Lie solvability in matrix algebras Micha� l Ziembowski Warsaw University of Technology Noncommutative and non-associative structures, braces and applications Malta, March 12, 2018 Based on a joint works with J. van den Berg, J. Szigeti and L. van Wyk M.Z. Lie solvability in matrix algebras
Problem: Give an example of a subalgebra of M n ( F ) which is commutative. M.Z. Lie solvability in matrix algebras
Problem: Give an example of a subalgebra of M n ( F ) which is commutative. Take any pair ( k 1 , k 2 ) of positive integers satisfying k 1 + k 2 = n and consider k 1 J = k 2 M.Z. Lie solvability in matrix algebras
Problem: Give an example of a subalgebra of M n ( F ) which is commutative. Take any pair ( k 1 , k 2 ) of positive integers satisfying k 1 + k 2 = n and consider k 1 J = k 2 R = FI n + J M.Z. Lie solvability in matrix algebras
R = FI n + J M.Z. Lie solvability in matrix algebras
R = FI n + J dim F ( R ) = 1 + k 1 k 2 M.Z. Lie solvability in matrix algebras
R = FI n + J dim F ( R ) = 1 + k 1 k 2 (Schur 1905, Jacobson 1944) The dimension over a field F of � � n 2 any commutative subalgebra of M n ( F ) is at most + 1, 4 where ⌊ ⌋ is the floor function. M.Z. Lie solvability in matrix algebras
R = FI n + J dim F ( R ) = 1 + k 1 k 2 (Schur 1905, Jacobson 1944) The dimension over a field F of � � n 2 any commutative subalgebra of M n ( F ) is at most + 1, 4 where ⌊ ⌋ is the floor function. Commutativity: ∀ r , s ∈ R , [ r , s ] def = rs − sr . M.Z. Lie solvability in matrix algebras
Define inductively the Lie central and Lie derived series of a ring R as follows: C 0 ( R ) := R , C q +1 ( R ) := [ C q ( R ) , R ] (central series) , (1) and D 0 ( R ) := R , D q +1 ( R ) := [ D q ( R ) , D q ( R )] (derived series) . (2) We say that R is Lie nilpotent (respectively, Lie solvable) of index q (for short, R is Ln q ; respectively, R is Ls q ) if C q ( R ) = 0 (respectively, D q ( R ) = 0). M.Z. Lie solvability in matrix algebras
Let k 1 , k 2 , . . . , k m +1 be a sequence of positive integers such that k 1 + k 2 + · · · + k m +1 = n . M.Z. Lie solvability in matrix algebras
Let k 1 , k 2 , . . . , k m +1 be a sequence of positive integers such that k 1 + k 2 + · · · + k m +1 = n . Let k 1 k 2 · · · · · · · J = · · · · · · · · k m k m +1 M.Z. Lie solvability in matrix algebras
Let k 1 , k 2 , . . . , k m +1 be a sequence of positive integers such that k 1 + k 2 + · · · + k m +1 = n . Let k 1 k 2 · · · · · · · J = · · · · · · · · k m k m +1 Let R = FI n + J (“TYPICAL EXAMPLE”) M.Z. Lie solvability in matrix algebras
k 1 k 2 · · · · · · · J = · · · · · · · · k m k m +1 M.Z. Lie solvability in matrix algebras
k 1 k 2 · · · · · · · J = · · · · · · · · k m k m +1 dim F R = k 1 ( n − k 1 ) + k 2 ( n − k 1 − k 2 ) + · · · + k m ( n − k 1 − k 2 − · · · − k m ) + 1 = � m +1 i , j =1 , i < j k i k j + 1 . M.Z. Lie solvability in matrix algebras
ℓ M ( ℓ, n ) def � = max k i k j + 1 : k 1 , k 2 , . . . , k ℓ are i , j =1 , i < j ℓ � � nonnegative integers such that k i = n . i =1 M.Z. Lie solvability in matrix algebras
ℓ M ( ℓ, n ) def � = max k i k j + 1 : k 1 , k 2 , . . . , k ℓ are i , j =1 , i < j ℓ � � nonnegative integers such that k i = n . i =1 If ℓ and n are positive integers with ℓ > n , then n 2 − n M ( ℓ, n ) = 1 � � + 1. 2 M.Z. Lie solvability in matrix algebras
ℓ M ( ℓ, n ) def � = max k i k j + 1 : k 1 , k 2 , . . . , k ℓ are i , j =1 , i < j ℓ � � nonnegative integers such that k i = n . i =1 If ℓ and n are positive integers with ℓ > n , then n 2 − n M ( ℓ, n ) = 1 � � + 1. 2 Let ℓ � n and � n � n = ℓ + r . ℓ M.Z. Lie solvability in matrix algebras
ℓ M ( ℓ, n ) def � = max k i k j + 1 : k 1 , k 2 , . . . , k ℓ are i , j =1 , i < j ℓ � � nonnegative integers such that k i = n . i =1 If ℓ and n are positive integers with ℓ > n , then n 2 − n M ( ℓ, n ) = 1 � � + 1. 2 Let ℓ � n and � n � n = ℓ + r . ℓ We get M ( ℓ, n ) for the sequence ( k 1 , k 2 , . . . , k ℓ ) ∈ N ℓ 0 defined in the following way: � n � , for 1 � i � ℓ − r ℓ def k i = � n � + 1 , for ℓ − r < i � ℓ. ℓ M.Z. Lie solvability in matrix algebras
Conjecture. (J. Szigeti, L. van Wyk) Let F be any field, m and n positive integers, and R an F -subalgebra of M n ( F ) with Lie nilpotence index m . Then dim F R � M ( m + 1 , n ) . M.Z. Lie solvability in matrix algebras
Theorem 1 Let F be any field, m and n positive integers, and R an F -subalgebra of M n ( F ) with Lie nilpotence index m . Then dim F R � M ( m + 1 , n ) . M.Z. Lie solvability in matrix algebras
PROBLEM: Every ring R that is Lie nilpotent of index m , is also Lie solvable of index m . Thus, it is natural to ask about the maximal dimension of Lie solvable of index m subalgebras of M n ( F ). M.Z. Lie solvability in matrix algebras
PROBLEM: Every ring R that is Lie nilpotent of index m , is also Lie solvable of index m . Thus, it is natural to ask about the maximal dimension of Lie solvable of index m subalgebras of M n ( F ). Unfortunately, we do not have good “typical example”. M.Z. Lie solvability in matrix algebras
FACTS: [ x 1 , y 1 ] [ x 2 , y 2 ] · · · [ x q , y q ] = 0 (3) Mal’tsev proved that all the polynomial identities of U q ( F ) are consequences of the identity in (3). We denote algebras satisfying (3) by D 2 q . M.Z. Lie solvability in matrix algebras
FACTS: [ x 1 , y 1 ] [ x 2 , y 2 ] · · · [ x q , y q ] = 0 (3) Mal’tsev proved that all the polynomial identities of U q ( F ) are consequences of the identity in (3). We denote algebras satisfying (3) by D 2 q . D 1 D (1 , 2) · · · D (1 , q ) . ... . 0 D 2 . D = (4) . ... ... . . D ( q − 1 , q ) 0 · · · 0 D q Each D i is a commutative F -subalgebra of M n i ( F ) for every i , and D ( j , k ) = M n j × n k ( F ) for all j and k such that 1 ≤ j < k ≤ q . D satisfies (3). M.Z. Lie solvability in matrix algebras
FACTS: In general Ln 2 �⇒ D 2 , D 2 �⇒ Ln 2 (5) M.Z. Lie solvability in matrix algebras
FACTS: In general Ln 2 �⇒ D 2 , D 2 �⇒ Ln 2 (5) The maximum dimension for D 2 F -subalgebra of M n ( F ) is � � 3 n 2 2 + . 8 M.Z. Lie solvability in matrix algebras
FACTS: In general Ln 2 �⇒ D 2 , D 2 �⇒ Ln 2 (5) The maximum dimension for D 2 F -subalgebra of M n ( F ) is � � 3 n 2 2 + . 8 The maximum dimension for Ln 2 F -subalgebra of M n ( F ) is � � n 2 1 + . 3 M.Z. Lie solvability in matrix algebras
FACTS: In general Ln 2 �⇒ D 2 , D 2 �⇒ Ln 2 (5) The maximum dimension for D 2 F -subalgebra of M n ( F ) is � � 3 n 2 2 + . 8 The maximum dimension for Ln 2 F -subalgebra of M n ( F ) is � � n 2 1 + . 3 Also, in general D 2 m , Ln m +1 ⇒ Ls m +1 (6) M.Z. Lie solvability in matrix algebras
FACTS: In general Ln 2 �⇒ D 2 , D 2 �⇒ Ln 2 (5) The maximum dimension for D 2 F -subalgebra of M n ( F ) is � � 3 n 2 2 + . 8 The maximum dimension for Ln 2 F -subalgebra of M n ( F ) is � � n 2 1 + . 3 Also, in general D 2 m , Ln m +1 ⇒ Ls m +1 (6) (Meyer, Szigeti, van Wyk) For any commutative ring R , the subring U ⋆ 3 ( U ⋆ 3 ( R )) of U ⋆ 9 ( R ) is Ls 2 , but it is neither Ln 2 nor D 2 , and so we have, in general, Ls 2 �⇒ Ln 2 or D 2 . (7) M.Z. Lie solvability in matrix algebras
Problem 2 Construct an example of Ls 2 subalgebra of M n ( F ) with dimension � � 3 n 2 bigger than 2 + . 8 Theorem 3 If A is an Ls m +1 (for some m ≥ 1 ) structural matrix subring of U n ( R ) , R a commutative ring and n ≥ 1 , then A is D 2 m . M.Z. Lie solvability in matrix algebras
Let k be a positive integer and n = 2 k + 1. Consider �� A 1 B � � A = : A 1 ∈ M k ( F ) , A 2 ∈ M k +1 ( F ) , A i − comm. . 0 A 2 M.Z. Lie solvability in matrix algebras
Let k be a positive integer and n = 2 k + 1. Consider �� A 1 B � � A = : A 1 ∈ M k ( F ) , A 2 ∈ M k +1 ( F ) , A i − comm. . 0 A 2 Theorem 4 If A is a D 2 subalgebra of U n ( F ) with maximum possible dimension for D 2 , such that A 1 , A 2 and B are independent, then A 1 and A 2 are commutative. M.Z. Lie solvability in matrix algebras
Thank you for your attention! M.Z. Lie solvability in matrix algebras
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