The original transcendence degree for (noncommutative) division algebras is the Gelfand-Kirillov transcendence degree, due to I. Gelfand and K. Kirillov ([ Sur les corps li´es aux algèbres envoloppantes des algèbres de Lie, Inst. Hautes Etudes Sci. Publ. Mat. 31 (1966), 5-19. ]) - in fact, it was introduced simultaneously with the now famous Gelfand-Kirillov dimension.
To the best of my knowledge, this is the most studied version of transcendence degrees for division algebras, considered by Borho and Kraft (1976), Lorenz (1984), Zhang (1996). Many other versions of transcendence degree for division algebras were also considered by Resco, Rosenberg, Schoenfield, Stafford, etc.
In 1998, J. J. Zhang introduced the lower transcendence degree [ On Lower Transcendence Degree. Adv. Math. 139 (1998), no. 2, 157-193. ]. It is conjecturally equal to the original Gelfand-Kirillov transcendence degree, but has better theoretical properties, and is connected with important open problems in ring theory, and also seems suitable as an invariant in noncommutative projective geometry. For a certain wide class of algebras (Zhang calls them LD-stable) it is easely computable: it is just the $GK$ dimension.
Another important notion of transcendence degree is the homological transcendence degree due to A. Yekutieli and J. J. Zhang [ Homological transcendence degree. Proc. London Math. Soc. (3) 93 (2006), no. 1, 105-137. ], motivated by homological considerations discussed by previous authors. This transcendence degree is both theoretically well behaved and at the same time is easely computable for algebras with good homological properties (such as AS regular graded algebras).
Since, so far, there is no uniform completely satisfactory notion of transcendence degree of division algebras, just these potential candidates, what are the pros and cons, and fruitful ramifications in mathematics, of the most used transcendence degrees in the noncommutative setting?
My question is mostly about
- Gelfand-Kirillov transcendence degree
- Lower transcendence degree
- Homological transcendence degree
but any relevant notion of noncommutative transcendence degree is welcome.