LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZATIONS Yu. A. Mikhalchishina «Knots, braids and automorphism groups» NOVOSIBIRSK – 2014 Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
The braid group. The Artin representation Recall the braid group B n , n ≥ 2, on n strands is defined by generators σ 1 , σ 2 , ..., σ n − 1 and the defining relations = for i = 1 , 2 , ..., n − 2 , σ i σ i + 1 σ i σ i + 1 σ i σ i + 1 = for | i − j | ≥ 2 . σ i σ j σ j σ i The group B n embeds in the automorphism group Aut ( F n ) of the free group F n = < x 1 , ..., x n > . Here the generator σ i , i = 1 , 2 , ..., n − 1, defines the automorphism σ i : B n → Aut ( F n ), x i → x i x i + 1 x − 1 , i σ i : x i + 1 → x i , x j → x j , j � = i , i + 1 . This representation is referred to as the Artin representation . Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
The Burau representation Using the Magnus approach and Fox derivatives from the Artin representation the Burau one is constructed. ρ B : B n → GL n ( Z [ t ± 1 ]) . I i − 1 0 0 0 0 1 − t t 0 ρ B ( σ i ) = , i = 1 , 2 , ..., n − 1 . 0 1 0 0 0 0 0 I n − i − 1 Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
Local representations of B n . Local homogeneous representations of B n Recall a representation ϕ : B n → GL n ( C ) is referred to as local if I i − 1 0 0 0 I i − 1 0 0 0 * * 0 0 0 ϕ ( σ i ) = = R i , 0 * * 0 0 0 I n − i − 1 0 0 0 I n − i − 1 i = 1 , 2 , ..., n − 1, where I m is the identity matrix of order m and R i is a matrix of order 2. A local representation is referred to as homogeneous if R 1 = R 2 = ... = R n − 1 . Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
Local representations of B 3 Theorem 1. Provided a local representation ϕ : B 3 → GL 3 ( C ) , ϕ coincides with one of the the two types of representations: ( 1 − d )( 1 − α + d α ) α ( 1 − d ) 0 c 1) ϕ ( σ 1 ) = c d 0 , 0 0 1 1 0 0 ( 1 − α )( 1 − d + d α ) ϕ ( σ 2 ) = 0 , where d , α � = 1 , c , γ � = 0; α γ 0 d ( 1 − α ) γ 1 0 0 0 b 0 bc 2) ϕ ( σ 1 ) = c 0 0 , ϕ ( σ 2 ) = 0 0 , where bc , γ � = 0. γ 0 0 1 0 0 γ Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
Local homogeneous representations of B n Theorem 2. Given ϕ : B n → GL n ( C ) a local homogeneous representation, ϕ coincides with one of the representations ϕ 1 , ϕ 2 , ϕ 3 defined as follows: ϕ j : B n → GL n ( C ) . 0 0 0 I i − 1 1 − α 0 0 α 1 ) ϕ 1 ( σ i ) = , γ � = 0 , i = 1 , 2 , ..., n − 1 ; γ 0 0 0 γ 0 0 0 I n − i − 1 I i − 1 0 0 0 1 − d 0 0 0 2 ) ϕ 2 ( σ i ) = c , c � = 0 , i = 1 , 2 , ..., n − 1 ; 0 c d 0 0 0 0 I n − i − 1 I i − 1 0 0 0 0 0 b 0 3 ) ϕ 3 ( σ i ) = , bc � = 0 , i = 1 , 2 , ..., n − 1 . 0 c 0 0 0 0 0 I n − i − 1 Note if γ = 1 , α = 1 − t in ϕ 1 we obtain the Burau representation. Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
Wada Representations Wada constructed four new local representations of B n in Aut ( F n ) defined as follows: x i → x k i x i + 1 x − k , i w ( k ) 1 ( σ i ) : x i + 1 → x i , x j → x j , j � = i , i + 1 . Note given k = 1, this is the Artin representation. x i → x i x − 1 i + 1 x i , w 2 ( σ i ) : x i + 1 → x i , x j → x j , j � = i , i + 1 . x i → x i x i + 1 x i , x i + 1 → x − 1 w 3 ( σ i ) : , i x j → x j , j � = i , i + 1 . x i → x 2 i x i + 1 , x i + 1 → x − 1 i + 1 x − 1 w 4 ( σ i ) : x i + 1 , i x j → x j , j � = i , i + 1 . Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
Linear representations corresponding to Wada representations Analogously as the Burau representation is constructed from the Artin’s one we use the Magnus approach to construct linear representations corresponding to Wada representations. w ( k ) 1 , w 2 , w 3 , w 4 : B n → Aut ( F n ) ρ ( k ) 1 , ρ 2 , ρ 3 , ρ 4 : B n → GL n ( Z [ t ± 1 1 , t ± 1 2 , ..., t ± 1 n ]) Obtained representations are as follows: I i − 1 0 0 0 1 − t k t k 0 0 ρ ( k ) 1 ( σ i ) = , i = 1 , 2 , ..., n − 1 . 0 1 0 0 0 0 0 I n − i − 1 Note given q = t k , we obtain the Burau representation. Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
Linear representations corresponding to Wada representations 0 0 0 I i − 1 0 1 − t t 0 ρ 2 ( σ i ) = , i = 1 , 2 , ..., n − 1 . 0 1 0 0 0 0 0 I n − i − 1 I i − 1 0 0 0 0 2 t i 0 ρ 3 ( σ i ) = , i = 1 , 2 , ..., n − 1 . − t − 1 0 0 0 i 0 0 0 I n − i − 1 I i − 1 0 0 0 0 2 1 0 ρ 4 ( σ i ) = , i = 1 , 2 , ..., n − 1 . 0 -1 0 0 0 0 0 I n − i − 1 Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
Local nonhomogeneous representations of B n Using the Theorem 1 result there was constructed the extensions of the local representation of B 3 to B n . The most interesting case looks as follows I i − 1 0 0 0 1 − α 0 0 α ϕ ( σ i ) = , γ i � = 0 , i = 1 , 2 , ..., n − 1 . γ i 0 0 0 γ i 0 0 0 I n − i − 1 Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
Linear local representations of B n Theorem. All linear local representations of B n are equivalent to the Burau one in some sense. In particular all linear local homogeneous representations of B n are equivalent to the Burau representations. So there does not exist a faithful local representation of B n to GL n ( C ) , n ≥ 5. Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
Generalizations of the braid group Recently the generalizations of braid group are studied such as the virtual braid group VB n , the welded braid group WB n and the singular braid group SB n . There exist the extensions of the Burau representation to these generalizations. Using the Theorem 2 results there were constructed another linear local representations of the virtual braid group VB n , the welded braid group WB n and the singular braid group SB n . Yu. A. Mikhalchishina LOCAL REPRESENTATIONS OF BRAID GROUP AND ITS GENERALIZA
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