.

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4

CENTRONE, LUCIO. "A NOTE ON GRADED GELFAND–KIRILLOV DIMENSION OF GRADED ALGEBRAS". Journal of Algebra and Its Applications 10, nr 05 (październik 2011): 865–89. http://dx.doi.org/10.1142/s0219498811004987.

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In this paper, we consider associative P.I. algebras over a field F of characteristic 0, graded by a finite group G. More precisely, we define the G-graded Gelfand–Kirillov dimension of a G-graded P.I. algebra. We find a basis of the relatively free graded algebras of the upper triangular matrices UTn(F) and UTn(E), with entries in F and in the infinite-dimensional Grassmann algebra, respectively. As a consequence, we compute their graded Gelfand–Kirillov dimension with respect to the natural gradings defined over these algebras. We obtain similar results for the upper triangular matrix algebra UTa, b(E) = UTa+b(E)∩Ma, b(E) with respect to its natural ℤa+b × ℤ2-grading. Finally, we compute the ℤn-graded Gelfand–Kirillov dimension of Mn(F) in some particular cases and with different methods.

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5

Bell, Jason P., T. H. Lenagan i Kulumani M. Rangaswamy. "Leavitt path algebras satisfying a polynomial identity". Journal of Algebra and Its Applications 15, nr 05 (30.03.2016): 1650084. http://dx.doi.org/10.1142/s0219498816500845.

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Leavitt path algebras [Formula: see text] of an arbitrary graph [Formula: see text] over a field [Formula: see text] satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When [Formula: see text] is a finite graph, [Formula: see text] satisfying a polynomial identity is shown to be equivalent to the Gelfand–Kirillov dimension of [Formula: see text] being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph [Formula: see text], the Leavitt path algebra [Formula: see text] has Gelfand–Kirillov dimension zero if and only if [Formula: see text] has no cycles. Likewise, [Formula: see text] has Gelfand–Kirillov dimension one if and only if [Formula: see text] contains at least one cycle, but no cycle in [Formula: see text] has an exit.

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6

Zhao, Xiangui, i Yang Zhang. "Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras". Algebra Colloquium 23, nr 04 (26.09.2016): 701–20. http://dx.doi.org/10.1142/s1005386716000596.

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Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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7

Moreno-Fernández, José M., i Mercedes Siles Molina. "Graph algebras and the Gelfand–Kirillov dimension". Journal of Algebra and Its Applications 17, nr 05 (26.04.2018): 1850095. http://dx.doi.org/10.1142/s0219498818500950.

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We study some properties of the Gelfand–Kirillov dimension in a non-necessarily unital context, in particular, its Morita invariance when the algebras have local units, and its commutativity with direct limits. We then give some applications in the context of graph algebras, which embraces, among some others, path algebras and Cohn and Leavitt path algebras. In particular, we determine the GK-dimension of these algebras in full generality, so extending the main result in A. Alahmadi, H. Alsulami, S. K. Jain and E. Zelmanov, Leavitt Path algebras of finite Gelfand–Kirillov dimension, J. Algebra Appl. 11(6) (2012) 1250225–1250231.

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8

Martinez, C. "Gelfand-Kirillov dimension in Jordan Algebras". Transactions of the American Mathematical Society 348, nr 1 (1996): 119–26. http://dx.doi.org/10.1090/s0002-9947-96-01528-0.

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9

Smith, S. Paul, i James J. Zhang. "A remark on Gelfand-Kirillov dimension". Proceedings of the American Mathematical Society 126, nr 2 (1998): 349–52. http://dx.doi.org/10.1090/s0002-9939-98-04074-x.

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10

Leroy, A., i I. Matczuk. "Gelfand-Kirillov dimension of certain localizations". Archiv der Mathematik 53, nr 5 (listopad 1989): 439–47. http://dx.doi.org/10.1007/bf01324719.

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11

ALAHMADI, ADEL, HAMED ALSULAMI, S. K. JAIN i EFIM ZELMANOV. "LEAVITT PATH ALGEBRAS OF FINITE GELFAND–KIRILLOV DIMENSION". Journal of Algebra and Its Applications 11, nr 06 (14.11.2012): 1250225. http://dx.doi.org/10.1142/s0219498812502258.

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12

Torrecillas, José Gómez. "Gelfand-Kirillov dimension of multi-filtered algebras". Proceedings of the Edinburgh Mathematical Society 42, nr 1 (luty 1999): 155–68. http://dx.doi.org/10.1017/s0013091500020083.

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We consider associative algebras filtered by the additive monoid ℕp. We prove that, under quite general conditions, the study of Gelfand-Kirillov dimension of modules over a multi-filtered algebra R can be reduced to the associated ℕp-graded algebra G(R). As a consequence, we show the exactness of the Gelfand-Kirillov dimension when the multi-filtration is finite-dimensional and G(R) is a finitely generated noetherian algebra. Our methods apply to examples like iterated Ore extensions with arbitrary derivations and “homothetic” automorphisms (e.g. quantum matrices, quantum Weyl algebras) and the quantum enveloping algebra of sl(v + 1)

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13

Bueso, José L., F. J. Castro Jiménez i Pascual Jara. "Effective computation of the Gelfand-Kirillov dimension". Proceedings of the Edinburgh Mathematical Society 40, nr 1 (luty 1997): 111–17. http://dx.doi.org/10.1017/s0013091500023476.

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14

Kobayashi, Shigeru, i Yuji Kobayashi. "On Algebras With Gelfand-Kirillov Dimension One". Proceedings of the American Mathematical Society 119, nr 4 (grudzień 1993): 1095. http://dx.doi.org/10.2307/2159971.

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15

Zhang, Yang, i Xiangui Zhao. "Gelfand-Kirillov dimension of differential difference algebras". ACM Communications in Computer Algebra 49, nr 1 (10.06.2015): 32. http://dx.doi.org/10.1145/2768577.2768642.

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16

Bell, Jason P. "Simple algebras of Gelfand-Kirillov dimension two". Proceedings of the American Mathematical Society 137, nr 03 (15.10.2008): 877–83. http://dx.doi.org/10.1090/s0002-9939-08-09724-4.

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17

Riley, David, i Hamid Usefi. "Lie algebras with finite Gelfand-Kirillov dimension". Proceedings of the American Mathematical Society 133, nr 6 (13.01.2005): 1569–72. http://dx.doi.org/10.1090/s0002-9939-05-07618-5.

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18

Lorenz, Martin. "On Gelfand-Kirillov dimension and related topics". Journal of Algebra 118, nr 2 (listopad 1988): 423–37. http://dx.doi.org/10.1016/0021-8693(88)90031-2.

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19

McConnell, J. C., i J. T. Stafford. "Gelfand-Kirillov dimension and associated graded modules". Journal of Algebra 125, nr 1 (sierpień 1989): 197–214. http://dx.doi.org/10.1016/0021-8693(89)90301-3.

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20

Chirvasitu, Alexandru, Chelsea Walton i Xingting Wang. "Gelfand-Kirillov dimension of cosemisimple Hopf algebras". Proceedings of the American Mathematical Society 147, nr 11 (10.06.2019): 4665–72. http://dx.doi.org/10.1090/proc/14616.

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21

Wu, Quanshui. "Gelfand-Kirillov dimension under base field extension". Israel Journal of Mathematics 73, nr 3 (październik 1991): 289–96. http://dx.doi.org/10.1007/bf02773842.

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22

Okninski, J. "Gelfand-Kirillov Dimension of Noetherian Semigroup Algebras". Journal of Algebra 162, nr 2 (grudzień 1993): 302–16. http://dx.doi.org/10.1006/jabr.1993.1255.

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23

Martinez, C., i E. Zelmanov. "Jordan Algebras of Gelfand–Kirillov Dimension One". Journal of Algebra 180, nr 1 (luty 1996): 211–38. http://dx.doi.org/10.1006/jabr.1996.0063.

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24

Vishne, Uzi. "Primitive Algebras with Arbitrary Gelfand-Kirillov Dimension". Journal of Algebra 211, nr 1 (styczeń 1999): 150–58. http://dx.doi.org/10.1006/jabr.1998.7567.

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25

Martínez-Villa, Roberto, i Øyvind Solberg. "Noetherianity and Gelfand–Kirillov dimension of components". Journal of Algebra 323, nr 5 (marzec 2010): 1369–407. http://dx.doi.org/10.1016/j.jalgebra.2009.12.013.

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26

Lenagan, T. H., Agata Smoktunowicz i Alexander A. Young. "Nil algebras with restricted growth". Proceedings of the Edinburgh Mathematical Society 55, nr 2 (23.02.2012): 461–75. http://dx.doi.org/10.1017/s0013091510001100.

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27

CEDÓ, FERRAN, ERIC JESPERS i JAN OKNIŃSKI. "SEMIPRIME QUADRATIC ALGEBRAS OF GELFAND–KIRILLOV DIMENSION ONE". Journal of Algebra and Its Applications 03, nr 03 (wrzesień 2004): 283–300. http://dx.doi.org/10.1142/s0219498804000848.

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We consider algebras over a field K with a presentation K<x1,…,xn:R>, where R consists of [Formula: see text] square-free relations of the form xixj=xkxl with every monomial xixj, i≠j, appearing in one of the relations. The description of all four generated algebras of this type that satisfy a certain non-degeneracy condition is given. The structure of one of these algebras is described in detail. In particular, we prove that the Gelfand–Kirillov dimension is one while the algebra is noetherian PI and semiprime in case when the field K has characteristic zero. All minimal prime ideals of the algebra are described. It is also shown that the underlying monoid is a semilattice of cancellative semigroups and its structure is described. For any positive integer m, we construct non-degenerate algebras of the considered type on 4m generators that have Gelfand–Kirillov dimension one and are semiprime noetherian PI algebras.

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28

Centrone, Lucio. "ℤ2-Graded Gelfand–Kirillov dimension of the Grassmann algebra". International Journal of Algebra and Computation 24, nr 03 (maj 2014): 365–74. http://dx.doi.org/10.1142/s0218196714500167.

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We consider the infinite dimensional Grassmann algebra E over a field F of characteristic 0 or p, where p > 2, and we compute its ℤ2-graded Gelfand–Kirillov (GK) dimension as a ℤ2-graded PI-algebra.

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29

Bell, Jason P. "Centralizers in domains of finite Gelfand-Kirillov dimension". Bulletin of the London Mathematical Society 41, nr 3 (28.04.2009): 559–62. http://dx.doi.org/10.1112/blms/bdp039.

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30

Wang, D. G., J. J. Zhang i G. Zhuang. "Connected Hopf algebras of Gelfand-Kirillov dimension four". Transactions of the American Mathematical Society 367, nr 8 (3.02.2015): 5597–632. http://dx.doi.org/10.1090/s0002-9947-2015-06219-9.

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31

BELL, JASON P., i LANCE W. SMALL. "CENTRALIZERS IN DOMAINS OF GELFAND–KIRILLOV DIMENSION 2". Bulletin of the London Mathematical Society 36, nr 06 (19.10.2004): 779–85. http://dx.doi.org/10.1112/s0024609304003534.

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32

Chakraborty, Partha Sarathi, i Bipul Saurabh. "Gelfand–Kirillov dimension of some simple unitarizable modules". Journal of Algebra 514 (listopad 2018): 199–218. http://dx.doi.org/10.1016/j.jalgebra.2018.08.007.

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33

Cui, Ranran, i Yanfeng Luo. "Gelfand-Kirillov dimension of some primitive abundant semigroups". Indian Journal of Pure and Applied Mathematics 44, nr 6 (grudzień 2013): 809–22. http://dx.doi.org/10.1007/s13226-013-0044-5.

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34

Bell, Allen D., i Gunnar Sigurdsson. "Catenarity and Gelfand-Kirillov dimension in Ore extensions". Journal of Algebra 127, nr 2 (grudzień 1989): 409–25. http://dx.doi.org/10.1016/0021-8693(89)90261-5.

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35

Iyudu, Natalia K., i Susan J. Sierra. "Enveloping algebras with just infinite Gelfand–Kirillov dimension". Arkiv för Matematik 58, nr 2 (2020): 285–306. http://dx.doi.org/10.4310/arkiv.2020.v58.n2.a4.

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36

Samet-Vaillant, A. Y. "C*-Algebras, Gelfand–Kirillov Dimension, and Følner Sets". Journal of Functional Analysis 171, nr 2 (marzec 2000): 346–65. http://dx.doi.org/10.1006/jfan.1999.3525.

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37

Andruskiewitsch, Nicolás, Iván Angiono i István Heckenberger. "Liftings of Jordan and Super Jordan Planes". Proceedings of the Edinburgh Mathematical Society 61, nr 3 (12.04.2018): 661–72. http://dx.doi.org/10.1017/s0013091517000402.

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AbstractWe classify pointed Hopf algebras with finite Gelfand–Kirillov dimension whose infinitesimal braiding has dimension 2 but is not of diagonal type, or equivalently is a block. These Hopf algebras are new and turn out to be liftings of either a Jordan or a super Jordan plane over a nilpotent-by-finite group.

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38

Nekrashevych, Volodymyr. "Growth of étale groupoids and simple algebras". International Journal of Algebra and Computation 26, nr 02 (marzec 2016): 375–97. http://dx.doi.org/10.1142/s0218196716500156.

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We study growth and complexity of étale groupoids in relation to growth of their convolution algebras. As an application, we construct simple finitely generated algebras of arbitrary Gelfand–Kirillov dimension [Formula: see text] and simple finitely generated algebras of quadratic growth over arbitrary fields.

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39

Small, L. W., J. T. Stafford i R. B. Warfield. "Affine algebras of Gelfand-Kirillov dimension one are PI". Mathematical Proceedings of the Cambridge Philosophical Society 97, nr 3 (maj 1985): 407–14. http://dx.doi.org/10.1017/s0305004100062976.

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The aim of this paper is to prove:Theorem.Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

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40

GOODEARL, K. R., i J. J. ZHANG. "NON-AFFINE HOPF ALGEBRA DOMAINS OF GELFAND–KIRILLOV DIMENSION TWO". Glasgow Mathematical Journal 59, nr 3 (20.03.2017): 563–93. http://dx.doi.org/10.1017/s0017089516000410.

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AbstractWe classify all non-affine Hopf algebras H over an algebraically closed field k of characteristic zero that are integral domains of Gelfand–Kirillov dimension two and satisfy the condition Ext1H(k, k) ≠ 0. The affine ones were classified by the authors in 2010 (Goodearl and Zhang, J. Algebra324 (2010), 3131–3168).

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41

SMOKTUNOWICZ, AGATA. "GK–DIMENSION OF ALGEBRAS WITH MANY GENERIC RELATIONS". Glasgow Mathematical Journal 51, nr 2 (maj 2009): 253–56. http://dx.doi.org/10.1017/s0017089508004667.

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AbstractWe prove some results on algebras, satisfying many generic relations. As an application we show that there are Golod–Shafarevich algebras which cannot be homomorphically mapped onto infinite dimensional algebras with finite Gelfand–Kirillov dimension. This answers a question of Zelmanov (Some open problems in the theory of infinite dimensional algebras, J. Korean Math. Soc. 44(5) 2007, 1185–1195).

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42

Bueso, J. L., J. Gómez-Torrecillas i F. J. Lobillo. "Re-filtering and exactness of the Gelfand–Kirillov dimension". Bulletin des Sciences Mathématiques 125, nr 8 (listopad 2001): 689–715. http://dx.doi.org/10.1016/s0007-4497(01)01090-9.

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43

Ferraro, Luigi, Jason Gaddis i Robert Won. "Simple Z-graded domains of Gelfand–Kirillov dimension two". Journal of Algebra 562 (listopad 2020): 433–65. http://dx.doi.org/10.1016/j.jalgebra.2020.06.030.

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44

Beaulieu, R. A., A. Jensen i S. Jøndrup. "Towards a more general notion of Gelfand-Kirillov dimension". Israel Journal of Mathematics 93, nr 1 (grudzień 1996): 73–92. http://dx.doi.org/10.1007/bf02761094.

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45

Bergen, Jeffrey, i Piotr Grzeszczuk. "Gelfand-Kirillov dimension of algebras with locally nilpotent derivations". Israel Journal of Mathematics 206, nr 1 (luty 2015): 313–25. http://dx.doi.org/10.1007/s11856-015-1152-1.

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46

Centrone, Lucio. "The graded Gelfand--Kirillov dimension of verbally prime algebras". Linear and Multilinear Algebra 59, nr 12 (grudzień 2011): 1433–50. http://dx.doi.org/10.1080/03081087.2011.559636.

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47

Goodearl, K. R., i J. J. Zhang. "Noetherian Hopf algebra domains of Gelfand–Kirillov dimension two". Journal of Algebra 324, nr 11 (grudzień 2010): 3131–68. http://dx.doi.org/10.1016/j.jalgebra.2009.11.001.

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48

GUPTA, ASHISH. "MODULES OVER QUANTUM LAURENT POLYNOMIALS". Journal of the Australian Mathematical Society 91, nr 3 (grudzień 2011): 323–41. http://dx.doi.org/10.1017/s1446788712000031.

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AbstractWe show that the Gelfand–Kirillov dimension for modules over quantum Laurent polynomials is additive with respect to tensor products over the base field. We determine the Brookes–Groves invariant associated with a tensor product of modules. We study strongly holonomic modules and show that there are nonholonomic simple modules.

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49

Preusser, Raimund. "The Gelfand–Kirillov dimension of a weighted Leavitt path algebra". Journal of Algebra and Its Applications 19, nr 03 (15.03.2019): 2050059. http://dx.doi.org/10.1142/s0219498820500590.

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We determine the Gelfand–Kirillov dimension of a weighted Leavitt path algebra [Formula: see text] where [Formula: see text] is a field and [Formula: see text] a row-finite weighted graph. Further we show that a finite-dimensional weighted Leavitt path algebra over [Formula: see text] is isomorphic to a finite product of matrix rings over [Formula: see text].

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50

Lenagan, T. H., i Agata Smoktunowicz. "An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension". Journal of the American Mathematical Society 20, nr 04 (1.10.2007): 989–1002. http://dx.doi.org/10.1090/s0894-0347-07-00565-6.

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