Higher braid groups and regular semigroups from polyadic-binary correspondence
aa r X i v : . [ m a t h . G R ] F e b H IGHER BRAID GROUPS AND REGULAR SEMIGROUPS FROMPOLYADIC - BINARY CORRESPONDENCE S TEVEN D UPLIJ
Center for Information Technology (WWU IT), Universit¨at M¨unster, R¨ontgenstrasse 7-13D-48149 M¨unster, Germany A BSTRACT . In this note we first consider a ternary matrix group related to the von Neumann regularsemigroups and to the Artin braid group (in an algebraic way). The product of a special kind ofternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groupswith their binary multiplication of components. We then generalize the construction to the higher aritycase, which allows us to obtain some higher degree versions (in our sense) of the regular semigroupsand braid groups. The latter are connected with the generalized polyadic braid equation and R -matrixintroduced by the author, which differ from any version of the well-known tetrahedron equation andhigher-dimensional analogs of the Yang-Baxter equation, n -simplex equations. The higher degree(in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these areconnected only in the non-higher case. C ONTENTS
1. I
NTRODUCTION
22. P
RELIMINARIES
23. T
ERNARY MATRIX GROUP CORRESPONDING TO THE REGULAR SEMIGROUP
34. P
OLYADIC MATRIX SEMIGROUP CORRESPONDING TO THE HIGHER REGULARSEMIGROUP
45. T
ERNARY MATRIX GROUP CORRESPONDING TO THE BRAID GROUP
76. T
ERNARY MATRIX GENERATORS
87. G
ENERATED k - ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP
EFERENCES E-mail address : [email protected]; [email protected]; https://ivv5hpp.uni-muenster.de/u/douplii . Date : of start January 8, 2021.
Date : of completion February 7, 2021.
Total : 27 references.2010
Mathematics Subject Classification.
Key words and phrases. regular semigroup, braid group, generator, relation, presentation, Coxeter group, symmetricgroup, polyadic matrix group, querelement, idempotence, finite order element. . I
NTRODUCTION
We begin by observing that the defining relations of the von Neumann regular semi-groups (e.g. G
RILLET [1995], H
OWIE [1976], P
ETRICH [1984]) and the Artin braid groupK
ASSEL AND T URAEV [2008], K
AUFFMAN [1991] correspond to such properties of ternary ma-trices (over the same set) as idempotence and the orders of elements (period). We then generalizethe correspondence thus introduced to the polyadic case and thereby obtain higher degree (in ourdefinition) analogs of the former. The higher (degree) regular semigroups obtained in this wayhave appeared previously in semisupermanifold theory D
UPLIJ [2000], and higher regular cate-gories in TQFT D
UPLIJ AND M ARCINEK [2002]. The representations of the higher braid relationsin vector spaces coincide with the higher braid equation and corresponding generalized R -matrixobtained in D UPLIJ [2018b], as do the ordinary braid group and the Yang-Baxter equation T
URAEV [1988]. The proposed constructions use polyadic group methods and differ from the tetrahedronequation Z
AMOLODCHIKOV [1981] and n -simplex equations H IETARINTA [1997] connected withthe braid group representations L
I AND H U [1995], H U [1997], also from higher braid groups ofM ANIN AND S CHECHTMAN [1989]. Finally, we define higher degree (in our sense) versions ofthe Coxeter group and the symmetric group and show that they are connected in the classical (i.e.non-higher) case only. 2. P
RELIMINARIES
There is a general observation N
IKITIN [1984], that a block-matrix, forming a semisimple p , k q -ring (Artinian ring with binary addition and k -ary multiplication) has the shape M p k ´ q ” M pp k ´ qˆp k ´ qq “ ¨˚˚˚˚˚˝ m p i ˆ i q . . .
00 0 m p i ˆ i q . . .
00 0 ... . . . ...... ... . . . m p i k ´ ˆ i k ´ q m p i k ´ ˆ i q . . . ˛‹‹‹‹‹‚ . ( )In other words, it is given by the cyclic shift p k ´ qˆp k ´ q matrix, in which identities are replacedby blocks of suitable sizes and with arbitrary entries.The set t M p k ´ qu is closed with respect to the product of k matrices, and we will therefore callthem k -ary matrices. They form a k -ary semigroup, and when the blocks are over an associativebinary ring, then total associativity follows from the associativity of matrix multiplication.Our proposal is to use single arbitrary elements (from rings with associative multiplication) inplace of the blocks m p i ˆ j q , supposing that the elements of the multiplicative part G of the rings formbinary (semi)groups having some special properties. Then we investigate the similar correspondencebetween the (multiplicative) properties of the matrices M p k ´ q , related to idempotence and order, andthe appearance of the relations in G leading to regular semigroups and braid groups, respectively. Wecall this connection a polyadic matrix-binary (semi)group correspondence (or in short the polyadic-binary correspondence ). – 2 – RELIMINARIES
In the lowest -arity case k “ , the ternary case, the ˆ matrices M p q are anti-triangle. From p M p qq “ M p q and p M p qq „ E p q (where E p q is the ternary identity, see below), we obtain thecorrespondences of the above conditions on M p q with the ordinary regular semigroups and braidgroups, respectively. In this way we extend the polyadic-binary correspondence on -arities k ě toget the higher relations p M p k ´ qq k “ " “ M p k ´ q corresponds to higher k -degree regular semigroups , “ q E p k ´ q corresponds to higher k -degree braid groups , ( )where E p k ´ q is the k -ary identity (see below) and q is a fixed element of the braid group.3. T ERNARY MATRIX GROUP CORRESPONDING TO THE REGULAR SEMIGROUP
Let G free “ t G | µ g u be a free semigroup with the underlying set G “ g p i q ( and the binarymultiplication. The anti-diagonal matrices over G free M g p q “ M p ˆ q ` g p q , g p q ˘ “ ˆ g p q g p q ˙ , g p q , g p q P G free ( )form a ternary semigroup M g ” M gk “ “ t M g p q | µ g u , where M g p q “ t M g p qu is the set ofternary matrices ( ) closed under the ternary multiplication µ g r M g p q , M g p q , M g p qs “ M g p q M g p q M g p q , @ M g p q , M g p q , M g p q P M g , ( )being the ordinary matrix product. Recall that an element M g p q P M g is idempotent, if µ g r M g p q , M g p q , M g p qs “ M g p q , ( )which in the matrix form ( ) leads to p M g p qq “ M g p q . ( )We denote the set of idempotent ternary matrices by M g id p q “ t M g id p qu . Definition 3.1.
A ternary matrix semigroup in which every element is idempotent ( ) is called an idempotent ternary semigroup .Using ( ) and ( ) the idempotence expressed in components gives the regularity conditions g p q g p q g p q “ g p q , ( ) g p q g p q g p q “ g p q , @ g p q , g p q P G free . ( ) Definition 3.2.
A binary semigroup G free in which any two elements are mutually regular ( )–( )is called a regular semigroup G reg . Proposition 3.3.
The set of idempotent ternary matrices ( ) form a ternary semigroup M g , id “t M id p q | µ u , if G reg is abelian.Proof. It follows from ( )–( ), that idempotence (and following from it regularity) is preservedwith respect to the ternary multiplication ( ), only when any g p q , g p q P G free commute. (cid:3) Definition 3.4.
We say that the set of idempotent ternary matrices M g id p q ( ) is in ternary-binarycorrespondence with the regular (binary) semigroup G reg and write this as M g id p q ≎ G reg ( )This means that such property of the ternary matrices as their idempotence ( ) leads to theregularity conditions ( )–( ) in the correspondent binary group G free .– 3 – ERNARY MATRIX GROUP CORRESPONDING TO THE REGULAR SEMIGROUP
Remark . The correspondence ( ) is not a homomorphism and not a bi-element mappingB
OROWIEC ET AL . [2006], and also not a heteromorphism in sense of D
UPLIJ [2018a], becausewe do not demand that the set of idempotent matrices M g id p q form a ternary semigroup (which ispossible in commutative case of G free only, see Proposition 3.3 ).4. P
OLYADIC MATRIX SEMIGROUP CORRESPONDING TO THE HIGHER REGULAR SEMIGROUP
We next extend the ternary-binary correspondence ( ) to the k -ary matrix case ( ) and therebyobtain higher k -regular binary semigroups .Let us introduce the p k ´ q ˆ p k ´ q matrix over a binary group G free of the form ( ) M g p k ´ q ” M pp k ´ qˆp k ´ qq ` g p q , g p q , . . . , g p k ´ q ˘ “ ¨˚˚˚˚˚˝ g p q . . .
00 0 g p q . . .
00 0 ... . . . ...... ... . . . g p k ´ q g p k ´ q . . . ˛‹‹‹‹‹‚ , ( )where g p i q P G free . Definition 4.1.
The set of k -ary matrices M g p k ´ q ( ) over G free is a k -ary matrix semigroup M gk “ t M g p k ´ q | µ gk u , where the multiplication µ gk r M g p k ´ q , M g p k ´ q , . . . , M gk p k ´ qs“ M g p k ´ q M g p k ´ q . . . M gk p k ´ q , M gi p k ´ q P M gk ( )is the ordinary product of k matrices M gi p k ´ q ” M pp k ´ qˆp k ´ qq ´ g p q i , g p q i , . . . , g p k ´ q i ¯ , see ( ).Recall that the polyadic power ℓ of an element M from a k -ary semigroup M k is defined by (e.g.P OST [1940]) M x ℓ y k “ p µ k q ℓ »—– ℓ p k ´ q` hkkkkikkkkj M, . . . , M fiffifl , ( )such that ℓ coincides with the number of k -ary multiplications . In the binary case k “ the polyadicpower is connected with the ordinary power p ( number of elements in the product) as p “ ℓ ` , i.e. M x ℓ y “ M ℓ ` “ M p . In the ternary case k “ we have x ℓ y “ ℓ ` , and so the l.h.s. of ( ) isof polyadic power ℓ “ . Definition 4.2.
An element of a k -ary semigroup M P M is called idempotent , if its first polyadicpower coincides with itself M x y k “ M, ( )and x ℓ y -idempotent , if M x ℓ y k “ M, M x ℓ ´ y k ‰ M. ( ) We use the following notation:Round brackets: p k q is size of matrix k ˆ k , also the sequential number of a matrix element.Square brackets: r k s is number of multipliers in the regularity and braid conditions.Angle brackets: x ℓ y k is the polyadic power (number of k -ary multiplications). – 4 – OLYADIC MATRIX SEMIGROUP CORRESPONDING TO THE HIGHER REGULAR SEMIGROUP
Definition 4.3. A k -ary semigroup M k is called idempotent ( ℓ -idempotent ), if each of its elements M P M k is idempotent ( x ℓ y -idempotent ). Assertion 4.4.
From M x y k “ M it follows that M x ℓ y k “ M , but not vice-versa, therefore all x y -idempotent elements are x ℓ y -idempotent, but an x ℓ y -idempotent element need not be x y -idempotent. Therefore, the definition given in( ) makes sense.
Proposition 4.5.
If a k -ary matrix M g p k ´ q P M gk is idempotent ( ), then its elements satisfythe p k ´ q relations g p q g p q , . . . g p k ´ q g p k ´ q g p q “ g p q , ( ) g p q g p q , . . . g p k ´ q g p q g p q “ g p q , ( ) ... g p k ´ q g p q g p q , . . . g p k ´ q g p k ´ q “ g p k ´ q , @ g p q , . . . , g p k ´ q P G free . ( ) Proof.
This follows from ( ), ( ) and ( ). (cid:3) Definition 4.6.
The relations ( )–( ) are called (higher) r k s -regularity (or higher k -degree regu-larity ). The case k “ is the standard regularity ( r s -regularity in our notation) ( )–( ). Proposition 4.7.
If a k -ary matrix M g p k ´ q P M gk is x ℓ y -idempotent ( ), then its elementssatisfy the following p k ´ q relations ℓ hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj` g p q g p q . . . g p k ´ q g p k ´ q ˘ . . . ` g p q g p q , . . . g p k ´ q g p k ´ q ˘ g p q “ g p q , ( ) ℓ hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj` g p q g p q , . . . g p k ´ q g p k ´ q g p q ˘ . . . ` g p q g p q . . . g p k ´ q g p k ´ q g p q ˘ g p q “ g p q , ( ) ... ℓ hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj` g p k ´ q g p q g p q . . . g p k ´ q g p k ´ q ˘ . . . ` g p k ´ q g p q g p q . . . g p k ´ q g p k ´ q ˘ g p k ´ q “ g p k ´ q , ( ) @ g p q , . . . , g p k ´ q P G free . Proof.
This also follows from ( ), ( ) and ( ). (cid:3) Definition 4.8.
The relations ( )–( ) are called (higher) r k s - x ℓ y - regularity . The case k “ ( )–( ) is the standard regularity ( r s - x y -regularity in this notation). Definition 4.9.
A binary semigroup G free , in which any k ´ elements are r k s -regular ( r k s - x ℓ y -regular), is called a higher r k s -regular ( r k s - x ℓ y -regular) semigroup G reg r k s ( G x ℓ y - reg r k s ).Similarly to Assertion 4.4 , it is seen that r k s - x ℓ y -regularity ( )–( ) follows from r k s -regularity ( )–( ), but not the other way around, and therefore we have Assertion 4.10.
If a binary semigroup G reg r k s is r k s -regular, then it is r k s - x ℓ y -regular as well, butnot vice-versa. Proposition 4.11.
The set of idempotent ( x ℓ y -idempotent) k -ary matrices M g id p k ´ q form a k -arysemigroup M g , id “ t M g id p k ´ q | µ gk u , if and only if G reg r k s ( G ℓ - reg r k s ) is abelian. – 5 – OLYADIC MATRIX SEMIGROUP CORRESPONDING TO THE HIGHER REGULAR SEMIGROUP
Proof.
It follows from ( )–( ) that the idempotence ( x ℓ y -idempotence) and the following r k s -regularity ( r k s - x ℓ y -regularity) are preserved with respect the k -ary multiplication ( ) only in thecase, when all g p q , . . . , g p k ´ q P G free mutually commute. (cid:3) By analogy with ( ), we have
Definition 4.12.
We will say that the set of k -ary p k ´ q ˆ p k ´ q matrices M g id p k ´ q ( ) overthe underlying set G is in polyadic-binary correspondence with the binary r k s -regular semigroup G reg r k s and write this as M g id p k ´ q ≎ G reg r k s . ( )Thus, using the idempotence condition for k -ary matrices in components (being simultaneouslyelements of a binary semigroup G free ) and the polyadic-binary correspondence ( ) we obtainthe higher regularity conditions ( )–( ) generalizing the ordinary regularity ( )–( ), whichallows us to define the higher r k s -regular binary semigroups G reg r k s ( G x ℓ y - reg r k s ). Example . The lowest nontrivial ( k ě ) case is k “ , where the ˆ matrices over G free areof the shape M p q “ M p ˆ q “ ¨˝ a
00 0 bc ˛‚ , a, b, c P G free , ( )and they form the -ary matrix semigroup M g . The idempotence p M p qq x y “ p M p qq “ M p q gives three r s -regularity conditions abca “ a, ( ) bcab “ b, ( ) cabc “ c. ( )According to the polyadic-binary correspondence ( ), the conditions ( )–( ) are r s -regularity relations for the binary semigroup G free , which defines to the higher r s -regular binarysemigroup G reg r s .In the case ℓ “ , we have p M p qq x y “ p M p qq “ M p q , which gives three r s - x y -regularityconditions (they are different from r s -regularity) abcabca “ a, ( ) bcabcab “ b, ( ) cabcabc “ c, ( )and these define the higher r s - x y -regular binary semigroup G x y - reg r s . Obviously, ( )–( )follow from ( )–( ), but not vice-versa.The higher regularity conditions ( )–( ) obtained above from the idempotence of polyadicmatrices using the polyadic-binary correspondence, appeared first in D UPLIJ [1998] and were thenused for transition functions in the investigation of semisupermanifolds D
UPLIJ [2000] and higherregular categories in TQFT D
UPLIJ AND M ARCINEK [2001, 2002].Now we turn to the second line of ( ), and in the same way as above introduce higher degreebraid groups. – 6 –
ERNARY MATRIX GROUP CORRESPONDING TO THE BRAID GROUP
5. T
ERNARY MATRIX GROUP CORRESPONDING TO THE BRAID GROUP
Recall the definition of the Artin braid group A
RTIN [1947] in terms of generators and relationsK
ASSEL AND T URAEV [2008] (we follow the algebraic approach, see, e.g. M
ARKOV [1945]).The
Artin braid group B n (with n strands and the identity e P B n ) has the presentation by n ´ generators σ , . . . , σ n ´ satisfying n p n ´ q { relations σ i σ i ` σ i “ σ i ` σ i σ i ` , ď i ď n ´ , ( ) σ i σ j “ σ j σ i , | i ´ j | ě , ( )where ( ) are called the braid relations , and ( ) are called far commutativity . A general elementof B n is a word of the form w “ σ p i . . . σ p r i r . . . σ p m i m , i m “ , . . . , n, ( )where p r P Z are (positive or negative) powers of the generators σ i r , r “ , . . . , m and m P N .For instance, B is generated by σ and σ satisfying one relation σ σ σ “ σ σ σ , and isisomorphic to the trefoil knot group. The group B has 3 generators σ , σ , σ satisfying σ σ σ “ σ σ σ , ( ) σ σ σ “ σ σ σ , ( ) σ σ “ σ σ . ( )The representation theory of B n is well known and well established K ASSEL AND T URAEV [2008], K
AUFFMAN [1991]. The connections with the Yang-Baxter equation were investigated,e.g. in T
URAEV [1988].Now we build a ternary group of matrices over B n having generators satisfying relations whichare connected with the braid relations ( )–( ). We then generalize our construction to a k -arymatrix group, which gives us the possibility to “go back” and define some special higher analogs ofthe Artin braid group.Let us consider the set of anti-diagonal ˆ matrices over B n M p q “ M p ˆ q ` b p q , b p q ˘ “ ˆ b p q b p q ˙ , b p q , b p q P B n . ( ) Definition 5.1.
The set of matrices M p q “ t M p qu ( ) over B n form a ternary matrix semigroup M k “ “ M “ t M p q | µ u , where k “ is the -arity of the following multiplication µ r M p q , M p q , M p qs ” M p q , M p q , M p q “ M p q , ( ) b p q b p q b p q “ b p q , ( ) b p q b p q b p q “ b p q , b p q i , b p q i P B n , M i p q “ ˜ b p q i b p q i ¸ ( )and the associativity is governed by the associativity of both the ordinary matrix product in the r.h.s.of ( ) and B n . Proposition 5.2. M p q is a ternary matrix group.Proof. Each element of the ternary matrix semigroup M p q P M is invertible (in the ternary sense)and has a querelement ¯ M p q (a polyadic analog of the group inverse D ¨ ORNTE [1929]) defined by µ “ M p q , M p q , ¯ M p q ‰ “ µ “ M p q , ¯ M p q , M p q ‰ “ µ “ ¯ M p q , M p q , M p q ‰ “ M p q . ( )– 7 – ERNARY MATRIX GROUP CORRESPONDING TO THE BRAID GROUP
It follows from ( )–( ), that ¯ M p q “ p M p qq ´ “ ˜ ` b p q ˘ ´ ` b p q ˘ ´ ¸ , b p q , b p q P B n , ( )where ` M p q ˘ ´ denotes the ordinary matrix inverse (but not the binary group inverse which doesnot exist in the k -ary case, k ě ). Non-commutativity of µ is provided by ( )–( ). (cid:3) The ternary matrix group M has the ternary identity E p q “ ˆ ee ˙ , e P B n , ( )where e is the identity of the binary group B n , and µ r M p q , E p q , E p qs “ µ r E p q , M p q , E p qs “ µ r E p q , E p q , M p qs “ M p q . ( )We observe that the ternary product µ in components is “naturally braided” ( )–( ). Thisallows us to ask the question: which generators of the ternary group M can be constructed usingthe Artin braid group generators σ i P B n and the relations ( )–( )?6. T ERNARY MATRIX GENERATORS
Let us introduce p n ´ q ternary ˆ matrix generators Σ ij p q “ Σ p ˆ q ij p σ i , σ j q “ ˆ σ i σ j ˙ , ( )where σ i P B n , i “ . . . , n ´ are generators of the Artin braid group. The querelement of Σ ij p q is defined by analogy with ( ) as ¯Σ ij p q “ p Σ ij p qq ´ “ ˆ σ ´ j σ ´ i ˙ . ( )Now we are in a position to present a ternary matrix group with multiplication µ in terms ofgenerators and relations in such a way that the braid group relations ( )–( ) will be reproduced. Proposition 6.1.
The relations for the matrix generators Σ ij p q corresponding to the braid grouprelations for σ i ( )–( ) have the form µ r Σ i,j ` p q , Σ i,j ` p q , Σ i,j ` p qs “ µ r Σ j ` ,i p q , Σ j ` ,i p q , Σ j ` ,i p qs“ q r s i E p q , ď i ď n ´ , ( ) µ r Σ ij p q , Σ ij p q , E p qs “ µ r Σ ji p q , Σ ji p q , E p qs , | i ´ j | ě , ( ) where q r s i “ σ i σ i ` σ i “ σ i ` σ i σ i ` , and E p q is the ternary identity ( ).Proof. Use µ as the triple matrix product ( )–( ) and the braid relations ( )–( ). (cid:3) Definition 6.2.
We say that the ternary matrix group M gen - Σ3 generated by the matrix generators Σ ij p q satisfying the relations ( )–( ) is in ternary-binary correspondence with the braid (binary)group B n , which is denoted as (cf. ( )) M gen - Σ3 ≎ B n . ( )Indeed, in components the relations ( ) give ( ), and ( ) leads to ( ).– 8 – ERNARY MATRIX GENERATORS
Remark . Note that the above construction is totally different from the bi-element representationsof ternary groups considered in B
OROWIEC ET AL . [2006] (for k -ary groups see D UPLIJ [2018a]).
Definition 6.4.
An element M p q P M is of finite polyadic (ternary) order , if there exists a finite ℓ such that M p q x ℓ y “ M p q ℓ ` “ E p q , ( )where E p q is the ternary matrix identity ( ). Definition 6.5.
An element M p q P M is of finite q -polyadic ( q -ternary) order , if there exists afinite ℓ such that M p q x ℓ y “ M p q ℓ ` “ qE p q , q P B n . ( )The relations ( ) therefore say that the ternary matrix generators Σ i,j ` p q are of finite q -ternaryorder. Each element of M gen - Σ3 is a ternary matrix word (analogous to the binary word ( )),being the ternary product of the polyadic powers ( ) of the ˆ matrix generators Σ p q ij and theirquerelements ¯Σ p q ij (on choosing the first or second row) W “ ˆ Σ i j p q ¯Σ i j p q ˙ x ℓ y , . . . , ˆ Σ i r j r p q ¯Σ i r j r p q ˙ x ℓ r y , . . . , ˆ Σ i m j m p q ¯Σ i m j m p q ˙ x ℓ m y “ ˆ Σ i j p q ¯Σ i j p q ˙ ℓ ` , . . . , ˆ Σ i r j r p q ¯Σ i r j r p q ˙ ℓ r ` , . . . , ˆ Σ i m j m p q ¯Σ i m j m p q ˙ ℓ m ` , ( )where r “ , . . . , m , i r , j r “ , . . . , n (from B n ), ℓ r , m P N . In the ternary case the total numberof multipliers in ( ) should be compatible with ( ), i.e. p ℓ ` q ` . . . ` p ℓ r ` q ` . . . `p ℓ m ` q “ ℓ W ` , ℓ W P N , and m is therefore odd. Thus, we have Remark . The ternary words ( ) in components give only a subset of the binary words ( ), andso M gen - Σ3 corresponds to B n , but does not present it. Example . For B we have only two ternary ˆ matrix generators Σ p q “ ˆ σ σ ˙ , Σ p q “ ˆ σ σ ˙ , ( )satisfying p Σ p qq x y “ p Σ p qq “ q r s E p q , ( ) p Σ p qq x y “ p Σ p qq “ q r s E p q , ( )where q r s “ σ σ σ “ σ σ σ , and both matrix relations ( )–( ) coincide in components. Example . For B , the ternary matrix group M gen - Σ3 is generated by more generators satisfyingthe relations p Σ p qq “ q p q E p q , ( ) p Σ p qq “ q p q E p q , ( ) p Σ p qq “ q p q E p q , ( ) p Σ p qq “ q p q E p q , ( ) Σ p q Σ p q E p q “ Σ p q Σ p q E p q , ( )– 9 – ERNARY MATRIX GENERATORS where q r s “ σ σ σ “ σ σ σ and q r s “ σ σ σ “ σ σ σ . The first two relations give the braidrelations ( )–( ), while the last relation corresponds to far commutativity ( ).7. G ENERATED k - ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP
The above construction of the ternary matrix group M gen - Σ3 corresponding to the braid group B n can be naturally extended to the k -ary case, which will allow us to “go in the opposite way” and buildso called higher degree analogs of B n (in our sense: the number of factors in braid relations morethan ). We denote such a braid-like group with n generators by B n r k s , where k is the number ofgenerator multipliers in the braid relations (as in the regularity relations ( )–( )). Simultaneously k is the -arity of the matrices ( ), we therefore call B n r k s a higher k -degree analog of the braidgroup B n . In this notation the Artin braid group B n is B n r s . Now we build B n r k s for any degree k exploiting the “reverse” procedure, as for k “ and B n in S ECTION
5. For that we need a k -ary generalization of the matrices over B n , which in the ternary case are the anti-diagonal matrices M p q ( ), and the generator matrices Σ ij p q ( ). Then, using the k -ary analog of multiplication( )–( ) we will obtain the higher degree (than ( )) braid relations which generate the so called higher k -degree braid group . In distinction to the higher degree regular semigroup constructionfrom S ECTION
4, where the k -ary matrices form a semigroup for the Abelian group G free , using thegenerator matrices, we construct a k -ary matrix semigroup (presented by generators and relations)for any (even non-commutative) matrix entries. In this way the polyadic-binary correspondence willconnect k -ary matrix groups of finite order with higher binary braid groups (cf. idempotent k -arymatrices and higher regular semigroups ( )).Let us consider a free binary group B free and construct over it a k -ary matrix group along thelines of N IKITIN [1984], similarly to the ternary matrix group M in ( )–( ). Definition 7.1.
A set M p k ´ q “ t M p k ´ qu of k -ary p k ´ q ˆ p k ´ q matrices M p k ´ q “ M pp k ´ qˆp k ´ qq ` b p q , b p q , . . . , b p k ´ q ˘ “ ¨˚˚˚˚˚˝ b p q . . .
00 0 b p q . . .
00 0 ... . . . ...... ... . . . b p k ´ q b p k ´ q . . . ˛‹‹‹‹‹‚ , ( ) b p j q P B free , j “ , . . . , k ´ , form a k -ary matrix semigroup M k “ t M p k ´ q | µ k u , where µ k is the k -ary multiplication µ k r M p k ´ q , M p k ´ q , . . . , M k p k ´ qs “ M p k ´ q M p k ´ q . . . M k p k ´ q “ M p k ´ q , ( ) b p q b p q , . . . b p k ´ q k ´ b p q k “ b p q , ( ) b p q b p q , . . . b p q k ´ b p q k “ b p q , ( )... b p k ´ q b p q , . . . b p k ´ q k ´ b p k ´ q k “ b p k ´ q , ( )where the r.h.s. of ( ) is the ordinary matrix multiplication of k -ary matrices ( ) M i p k ´ q “ M pp k ´ qˆp k ´ qq ´ b p q i , b p q i , . . . , b p k ´ q i ¯ , i “ , . . . , k .– 10 – ENERATED k - ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP
Proposition 7.2. M k is a k -ary matrix group.Proof. Because B free is a (binary) group with the identity e P B free , each element of the k -ary matrix semigroup M p k ´ q P M k is invertible (in the k -ary sense) and has a querelement ¯ M p k ´ q (see D ¨ ORNTE [1929]) defined by (cf. ( )) µ k »– k ´ hkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkj M p k ´ q , . . . , M p k ´ q , ¯ M p k ´ q fifl “ . . . “ M p k ´ q , ( )where ¯ M p k ´ q can be on any place, and so we have k conditions (cf. ( ) for k “ ). (cid:3) The k -ary matrix group has the polyadic identity E p k ´ q “ E pp k ´ qˆp k ´ qq “ ¨˚˚˚˚˚˝ e . . .
00 0 e . . .
00 0 ... . . . ...... ... . . . ee . . . ˛‹‹‹‹‹‚ , e P B free , ( )satisfying µ k r M p k ´ q , E p k ´ q , . . . , E p k ´ qs “ . . . “ M p k ´ q , ( )where M p k ´ q can be on any place, and so we have k conditions (cf. ( )). Definition 7.3.
An element of a k -ary group M p k ´ q P M k has the polyadic order ℓ , if p M p k ´ qq x ℓ y k ” p M p k ´ qq ℓ p k ´ q` “ E p k ´ q , ( )where E p k ´ q P M k is the polyadic identity ( ), for k “ see ( ). Definition 7.4.
An element ( p k ´ q ˆ p k ´ q -matrix over B free ) M p k ´ q P M t k u is of finite q -polyadic order , if there exists a finite ℓ such that p M p k ´ qq x ℓ y k ” p M p k ´ qq ℓ p k ´ q` “ q E p k ´ q , q P B free . ( )Let us assume that the binary group B free is presented by generators and relations (cf. the Artinbraid group ( )–( )), i.e. it is generated by n ´ generators σ i , i “ , . . . , n ´ . An element of B gen - σ n ” B free p e , σ i q is the word of the form ( ). To find the relations between σ i we constructthe corresponding k -ary matrix generators analogous to the ternary ones ( ). Then using a k -aryversion of the relations ( )–( ) for the matrix generators, as the finite order conditions ( ),we will obtain the corresponding higher degree braid relations for the binary generators σ i , and cantherefore present a higher degree braid group B n r k s in the form of generators and relations.Using n ´ generators σ i of B gen - σ n we build p n ´ q k polyadic (or k -ary) p k ´ q ˆ p k ´ q -matrix generators having k ´ indices i , . . . , i k ´ “ , . . . , n ´ , as follows Σ i ,...,i k ´ p k ´ q ” Σ pp k ´ qˆp k ´ qq i ,...,i k ´ ` σ i , . . . , σ i k ´ ˘ “ ¨˚˚˚˚˚˝ σ i . . .
00 0 σ i . . .
00 0 ... . . . ...... ... . . . σ i k ´ σ i k ´ . . . ˛‹‹‹‹‹‚ . ( )For the matrix generator Σ i ,...,i k ´ p k ´ q ( ) its querelement ¯Σ i ,...,i k ´ p k ´ q is defined by( ). – 11 – ENERATED k - ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP
We now build a k -ary matrix analog of the braid relations ( ), ( ) and of far commutativity( ), ( ). Using ( ) we obtain p k ´ q conditions that the matrix generators are of finite polyadicorder (analog of ( )) µ k r Σ i,i ` ,...,i ` k ´ p k ´ q , Σ i,i ` ,...,i ` k ´ p k ´ q , . . . , Σ i,i ` ,...,i ` k ´ p k ´ qs ( ) “ µ k r Σ i ` ,i ` ,...,i ` k ´ ,i p k ´ q , Σ i ` ,i ` ,...,i ` k ´ ,i p k ´ q , . . . , Σ i ` ,i ` ,...,i ` k ´ ,i p k ´ qs ( )... µ k r Σ i ` k ´ ,i,i ` ,...,i ` k ´ p k ´ q , Σ i ` k ´ ,i,i ` ,...,i ` k ´ p k ´ q , . . . , Σ i ` k ´ ,i,i ` ,...,i ` k ´ p k ´ qs ( ) “ q r k s i E p k ´ q , ď i ď n ´ k ` , ( )where E p k ´ q are polyadic identities ( ) and q r k s i P B gen - σ n .We propose a k -ary version of the far commutativity relation ( ) in the following form µ k »– k ´ hkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkj Σ i ,...,i k ´ p k ´ q , . . . , Σ i ,...,i k ´ p k ´ q ,E p k ´ q fifl “ . . . ( ) “ µ k »– k ´ hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj Σ τ p i q ,τ p i q ,...,τ p i k ´ q p k ´ q , . . . , Σ τ p i q ,τ p i q ,...,τ p i k ´ q p k ´ q ,E p k ´ q fifl , ( )if all | i p ´ i s | ě k ´ , p, s “ , . . . , k ´ , ( )where τ is an element the permutation symmetry group τ P S k ´ .In matrix form we can define Definition 7.5. A k -ary (generated) matrix group M gen - Σ k is presented by the p k ´ q ˆ p k ´ q matrix generators Σ i ,...,i k ´ p k ´ q ( ) and the relations (we use ( )) p Σ i,i ` ,...,i ` k ´ p k ´ qq k ( ) “ p Σ i ` ,i ` ,...,i ` k ´ ,i p k ´ qq k ( )... p Σ i ` k ´ ,i,i ` ,...,i ` k ´ p k ´ qq k ( ) “ q r k s i E p k ´ q , ď i ď n ´ k ` , and ´ Σ p k ´ q i ,i ,...,i k ´ ¯ k ´ E p k ´ q “ ´ Σ p k ´ q τ p i q ,τ p i q ,...,τ p i k ´ q ¯ k ´ E p k ´ q , ( )if all | i p ´ i s | ě k ´ , p, s “ , . . . , k ´ , where τ P S k ´ and q r k s i P B gen - σ n .Each element of M gen - Σ k is a k -ary matrix word (analogous to the binary word ( )) being the k -ary product of the polyadic powers ( ) of the matrix generators Σ i ,...,i k ´ p k ´ q and their querele-ments ¯Σ i ,...,i k ´ p k ´ q as in ( ). – 12 – ENERATED k - ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP
Similarly to the ternary case k “ (S ECTION
5) we now develop the k -ary “reverse” procedureand build from B gen - σ n the higher k -degree braid group B n r k s using ( ). Because the presen-tation of M gen - Σ k by generators and relations has already been given in ( )–( ), we need toexpand them into components and postulate that these new relations between the (binary) generators σ i present a new higher degree analog of the braid group. This gives Definition 7.6. A higher k -degree braid (binary) group B n r k s is presented by p n ´ q generators σ i ” σ r k s i (and the identity e ) satisfying the following relations ‚ p k ´ q higher braid relations k hkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkj σ i σ i ` . . . σ i ` k ´ σ i ` k ´ σ i ( ) “ σ i ` σ i ` . . . σ i ` k ´ σ i σ i ` ( )... “ σ i ` k ´ σ i σ i ` σ i ` . . . σ i σ i ` σ i ` k ´ ” q r k s i , q r k s i P B n r k s , ( ) i P I braid “ t , . . . , n ´ k ` u , ( ) ‚ p k ´ q -ary far commutativity k ´ hkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkj σ i σ i . . . σ i k ´ σ i k ´ σ i k ´ ( )... “ σ τ p i q σ τ p i q . . . σ τ p i k ´ q σ τ p i k ´ q σ τ p i k ´ q , ( )if all | i p ´ i s | ě k ´ , p, s “ , . . . , k ´ , ( ) I far “ t n ´ k, . . . , n ´ u , ( )where τ is an element of the permutation symmetry group τ P S k ´ .A general element of the higher k -degree braid group B n r k s is a word of the form w “ σ p i . . . σ p r i r . . . σ p m i m , i m “ , . . . , n, ( )where p r P Z are (positive or negative) powers of the generators σ i r , r “ , . . . , m and m P N . Remark . The ternary case k “ coincides with the Artin braid group B r s n “ B n ( )–( ). Remark . The representation of the higher k -degree braid relations in B n r k s in the tensor productof vector spaces (similarly to B n and the Yang-Baxter equation T URAEV [1988]) can be obtainedusing the n -ary braid equation introduced in D UPLIJ [2018b] (
Proposition 7.2 and next there).
Definition 7.9.
We say that the k -ary matrix group M gen - Σ k generated by the matrix generators Σ i ,i ,...,i k ´ p k ´ q satisfying the relations ( )–( ) is in polyadic-binary correspondence withthe higher k -degree braid group B n r k s , which is denoted as (cf. ( )) M gen - Σ k ≎ B n r k s . ( ) Example . Let k “ , then the -ary matrix group M gen - Σ4 is generated by the matrix generators Σ i ,i ,i p q satisfying ( )–( ) ‚ -ary relations of q -polyadic order ( ) p Σ i,i ` ,i ` p qq “ p Σ i ` ,i,i ` p qq “ p Σ i ` ,i ` ,i p qq “ q r s i E p q , ď i ď n ´ , ( )– 13 – ENERATED k - ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP ‚ far commutativity p Σ i ,i ,i p qq E p q “ p Σ i ,i ,i p qq E p q “ p Σ i ,i ,i p qq E p q“ p Σ i ,i ,i p qq E p q “ p Σ i ,i ,i p qq E p q “ p Σ i ,i ,i p qq E p q , ( ) | i ´ i | ě , | i ´ i | ě , | i ´ i | ě . Let σ i ” σ r s i P B n r s , i “ , . . . , n ´ , then we use the -ary ˆ matrix presentation for thegenerators (cf. Example ) Σ i ,i ,i p q ” Σ p ˆ q p σ i , σ i , σ i q “ ¨˝ σ i
00 0 σ i σ i ˛‚ , i , i , i “ , . . . , n ´ . ( )The querelement ¯Σ i ,i ,i p q satisfying p Σ i ,i ,i p qq ¯Σ i ,i ,i p q “ Σ i ,i ,i p q , ( )has the form ¯Σ i ,i ,i p q “ ¨˝ σ ´ i σ ´ i
00 0 σ ´ i σ ´ i σ ´ i σ ´ i ˛‚ . ( )Expanding ( )–( ) in components, we obtain the relations for the higher -degree braidgroup B n r s as follows ‚ higher -degree braid relations σ i σ i ` σ i ` σ i “ σ i ` σ i ` σ i σ i ` “ σ i ` σ i σ i ` σ i ` ” q r s i , ď i ď n ´ , ( ) ‚ ternary far (total) commutativity σ i σ i σ i “ σ i σ i σ i “ σ i σ i σ i “ σ i σ i σ i “ σ i σ i σ i “ σ i σ i σ i , ( ) | i ´ i | ě , | i ´ i | ě , | i ´ i | ě . ( )In the higher -degree braid group the minimum number of generators is , which follows from( ). In this case we have a braid relation for i “ only and no far commutativity relations,because of ( ). Then Example . The higher -degree braid group B r s is generated by generators σ , σ , σ ,which satisfy only the braid relation σ σ σ σ “ σ σ σ σ “ σ σ σ σ . ( )If n ď , then there will no far commutativity relations at all, which follows from ( ), and sothe first higher -degree braid group containing far commutativity should have n “ elements. Example . The higher -degree braid group B r s is generated by generators σ , . . . , σ ,which satisfy the braid relations with i “ , . . . , σ σ σ σ “ σ σ σ σ “ σ σ σ σ , ( ) σ σ σ σ “ σ σ σ σ “ σ σ σ σ , ( ) σ σ σ σ “ σ σ σ σ “ σ σ σ σ , ( ) σ σ σ σ “ σ σ σ σ “ σ σ σ σ , ( ) σ σ σ σ “ σ σ σ σ “ σ σ σ σ , ( )– 14 – ENERATED k - ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP together with the ternary far commutativity relation σ σ σ “ σ σ σ “ σ σ σ “ σ σ σ “ σ σ σ “ σ σ σ . ( ) Remark . In polyadic group theory there are several possible modifications of the commutativ-ity property, but nevertheless we assume here the total commutativity relations in the k -ary matrixgenerators and the corresponding far commutativity relations in the higher degree braid groups.If B n r k s Ñ Z is the abelianization defined by σ ˘ i Ñ ˘ , then σ pi “ e , if and only if p “ , and σ i are of infinite order. Moreover, we can prove (as in the ordinary case k “ D YER [1980])
Theorem 7.14.
The higher k -degree braid group B n r k s is torsion-free. Recall (see, e.g. K
ASSEL AND T URAEV [2008]) that there exists a surjective homomorphism ofthe braid group onto the finite symmetry group B n Ñ S n by σ i Ñ s i “ p i, i ` q P S n . Thegenerators s i satisfy ( )–( ) together with the finite order demand s i s i ` s i “ s i ` s i s i ` , ď i ď n ´ , ( ) s i s j “ s j s i , | i ´ j | ě , ( ) s i “ e, i “ , . . . , n ´ , ( )which is called the Coxeter presentation of the symmetry group S n . Indeed, multiplying both sidesof ( ) from the right successively by s i ` , s i , and s i ` , using ( ), we obtain p s i s i ` q “ , and( ) on s i and s j , we get p s i s j q “ . Therefore, a Coxeter group B RIESKORN AND S AITO [1972]corresponding ( )–( ) is presented by the same generators s i and the relations p s i s i ` q “ , ď i ď n ´ , ( ) p s i s j q “ , | i ´ j | ě , ( ) s i “ e, i “ , . . . , n ´ . ( )A general Coxeter group W n “ W n p e, r i q is presented by n generators r i and the relationsB J ¨ ORNER AND B RENTI [2005] p r i r j q m ij “ e, m ij “ " , i “ j, ě , i ‰ j. ( )By analogy with ( )–( ), we make the following Definition 7.15.
A higher analog of S n , the k -degree symmetry group S n r k s “ S r k s n p e , s i q , ispresented by generators s i , i “ , . . . , n ´ satisfying ( )–( ) together with the additionalcondition of finite p k ´ q -order s p k ´ q i “ e , i “ , . . . , n . Example . The lowest higher degree case is S r s which is presented by three generators s , s , s satisfying (see ( )) s s s s “ s s s s “ s s s s , ( ) s “ s “ s “ e . ( )In a similar way we define a higher degree analog of the Coxeter group ( ).– 15 – ENERATED k - ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP
Definition 7.17.
A higher k -degree Coxeter group W n r k s “ W r k s n p e , r i q is presented by n genera-tors r i obeying the relations ` r i , r i , . . . , r i k ´ ˘ m i i ,...,ik ´ “ e, ( ) m i i ,..., ik ´ “ " , i “ i “ . . . “ i k ´ , ě k ´ , | i p ´ i s | ě k ´ , p, s “ , . . . , k ´ . ( )It follows from ( ) that all generators are of p k ´ q order r k ´ i “ e . A higher k -degree Coxetermatrix is a hypermatrix M r k ´ s n,Cox ¨˝ k ´ hkkkkkkkkikkkkkkkkj n ˆ n ˆ . . . ˆ n ˛‚ having on the main diagonal and other entries m i i ,..., ik ´ . Example . In the lowest higher degree case k “ and all m i i ,..., ik ´ “ we have (instead ofcommutativity in the ordinary case k “ ) p r i r j q “ r j r i , ( ) r i r j r i “ r j r i r j . ( ) Example . A higher -degree analog of ( )–( ) is given by p r i r i ` r i ` q “ , ď i ď n ´ , ( ) p r i r i r i q “ , | i ´ i | ě , | i ´ i | ě , | i ´ i | ě , ( ) r i “ e , i “ , . . . , n ´ . ( )It follows from ( ), that p r i r i r i q “ r i r i r i , ( )which cannot be reduced to total commutativity ( ). From the first relation ( ) we obtain r i r i ` r i ` r i “ r i ` r i ` , ( )which differs from the higher -degree braid relations ( ). Example . In the simplest case the higher -degree Coxeter group W r s has generator r , r , r satisfying p r r r q “ r “ r “ r “ e . ( ) Example . The minimal case, when the conditions ( ) appear is W r s r r r r “ r r , ( ) r r r r “ r r , ( ) r r r r “ r r , ( ) r r r r “ r r , ( ) r r r r “ r r , ( )and an analog of commutativity p r r r q “ r r r . ( )Thus, we arrive at Theorem 7.22.
The higher k -degree Coxeter group can present the k -degree symmetry group in thelowest case only, if and only if k “ . – 16 –s a further development, it would be interesting to consider the higher degree (in our sense)groups constructed here from a geometric viewpoint (e.g., B IRMAN [1976], K
AUFFMAN [1991]).
Acknowledgement . The author is grateful to Mike Hewitt, Thomas Nordahl, Vladimir Tkach andRaimund Vogl for the numerous fruitful discussions and valuable support.R
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