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It is almost two decades since the now classical books by McConnell and Robinson's

  • [ Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition. Graduate Studies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001 ],

and Krause and Lenagan's

  • [ Growth of algebras and Gelfand-Kirillov dimension. Revised edition. Graduate Studies in Mathematics, 22. American Mathematical Society, Providence, RI, 2000. ],

which are were (and still are in my opinion), the standard references on almost everything related to the Gelfand-Kirillov dimension, appeared.

Time has passed, and a lot of new work on this dimensional invariant has been done.

I am looking for references, surveys and pherhaps lecture notes on the Gelfand-Kirillov dimension which covers relevant developments regarding this invariant in the last 20 years.

Regarding its computational aspects, one has for instance

  • J. Bueso, J. Gomés-Torrecillas, A. Verschoren, [ Algorithmic methods in non-commutative algebra. Applications to quantum groups. Mathematical Modelling: Theory and Applications, 17. Kluwer Academic Publishers, Dordrecht, 2003 ],

but it does not cover all aspects of recent developments.

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2 Answers 2

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This list is certainly far from being complete, but it contains some important results obtained in the last 20 years.

The following thesis discusses some recent results obtained by Bell (see Section 5):

Michelle Roshan Marie Ashburner (2008). A Survey of the Classification of Division Algebras over Fields. Master Thesis, University of Waterloo

This is a survey on GK dimension of graded PI-algebras:

L. Centrone, On some recent results about the graded Gelfand-Kirillov dimension of graded PI-algebras, Serdica Math. J. 38(1-3) (2012), 43-68.

Centrone also wrote other papers where he proved some interesting results on GK dimension. For instance, he wrote:

L. Centrone, The graded Gelfand-Kirillov dimension of verbally prime algebras, Linear Multilinear Algebra 59(12) (2011), 1433-1450.

and

L. Centrone, A note on graded Gelfand-Kirillov dimension of graded algebras, J. Algebra Appl. 10(5) (2011), 865-889.

For some results on Hopf algebras with finite GK dimension, see:

Zhang, G. (2013). Hopf algebras of finite Gelfand-Kirillov dimension. PhD Thesis, University of Washington

To conclude, GK dimension has been recently extended to algebras over commutative domains by Zhang and Bell. Now, GK dimensions can be studied on many new structures. In the following paper, GK is studied for skew PBW extensions

Reyes, A.: Gelfand–Kirillov dimension of skew PBW extensions. Rev. Col. Mat. 47(1), 95–111 (2013)

while in this one it has been studied for rings:

Lezama, O., Venegas, H. Gelfand–Kirillov dimension for rings. São Paulo J. Math. Sci. 14, 207–222 (2020).

I'm not aware of any survey discussing all these new developments.

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  • $\begingroup$ This is indeed a very good list of references. Thank you very much! $\endgroup$
    – jg1896
    Commented Aug 2, 2020 at 20:35
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Complementing Manuel Norman's excelent answer, recently I've found a very nice survey about the Gelfand-Kirillov dimension, from 2015, by Jason Bell, called Growth Functions.

This survey discusses many important results involving the Gelfand-Kirillov dimension. Among the more recent ones we have:

  • Agata Smoktunowicz result (see paper here) that there are no graded domains of GK dimension strictly between $2$ and $3$. This was conjectured by Artin and Stafford, who obtained the weaker result that there are no graded domains of GK dimension strictly between $2$ and $\frac{11}{5}$. Smoktunowicz result is at the same time an extension of Bergman's Gap theorem and, in the context of noncommutative projective algebraic geometry, a proof of the (extremely desirable) fact that the are no noncommutative projective schemes with dimension between $2$ and $3$.

  • A result by Jason Bell (see the paper here) that shows that if a complex domain of quadratic growth is either PI (and hence close to being commutative) or primitive (and hence as far from being commutative as possible.

  • A result by Lenagan and Smoktunowicz (see the paper here) that shows that there exists algebras with finite Gelfand-Kirillov dimension for which Kurosh Problem has a negative solution.

I finish this answer with what I discovered to be a challanging open problem in the area: for left and right Noetherian domains, is the Gelfand-Kirillov necessarily an integer? There is a discussion of a similar problem in this MO post.

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