On homomorphisms of crossed products of locally indicable groups to - PowerPoint PPT Presentation

On homomorphisms of crossed products of locally indicable groups to division rings Andrei Jaikin-Zapirain UAM and ICMAT Spa, June 2019 Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings A commutative domain R can be embedded in a field → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . R ֒ Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR � = { 0 } if a , b � = 0. → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . The Ore condition ⇒ R ֒ A Noetherian domain is embedded into a division ring. A. I. Malcev (1937): There exists a noncomutative domain that cannot be embedded in a division ring. A. J. Bowtell’s example(1967): Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings A commutative domain R can be embedded in a field → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . R ֒ Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR � = { 0 } if a , b � = 0. → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . The Ore condition ⇒ R ֒ A Noetherian domain is embedded into a division ring. A. I. Malcev (1937): There exists a noncomutative domain that cannot be embedded in a division ring. A. J. Bowtell’s example(1967): Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings A commutative domain R can be embedded in a field → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . R ֒ Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR � = { 0 } if a , b � = 0. → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . The Ore condition ⇒ R ֒ A Noetherian domain is embedded into a division ring. A. I. Malcev (1937): There exists a noncomutative domain that cannot be embedded in a division ring. A. J. Bowtell’s example(1967): Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings A commutative domain R can be embedded in a field → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . R ֒ Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR � = { 0 } if a , b � = 0. → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . The Ore condition ⇒ R ֒ A Noetherian domain is embedded into a division ring. A. I. Malcev (1937): There exists a noncomutative domain that cannot be embedded in a division ring. A. J. Bowtell’s example(1967): Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings A commutative domain R can be embedded in a field → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . R ֒ Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR � = { 0 } if a , b � = 0. → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . The Ore condition ⇒ R ֒ A Noetherian domain is embedded into a division ring. A. I. Malcev (1937): There exists a noncomutative domain that cannot be embedded in a division ring. A. J. Bowtell’s example(1967): Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings A commutative domain R can be embedded in a field → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . R ֒ Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR � = { 0 } if a , b � = 0. → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . The Ore condition ⇒ R ֒ A Noetherian domain is embedded into a division ring. A. I. Malcev (1937): There exists a noncomutative domain that cannot be embedded in a division ring. A. J. Bowtell’s example(1967): Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings A commutative domain R can be embedded in a field → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . R ֒ Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR � = { 0 } if a , b � = 0. → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . The Ore condition ⇒ R ֒ A Noetherian domain is embedded into a division ring. A. I. Malcev (1937): There exists a noncomutative domain that cannot be embedded in a division ring. A. J. Bowtell’s example(1967): Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings A commutative domain R can be embedded in a field → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . R ֒ Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR � = { 0 } if a , b � = 0. → Q ( R ) = { ab − 1 : a ∈ R , 0 � = b ∈ R } . The Ore condition ⇒ R ֒ A Noetherian domain is embedded into a division ring. A. I. Malcev (1937): There exists a noncomutative domain that cannot be embedded in a division ring. A. J. Bowtell’s example(1967): Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings Can we embed a group ring K [ G ] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K [ G ] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K [ G ] and E ∗ G are domains. G. Higman (1940): if G is left-orderable, then K [ G ] (and E ∗ G ) are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K [ G ] (and E ∗ G ) can be embedded in a division ring. Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings Can we embed a group ring K [ G ] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K [ G ] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K [ G ] and E ∗ G are domains. G. Higman (1940): if G is left-orderable, then K [ G ] (and E ∗ G ) are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K [ G ] (and E ∗ G ) can be embedded in a division ring. Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings Can we embed a group ring K [ G ] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K [ G ] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K [ G ] and E ∗ G are domains. G. Higman (1940): if G is left-orderable, then K [ G ] (and E ∗ G ) are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K [ G ] (and E ∗ G ) can be embedded in a division ring. Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings Can we embed a group ring K [ G ] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K [ G ] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K [ G ] and E ∗ G are domains. G. Higman (1940): if G is left-orderable, then K [ G ] (and E ∗ G ) are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K [ G ] (and E ∗ G ) can be embedded in a division ring. Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings Can we embed a group ring K [ G ] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K [ G ] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K [ G ] and E ∗ G are domains. G. Higman (1940): if G is left-orderable, then K [ G ] (and E ∗ G ) are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K [ G ] (and E ∗ G ) can be embedded in a division ring. Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings Can we embed a group ring K [ G ] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K [ G ] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K [ G ] and E ∗ G are domains. G. Higman (1940): if G is left-orderable, then K [ G ] (and E ∗ G ) are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K [ G ] (and E ∗ G ) can be embedded in a division ring. Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings