Introduction and Motivations Group Identities for U + ( FG ) Group Identities for Un ( FG ) Group identities for unitary units of group rings Ernesto Spinelli Universit` a di Roma “La Sapienza” Dipartimento di Matematica “G. Castelnuovo” Joint work with Greg Lee and Sudarshan Sehgal Spa, June 18-24, 2017 Groups, Rings and the Yang-Baxter equation
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Group identities Let � x 1 , x 2 , . . . � be the free group on a countable infinitude of generators. Definition A subset S of a group G satisfies a group identity (and write S is GI) if there exists a non-trivial reduced word w ( x 1 , . . . , x n ) ∈ � x 1 , x 2 , . . . � such that w ( g 1 , . . . , g n ) = 1 for all g i ∈ S .
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Examples Let us define ( x 1 , x 2 ) := x − 1 1 x − 1 2 x 1 x 2 and recursively ( x 1 , . . . , x n + 1 ) := (( x 1 , . . . , x n ) , x n + 1 ) .
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Examples Let us define ( x 1 , x 2 ) := x − 1 1 x − 1 2 x 1 x 2 and recursively ( x 1 , . . . , x n + 1 ) := (( x 1 , . . . , x n ) , x n + 1 ) . A group G is abelian if it satisfies ( x 1 , x 2 ) = 1
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Examples Let us define ( x 1 , x 2 ) := x − 1 1 x − 1 2 x 1 x 2 and recursively ( x 1 , . . . , x n + 1 ) := (( x 1 , . . . , x n ) , x n + 1 ) . A group G is abelian if it satisfies ( x 1 , x 2 ) = 1 G is nilpotent if it satisfies ( x 1 , x 2 , . . . , x n ) = 1 for some n ≥ 2
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Examples Let us define ( x 1 , x 2 ) := x − 1 1 x − 1 2 x 1 x 2 and recursively ( x 1 , . . . , x n + 1 ) := (( x 1 , . . . , x n ) , x n + 1 ) . A group G is abelian if it satisfies ( x 1 , x 2 ) = 1 G is nilpotent if it satisfies ( x 1 , x 2 , . . . , x n ) = 1 for some n ≥ 2 G is bounded Engel if it satisfies ( x 1 , x 2 , . . . , x 2 ) = 1 for � �� � n some n ≥ 1
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Hartley’s Conjecture Hartley’s Conjecture Let F be a field and G a torsion group. U ( FG ) is GI = ⇒ FG is PI
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Hartley’s Conjecture Hartley’s Conjecture Let F be a field and G a torsion group. U ( FG ) is GI = ⇒ FG is PI Let F � X � be the free associative algebra generated by a countable set X := { x 1 , x 2 , . . . } over F . Definition A subset S of an F -algebra A is said to satisfy a polynomial identity (and write S is PI) if there exists 0 � = f ( x 1 , . . . , x n ) ∈ F � X � such that f ( a 1 , . . . , a n ) = 0 for all a i ∈ S .
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Solution of the Conjecture Solution of Hartley’s Conjecture Giambruno-Jespers-Valenti (1994): char F = 0 or F infinite, char F = p ≥ 2 and P = 1
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Solution of the Conjecture Solution of Hartley’s Conjecture Giambruno-Jespers-Valenti (1994): char F = 0 or F infinite, char F = p ≥ 2 and P = 1 Giambruno-Sehgal-Valenti (1997): F infinite
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Solution of the Conjecture Solution of Hartley’s Conjecture Giambruno-Jespers-Valenti (1994): char F = 0 or F infinite, char F = p ≥ 2 and P = 1 Giambruno-Sehgal-Valenti (1997): F infinite Liu (1999): F finite
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Solution of the Conjecture Solution of Hartley’s Conjecture Giambruno-Jespers-Valenti (1994): char F = 0 or F infinite, char F = p ≥ 2 and P = 1 Giambruno-Sehgal-Valenti (1997): F infinite Liu (1999): F finite Characterization of when U ( FG ) is GI Passman (1997): F infinite and G torsion
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Solution of the Conjecture Solution of Hartley’s Conjecture Giambruno-Jespers-Valenti (1994): char F = 0 or F infinite, char F = p ≥ 2 and P = 1 Giambruno-Sehgal-Valenti (1997): F infinite Liu (1999): F finite Characterization of when U ( FG ) is GI Passman (1997): F infinite and G torsion Liu-Passman (1999): F finite and G torsion
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Solution of the Conjecture Solution of Hartley’s Conjecture Giambruno-Jespers-Valenti (1994): char F = 0 or F infinite, char F = p ≥ 2 and P = 1 Giambruno-Sehgal-Valenti (1997): F infinite Liu (1999): F finite Characterization of when U ( FG ) is GI Passman (1997): F infinite and G torsion Liu-Passman (1999): F finite and G torsion Giambruno-Sehgal-Valenti (2000): G non-torsion
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Classical and F -linear involutions Let G be a group endowed with an involution ⋆ . Let us consider the F -linear extension of ⋆ to FG setting � � � ⋆ � a g g ⋆ . a g g := g ∈ G g ∈ G This extension, which we denote again by ⋆ , is an involution of FG wich fixes the ground field F elementwise.
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Classical and F -linear involutions Let G be a group endowed with an involution ⋆ . Let us consider the F -linear extension of ⋆ to FG setting � � � ⋆ � a g g ⋆ . a g g := g ∈ G g ∈ G This extension, which we denote again by ⋆ , is an involution of FG wich fixes the ground field F elementwise. As is well-known, any group G has a natural involution which is given by the map ∗ : g �→ g − 1 . Definition Let FG be the group algebra of a group G over a field F . If G is endowed with an involution ⋆ , its linear extension to the group algebra FG is called a F-linear involution of FG . In particular, if ⋆ = ∗ the induced involution is called the classical involution .
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Symmetric and unitary units Let us consider U + ( FG ) := { x | x ∈ U ( FG ) x = x ⋆ } , xx ⋆ = x ⋆ x = 1 } . Un ( FG ) := { x | x ∈ FG Un ( FG ) is a subgroup of U ( FG ) , whereas U + ( FG ) is a subset of U ( FG ) . Definition Let FG be the group algebra of a group G over a field F endowed with an F -linear involution. The elements of U + ( FG ) are called the symmetric units of FG (with respect to ⋆ ) and those of Un ( FG ) are called the unitary units of FG .
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Symmetric and unitary units Let us consider U + ( FG ) := { x | x ∈ U ( FG ) x = x ⋆ } , xx ⋆ = x ⋆ x = 1 } . Un ( FG ) := { x | x ∈ FG Un ( FG ) is a subgroup of U ( FG ) , whereas U + ( FG ) is a subset of U ( FG ) . Definition Let FG be the group algebra of a group G over a field F endowed with an F -linear involution. The elements of U + ( FG ) are called the symmetric units of FG (with respect to ⋆ ) and those of Un ( FG ) are called the unitary units of FG .
Introduction and Motivations Group Identities Group Identities for U + ( FG ) Hartley’s Conjecture Group Identities for Un ( FG ) Developments arising: Involutions and Related Problems Constraints on subsets of U ( FG ) We want to determine if we can decide the structure of G by imposing constraints upon subsets of the unit group U ( FG ) .
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