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Agata Smoktunowicz

Agata Smoktunowicz. University of Edinburgh, Edinburgh, Scotland, UK. Agata Smoktunowicz. Some results on algebras with finite Gelfand-Kirillov dimension. To Barbara and Carl. Outline. Some results on algebras with finite GK dimension.

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Agata Smoktunowicz

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  1. Agata Smoktunowicz University of Edinburgh, Edinburgh, Scotland, UK

  2. Agata Smoktunowicz Some results on algebras with finite Gelfand-Kirillov dimension.

  3. To Barbara and Carl

  4. Outline • Some results on algebras with finite GK dimension. • Non-commutative rings, definitions, nil rings, the Jacobson radical, basic results. • 3. Some recent results on growth of Jacobson radical algebras and nil algebras. • 4. Some open questions.

  5. Gelfand-Kirillov dimension Gelfand-Kirillov dimension is analogue of word growth for algebras.

  6. Gelfand-Kirilov dimension • Can take non-integer values. • .

  7. Gelfand-Kirilov dimension • Can take non-integer values. • Gelfand-Kirilov dimension 0-finite dimensional • algebras. The next possible value is 1.

  8. Gelfand-Kirilov dimension • Can take non-integer values. • Gelfand-Kirilov dimension 0-finite dimensional • algebras. The next possible value is 1. • Bergman 1978: No values between 1 and 2.

  9. Gelfand-Kirilov dimension • Can take non-integer values. • Gelfand-Kirilov dimension 0-finite dimensional • algebras. The next possible value is 1. • Bergman 1978: No values between 1 and 2. • Small, Stafford, Warfield 1985: algebras with GK dim 1 are close to being commutative.

  10. Gelfand-Kirillov dimension  Graded algebras correspond to projective geometry.

  11. Gelfand-Kirillov dimension • Graded domains correspond to projective geometry • Artin, Stafford 1995: Graded domains with GK dimension 2 related with automorphisms of elliptic curves described when primitive, PI, Noetherian

  12. Gelfand-Kirilov dimension • (Smoktunowicz 2006) Let K be a field. • then, there are no graded (by natural numbers) • domains with Gelfand-Kirillov dimension • strictly between two and three.

  13. Gelfand-Kirilov dimension • (Smoktunowicz 2006) Let K be a field. • then, there are no graded (by natural numbers) • domains with Gelfand-Kirillov dimension • strictly between two and three. • OPEN QUESTION : Is there a domain • with non-zero, finite but not integer Gelfand- • Kirillov dimension? • .

  14. Gelfand-Kirilov dimension • (Smoktunowicz 2006) Let K be a field. • then, there are no graded (by natural numbers) • domains with Gelfand-Kirillov dimension • strictly between two and three. • OPEN QUESTION : Is there a Noetherian • graded algebra with Gelfand-Kirillov dimension • strictly between two and three? • .

  15. Gelfand-Kirilov dimension • OPEN QUESTION : Is there a Noetherian • graded algebra with Gelfand-Kirillov dimension • strictly between two and three?

  16. Related result… • FAITH—UTUMI THEOREM • If a prime right Goldie ring has a right quotient • ring D[n] (n by n matrices over D), then there exists • a right Ore domain F with right quotient skew field D • such that contains a ring F[n] of all n by n matrices • over F. • In other words every prime Goldie ring R is sandwiched • between F[n] and D[n].

  17. Carl Faith with his wife Molly All the best, Carl and Molly!

  18. Let K be a field and let A be a prime Noetherian affine algebra with Gelfand-Kirillov dimension two. Does it follow that A is either primitive or PI? Lance Small’s question

  19. Small’s question • Let K be a field and let A be a prime Noetherian • affine algebra with Gelfand-Kirillov dimension • two. Does it follow that A is either primitive or PI? • True for graded algebras ( Artin-Stafford, 1996)

  20. Small’s question • Let K be a field and let A be a prime Noetherian • affine algebra with Gelfand-Kirillov dimension • two. Does it follow that A is either primitive or PI? • True for graded algebras ( Artin-Stafford, 1996) • If A not Noetherian not true (Bell 2005).

  21. Open question • Let K be a field and let A be a prime semiprimitive • affine algebra with Gelfand-Kirillov dimension • two. Does it follow that A is either primitive or PI?

  22. Open question • Let K be a field and let A be a prime semiprimitive • affine algebra with Gelfand-Kirillov dimension • two. Does it follow that A is either primitive or PI? • True for monomial algebras (Bell, Smoktunowicz, 2007).

  23. Related open question • (Bell) Let K be a field and let A be a prime Semiprimitiveaffine monomial algebra. Does it follow that A is either primitive or PI?

  24. Related result • (Beidar, Fong, 2000S) Let K be a field and let A be a • prime monomial algebra. Then the • Jacobson radical of A is locally nilpotent. • The Jacobson radical of a monomial affine prime • algebra need not be nilpotent as shown by Zelmanov.

  25. Some results on algebras with finite GK dimension • (Smith, Zhang ) If A is a finitely generated non-PI • algebra over a field K which is a domain of GK dimension • d and Z is the centre of the quotient division algebra of • A then the transcendence degree of Z is at most d-2.

  26. Some results on algebras with finite GK dimension • (Smith, Zhang ) If A is a finitely generated non-PI • algebra over a field K which is a domain of GK dimension • d and Z is the centre of the quotient division algebra of • A then the transcendence degree of Z is at most d-2. • (Bell, Smoktunowicz, 2007) If A is a finitely generated • non-PI algebra of GK -dimension d and without • locallly nilpotent ideals and Z is the extended • centre of A then the transcendence degree of Z is at most • GKDim(A)-2.

  27. Some results on algebras with finite GK dimension • (Bell, Smoktunowicz, 2007) • Let K be a field. Then there is a finitely generated • non-PI prime K-algebra of GK dimension two • whose extended centre has infinite transcendence • degree over K.

  28. Some results on algebras with finite GK dimension • (Bell, Smoktunowicz, 2007) • If A is a finitely generated prime non-PI algebra with • quadratic growth over a field K and Z is the extended • centre of A then Z is algebraic over K.

  29. Applications to graded algebras • (Bell, Smoktunowicz ) If A is an affine prime algebra with • quadratic growth and A is graded by natural numbers • and I is a prime ideal in A then either A/I is PI or I • is homogeneous.

  30. Applications to graded algebras • (Bell, Smoktunowicz ) If A is an affine prime algebra with • quadratic growth and A is graded by natural numbers • and I is a prime ideal in A then either A/I is PI or I • is homogeneous. • (Bell, Smoktunowicz, 2007) If A is a finitely generated • prime non-PI algebra of quadratic growth and A is graded • by then natural numbers then intersection of nonzero • prime ideals P in A such that A/P has GK -dimension 2 is • non-empty provided that there is at least one such ideal.

  31. Some results on monomial algebras • (Bell 2007 ) • Let K be a field and let A be prime monomial K-algebra • with quadratic growth. Then the set of primes P such that • GKdim(A/P)=1 is finite, moreover all such primes are • monomial ideals. In particular A has bounded matrix • images .

  32. NIL ALGEBRAS  Let K be a field. A k-algebra A is a ring which is a vector space over k.

  33. NIL ALGEBRAS •  Let K be a field. A k-algebra A is a ring • which is a vector space over k. • An algebra is nil if each element is nilpotent. Element r is nilpotent if some power of r is zero.

  34. NIL ALGEBRAS •  Let K be a field. A k-algebra A is a ring • which is a vector space over k. • An algebra is nil if each element is nilpotent. Element r is nilpotent if some power of r is zero. •  An algebra is nilpotent if there is an n such that • any product of n elements is equal to zero.

  35. NIL ALGEBRAS •  Let K be a field. A k-algebra A is a ring • which is a vector space over k. • An algebra is nil if each element is nilpotent. Element r is nilpotent if some power of r is zero. •  An algebra is nilpotent if there is an n such that • any product of n elements is equal to zero. •  Obviously, nilpotent implies nil.

  36. MORE DEFINITIONS A more appropriate definition in many situations is the following:

  37. MORE DEFINITIONS A more appropriate definition in many situations is the following: An algebra R is locally nilpotentif each finitely generated sub algebra of R is nilpotent.

  38. MORE DEFINITIONS A more appropriate definition in many situations is the following: An algebra R is locally nilpotentif each finitely generated sub algebra of R is nilpotent.  An algebra R is Jacobson radicalif each r in R there is s in R such that r+s+rs =0.

  39. KEY QUESTION Why nil rings are important?

  40. Wedderburn’s radical • In developing the theory of finite-dimensional algebras over a field, Wedderburn defined for every such algebra A an ideal, Rad(A), which is the largest nilpotent ideal of A, i.e. The sum of all nilpotent ideals of A. Joseph Wedderburn

  41. Wedderburn’s radical • A finitely dimensional algebra is semisimple if its radical is zero. Joseph Wedderburn

  42. Wedderburn’s radical • A finitely dimensional algebra is semisimple if its radical is zero. • Such an algebra is the direct product of matrix algebras over division algebras. Joseph Wedderburn

  43. Radicals The problem of finding the appropriate generalization of Wedderburn’s radical for arbitrary rings remained unsolved for almost forty years.

  44. Radicals The problem of finding the appropriate generalization of Wedderburn’s radical for arbitrary rings remained unsolved for almost forty years. Finally, in a fundamental paper in 1945, N.Jacobson initiated the general notion of the radical of an arbitrary ring R.

  45. Radicals The problem of finding the appropriate generalization of Wedderburn’s radical for arbitrary rings remained unsolved for almost forty years. Finally, in a fundamental paper in 1945, N.Jacobson initiated the general notion of the radical of an arbitrary ring R. By definition, the Jacobson radical of R is the intersection of the maximal left (or right) ideals of R.

  46. Jacobson radical • Finally, in a fundamental paper in 1945, N.Jacobson initiated the general notion of the radical of an arbitrary ring R. Nathan Jacobson

  47. TYPES OF IDEALS Nil radical is the largest nil ideal in ring R.

  48. TYPES OF IDEALS Nil radical is the largest nil ideal in ring R. Nil radical is always Jacobson radical.

  49. TYPES OF IDEALS Nil radical is the largest nil ideal in ring R. Nil radical is always Jacobson radical. On the other hand a Jacobson radical of finitely generated algebra over an uncountable field is always nil, as showed by Amitsur in 1973.

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