Structure Groups (and Rings) Wolfgang Rump Instead of “Groups and associated Structures” I will talk on Structures and associated Groups . 1. Structures and groups To analyse a mathematical structure X in terms of group theory, one would associate a group G ( X ) in a functorial way. In several cases, X is connected with G ( X ) by a natural map q X : X − → G ( X ) , and then G ( X ) is called the structure group of X . This is particularly nice when q X is an embedding, and it is optimal if G ( X ) is a classifying invariant of X . Examples. • The knot group of a link (or its associated rack) • The Artin-Tits group of a Coxeter system • Structure group of a solution to the Yang-Baxter equation • Structure group of an orthomodular lattice (OML)
Questions: • Is there a systematic way to get a structure group? • Can the relationship X → G ( X ) be characterized? 2. L -algebras To answer both questions in one step, we would need an intermediate structure X which • can be associated to various structures Y so that • X has a well-understood relation to G ( X ) = G ( Y ). Cox Y 1 ◗◗◗◗◗◗◗ YBE Y 2 PPPPPPP ◗ P ✲ G ( X ) = G ( Y i ) X MV Y 3 ✏✏✏✏✏✏✏ ✏ ✑ ✑✑✑✑✑✑✑ OML Y 4 LEA Y 5 Definition 1. An L -algebra is a set X with an operation · satisfying 1 · x = x, x · 1 = x · x = 1 (1) ( x · y ) · ( x · z ) = ( y · x ) · ( y · z ) (2) x · y = y · x = 1 = ⇒ x = y (3) An element 1 satisfying (1) is called a logical unit . 2
Logical interpretation: ( x · y ) · ( x · z ) = ( y · x ) · ( y · z ) ( x → y ) → ( x → z ) ⇔ ( y → x ) → ( y → z ) Any L -algebra X has a partial order ( entailment ) x � y : ⇐ ⇒ x · y = 1 with greatest element 1. For y ∈ X , consider the downset ↓ y := { x ∈ X | x � y } in X . The map ε y : ↓ y → X with ε y ( x ) := y · x is always injective: 1 � ✒ � � X � � y ε y � ✒ � ✒ � ↓ y � � � � � Definition 2. An L -algebra is said to be self-similar if the ε y are bijective. Then the operation xy := ε − 1 y ( x ) is associative! A self-similar L -algebra X can be described equationally as a monoid with a second operation · satisfying y · xy = x (4) xy · z = x · ( y · z ) (5) ( x · y ) x = ( y · x ) y. (6) 3
Again, (4)-(6) is pure logic: xy is a non-commutative conjunction: xy · z = x · ( y · z ) , (5) while the commutative operation x ∧ y := ( x · y ) x = ( y · x ) y (6) is the glb of x and y - classical conjunction! We have x · ( y ∧ z ) = ( x · y ) ∧ ( x · z ) x · 1 = 1 (7) ( x ∧ y ) · z = ( x · y ) · ( x · z ) 1 · x = x, (8) that is, X is a semibrace ! 1 The picture ✒ � � � � X � y ր ✒ � ↓ y � � � � suggests that X may embed into a larger L -algebra. Indeed: Theorem 1. Every L -algebra X is an L -subalgebra of a (canonical) self-similar L -algebra S ( X ) . The ∧ -closure C ( X ) in S ( X ) is a semibrace. S ( X ) is called the self-similar closure of X . So there are canonical inclusions X ⊂ C ( X ) ⊂ S ( X ). The inclusion X ⊂ S ( X ) can be exactly described: 4
Theorem 2. Let X be an L -subalgebra of a self- similar L -algebra A . Then A = S ( X ) if and only if the monoid A is generated by X . This can be used to find the self-similar closure S ( X ) within an ambient self-similar L -algebra of X . 3. The structure group After embedding an L -algebra X into S ( X ), we now pass to the structure group. Let A be a self-similar L -algebra. Then y · xy = x (4) shows that A is right cancellative: xz = yz ⇒ x = y . And ( x · y ) x = ( y · x ) y (6) implies that A satisfies the left Ore condition. So A admits a group G ( A ) of left fractions, and there is a canonical map for any L -algebra X : q X : X ֒ → S ( X ) → G ( S ( X )) = G ( X ) . We call G ( X ) the structure group of X . a The basic relation (6) can be � ❅ x y � ❅ visualized as a mesh in the � ❅ � ❅ xa ya Cayley graph of the monoid ❅ ❅ � � x · y y · x S ( X ), labelled by elements ❅ � ❅ � ( x ∧ y ) a of the L -algebra X . 5
The situation is particularly simple if X has a trivial partial order: 1 ✟ ❍ ✟✟✟✟✟✟✟✟ ❍ � � ❍ ❅ ❅ · · · ❍ · · · � ❍ ❅ ❍ � ❍ ❅ ❍ � ❍ ❅ x y z t u Definition 3. An L -algebra is said to be discrete if x < y implies that y = 1. Theorem 3. Let X be a discrete L -algebra. The map q X : X → G ( X ) is injective if and only if x · y = y · x = ⇒ x = y (9) holds for all x, y ∈ X . Example 1. The Yang-Baxter equation . Set- theoretic solutions are maps s : X 2 → X 2 , satisfying ( s × 1 X )(1 X × s )( s × 1 X ) = (1 X × s )( s × 1 X )(1 X × s ) . A non-degenerate involutive solution is equivalent to a non-degenerate cycle set ( X ; · ), that is, the maps x �→ y · x and x �→ x · x are bijective for y ∈ X , and ( x · y ) · ( x · z ) = ( y · x ) · ( y · z ) (2) is satisfied. Adjoining an element 1 to X with 1 · x = x, x · 1 = x · x = 1 , (1) we get a discrete L -algebra � X := X ⊔ { 1 } . 6
Theorem 4. The structure group G ( � X ) coincides with the structure group G X of Etingof et al. (1999). Corollary. Non-degenerate cycle sets correspond to a special class of discrete L -algebras. A group G with a lattice order, invariant under right multiplication, is said to be a right ℓ -group . The elements a ∈ G with b � c ⇐ ⇒ ab � ac are said to be a normal . The normal elements form a (two-sided) ℓ -group N ( G ), the quasi-centre of G . An element u ∈ N ( G ) is said to be a strong order unit if each a ∈ G satisfies a � u n for some n ∈ N . Definition 4. A right ℓ -group G with a strong order unit is said to be a Garside group if the set X ( G ) of maximal elements in G − is finite, and the intervals in G are of bounded length. Examples. 2. Artin’s braid group B n is a Garside group (F. A. Garside 1969). In general, the lattice of a Garside group is very far from modular . 3. The structure group G X of a finite cycle set X is a distributive Garside group (Chouraqui 2010). 7
Theorem 5. The map G �→ X ( G ) establishes a one-to-one correspondence between modular Gar- side groups and finite discrete L -algebras satisfying x · y = y · x = ⇒ x = y. (9) Recall that (9) says that X = X ( G ) embeds into its structure group G = G ( X ). Example 4. “God made the integers, all else is the work of man.” (Kronecker 1886) To challenge Kronecker, let us consider the 2-element Boolean algebra B = { 0 , 1 } , the standard model of classical logic. (A proposition of classical logic is true if and only if any valuation in B yields the value 1.) We claim that B is more fundamental than the ring Z of integers. Note that B is an L -algebra. Question. What is the structure group of B ? 8
Answer: B is the L -algebra of the 1-element cycle set. So G ( B ) = Z , a trivial brace! To Kronecker: Z = End( Z ; +). So our number system starts with B : B ⊂ Z ⊂ Q ⊂ R ⊂ C ⊂ H ⊂ O It is based on pure logic (and L -algebras). Example 5. Let G be a nilpotent group of class 2. (Heineken-Liebeck 1974: Every finite group is of the form Aut( G ) / Aut c ( G ) with G finite, [[ GG ] G ] = 1.) Conjugation x ∗ y := xyx − 1 makes G into a rack : x ∗ ( y ∗ z ) = ( x ∗ y ) ∗ ( x ∗ z ) . Moreover, ( G ; ∗ ) is a cycle set . Since ( a ∗ b ) a = ab , it is not a brace ! A (non-canonical) affine structure exists (Cedo et al. 2010). (For the dihedral group D 4 , there are 8 such braces.) Here is a canonical one: Assume that G is a p -group ( p odd). By the Baer correspondence, G can be made into a Lie ring with � [ x, y ] = xyx − 1 y − 1 . x + y = [ y, x ] xy, � With this addition, G is a brace: x · y = [ y, x ]( x ∗ y ). Addition can be viewed as a geometric mean: xy � x + y � yx ✲ ✲ [ y, x ] [ y, x ] Note: A non-degenerate cycle set is a rack if and only if it is 2-retractible: ( x · y ) · z = y · z . 9
Recall the correspondence: → finite discrete L -algebras finite cycle sets ֒ x · y = y · x = ⇒ x = y �≀ �≀ distributive Garside groups ֒ → modular Garside groups Consider the 4-element discrete L - Example 6. algebra X = { 1 , x, y, z } given by · x y z x 1 y y y z 1 z z x x 1 The elements of two consecutive layers of the lattice G ( X ) is a directed infinite trivalent graph : ❅ ■ ✒ � ❅ ■ ✒ � ❅ � ❅ � z z y ❄ ❄ y ✟✟✟✟ ✯ ❨ ❍ ✟✟✟✟ ✯ ❍ ❨ ❍ ❍ ❍ ❨ ❍ ❍ ✟✟ ✯ x x ❍ ❍ ❍ ❄ ❄ z ✸❄ ✑✑✑✑✑✑ ◗ ❦ ◗ ❄ y ◗ ❄ x ✟✟ ✯ ❍ ❨ x y ✟✟✟✟ ✯ ❍ ❨ ◗ ❍ ❍ ◗ ❍ ❍ ◗ z z ❄ ❄ ✒ � ■ ❅ ✒ � ■ ❅ � ❅ � ❅ 10
Recommend
More recommend
