Structure Groups (and Rings) Wolfgang Rump Instead of Groups and - - PDF document

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 qX : X − → G(X), and then G(X) is called the structure group of X. This is particularly nice when qX 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 Y1 YBE Y2 MV Y3 OML Y4 LEA Y5 X

PPPPPPP P ◗◗◗◗◗◗◗ ◗ ✏✏✏✏✏✏✏ ✏ ✑✑✑✑✑✑✑ ✑ ✲ G(X) = G(Yi)

Definition 1. An L-algebra is a set X with an

• peration · 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 εy : ↓y → X with εy(x) := y · x is always injective:

• ✒
• ✒

1 y ↓y

• ✒

εy X Definition 2. An L-algebra is said to be self-similar if the εy 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! The picture

• ✒
• ✒

1 y ↓y ր X suggests that X may embed into a larger L-algebra. Indeed: Theorem 1. Every L-algebra X is an L-subalgebra

• f 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: qX : X ֒ → S(X) → G(S(X)) = G(X). We call G(X) the structure group of X. a xa ya (x ∧ y)a

• ❅

❅ ❅ ❅

• ❅

❅ ❅ ❅

x y x · y y · x The basic relation (6) can be visualized as a mesh in the Cayley graph of the monoid S(X), labelled by elements

• f the L-algebra X.

5

The situation is particularly simple if X has a trivial partial order: x y z t u 1

✟✟✟✟✟✟✟✟ ✟

• ❅

❅ ❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

· · · · · · 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 qX : 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: X2 → X2, satisfying (s×1X)(1X ×s)(s×1X) = (1X ×s)(s×1X)(1X ×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}.

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Theorem 4. The structure group G( X) coincides with the structure group GX 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 un 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)

• f maximal elements in G− is finite, and the intervals

in G are of bounded length.

• Examples. 2. Artin’s braid group Bn 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 GX of a finite cycle set X

is a distributive Garside group (Chouraqui 2010).

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Theorem 5. The map G → X(G) establishes a

• ne-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?

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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)/Autc(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 D4, 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 =

• [y, x]xy,

[x, y] = xyx−1y−1. 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.

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Recall the correspondence: finite cycle sets ֒ → finite discrete L-algebras x · y = y · x = ⇒ x = y ≀ ≀ distributive Garside groups ֒ → modular Garside groups Example 6. Consider the 4-element discrete L- 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 y x y z z x x y z x z y

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Interpret the arrows as inclusions. To obtain the struc- ture group G(X), replace each edge by an infinite zigzag

❳❳❳❳❳❳❳ ❳ ③ ✘✘✘✘✘✘✘ ✘ ✿ ❳❳❳❳❳❳❳ ❳ ③ ✘✘✘✘✘✘✘ ✘ ✿ ❳❳❳❳❳❳❳ ❳ ③ ✘✘✘✘✘✘✘ ✘ ✿ ❳❳❳❳❳❳❳ ❳ ③

. . . . . . x level -1 level 0 and complete the meshes, using the relation

✘✘✘✘✘✘✘✘ ✿ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ② ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ② ✘✘✘✘✘✘✘ ✘ ✿

a b a · b b · a (A similar construction yields the derived category of a finite dimensional hereditary algebra over a field.) The semibrace C(X) looks as follows:

• ❅

❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ ❅ s s s s s

x y z y z x 1 u−1 The “Garside element” u is a strong order unit.

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• 4. The geometry of a discrete L-algebra

Let x, y be elements of an L-algebra X. We say that x is left orthogonal to y (x⊤y) if x·y = y. We call an L-algebra X symmetric if x⊤y ⇔ y⊤x. The L-algebras in the correspondence finite discrete L-algebras x · y = y · x = ⇒ x = y ↔ modular Garside groups can be viewed as finite quantum sets (requested by Manin 1976 as a model for sets of quantum particles). The length #X of C(X) stands for the cardinality

• f the quantum set. Elements of the quantum set

belong to S1(X) := X {1}. They are discernable if they are two-sided orthogonal. In the above example C(X):

• ❅

❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅ r r r r r

x y z y z x 1

• rthogonality is not symmetric. In an ordinary set

(trivial L-algebra), elements are pairwise orthogonal. Definition 5. Let X be a discrete L-algebra. For distinct x, y ∈ X we define the the connecting line ℓ(x, y) := {z ∈ X | z · x and z · y comparable}.

12

If z, t ∈ ℓ(x, y) are distinct, then ℓ(z, t) = ℓ(x, y), which justifies the concept. Definition 6. We call an L-algebra X sharp if for all x, y ∈ X: x · (x · y) = x · y Recall that a projective space is given by a set X of points and subsets ℓ ⊂ X called lines with

• A. Each line is determined by any two distinct points.
• B. The Veblen-Young axiom:

✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ PPPPPPPPPPP P r r r r r ❜

y q x p r z If p, q, r ∈ X are not collinear, while x, y ∈ X are distinct points with p, q, x and q, r, y collinear, there is a point z ∈ X such that p, r, z and x, y, z are collinear. The subsets closed with respect to lines are called subspaces of X. They form a complete atomistic complemented modular lattice L(X) (O. Frink 1946). The maximal proper subspaces are called hyperplanes. A self-map A → A⊥ of L(X) is said to a polarity if the following are satisfied:

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(1) X⊥ = ∅. (2) A ⊂ B = ⇒ A⊥ ⊃ B⊥. (3) For all x ∈ X, the subspace {x}⊥ is a hyperplane, and {x}⊥⊥ = {x}. For brevity, we call a projective space X with polarity (1)-(3) a polarity space (X; ⊥). We call (X; ⊥) elliptic if x / ∈ x⊥ for all x ∈ X. Theorem 6. Let X be a sharp symmetric discrete L-algebra. Then

• S1(X), ⊤
• is an elliptic polarity

space, and every elliptic polarity space arises in this way. Any projective space decomposes into irreducible

• nes where lines have at least three points.

Corollary 1. If S1(X) is irreducible and #X > 2, there exists a vector space V over a skew-field K so that S1(X) = P(V ) = projective space over V . Corollary 2. If, in addition, #X < ∞, there exists an involution λ → λ∗ of K and a non- degenerate Hermitean form σ: V × V → K which induces the polarity. The other other extreme:

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Corollary 3. If #X < ∞ and all lines have just two points, L(X) is distributive, and S1(X) is a cycle set. The modular Garside groups with their discrete L- algebras are included by the following Theorem 7. Every sharp symmetric discrete L- algebra satisfies x · y = y · x = ⇒ x = y. (9) Example 7. The ideals of a Frobenius algebra A form an L-algebra with I · J := (I ∩ J) + I⊥, where I⊥ denotes the right annihilator of I. Here the maximal ideals, together with A, form a sharp discrete L-subalgebra which is symmetric if and only if A is a symmetric algebra.

• 5. The L-algebra of a Euclidean space

To clarify the role of the structure group, let us explain its architecture in a very familiar special case. Let E be a Euclidean space, with scalar product , . Let L(E) be the lattice of all subspaces, and L1(E) the set of subspaces of codimension 1. Any

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U ∈ L(E) has an orthogonal complement U ⊥. Then L(E) is an L-algebra with U · V := (U ∩ V ) + U ⊥, (10) and L1(E) is a sharp symmetric discrete L-subalgebra.

• Eq. (10) gives U⊤V

⇐ ⇒ V ⊤U ⇐ ⇒ U ⊥ ⊂ V , while sharpness of L1(E) follows by modularity. The hyperplanes of E form a projective space, a Grassmann variety, a quantum set S1(L1(E)) with #S1(L1(E)) = dim E points. To determine the structure group G(L1(E)), consider the orthogonal group O(E), the group of endomorphisms α of E with α(x), α(y) = x, y for all x, y ∈ E, and the ring Rq := R[q, q−1] of real Laurent polynomials. The map q → q−1 extends to an involution f → f ∗ of

• Rq. Accordingly, we extend E to an Rq-module Eq :=

Rq ⊗ E. The elements of Eq are Laurent polynomials x =

n

• i=−m

qixi with xi ∈ E. We also extend the bilinear form to Eq: x, yq :=

∞

• j=−∞

qj

i

xi, yi−j. This form is Rq-linear in the first, antilinear in the second variable: x, fyq = f ∗x, yq.

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Moreover, the extended form , q is hermitean: y, xq = x, y∗

q

and anisotropic: x, xq = 0 = ⇒ x = 0. Every α ∈ EndRq(Eq) can be written as α =

∞

• i=−∞

qiαi with αi ∈ End(E). There is an adjoint operator α∗ :=

∞

• i=−∞

q−iα∗

i

satisfying α(x), yq = x, α∗(y)q. Definition 7. The para-unitary group PUq(E) consists of the operators α ∈ EndRq(Eq) satisfying α∗ = α−1, that is, α(x), α(x)q = x, yq. Specializing q = 1 gives a natural epimorphism ε1: PUq(E) − → → O(E). The kernel is called the pure para-unitary group PPUq(E). So we have a short exact sequence PPUq(E) ֒ → PUq(E) − → → O(E).

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Define PPU −

q (E) := {α ∈ PPUq(E) | ∀ i < 0: αi = 0}.

So the α ∈ PPU −

q (E) are polynomials

α = 1E + qα1 + q2α2 + · · · + qnαn with αi ∈ End(E) such that (1 + qα1 + · · · + qnαn)(1 + q−1α∗

1 + · · · + q−nα∗ n) = 1.

Theorem 8. PPU −

q (E) is the negative cone of a

lattice order on PPUq(E) which makes PPUq(E) into a right ℓ-group.

• Proof. We represent the elements of PPU −

q (E) by

submodules of a fixed PPU −

q (E)-module. Precisely,

consider the PPU −

q (E)-submodule E− q := R[q] ⊗ E

• f Eq and its factor module

Q := Eq/E−

q .

Thus each element of Q is of the form x = q−1x1 + q−2x2 + · · · + q−nxn with xi ∈ E. Let sub(Q) be the lattice of finitely generated R[q]-submodules of Q. Note that each U ∈ sub(Q) is finite dimensional over R. For α ∈ PPU −

q (E) and each x ∈ Q we have x =

αα∗(x). Hence α is surjective on Q. It turns out that α is determined by its kernel:

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Theorem 9. The kernel map Ker: PPU −

q (E) −

→ sub(Q) is a lattice anti-isomorphism.

PPPPPPPP P q PPPPPPPP P q PPP P q

Q Q . . . . . . α

• Ker α

Theorem 9 gives a binary operation on the lattice of submodules of the q-graded module Q. Note that E can be viewed as the first layer Ker q of Q. For a subspace U of E, let πU ∈ End(E) denote the orthogonal projection onto U. Define ̺U := πU + qπU⊥ Then ̺U ∈ PPU −

q (E): Since π∗ U = πU = π2 U, we have

indeed (πU + qπU⊥)(πU + q−1πU⊥) = πU + πU⊥ = 1E. Furthermore, Ker ̺U = q−1U ⊥. The equation ̺U̺∗

U = 1 can be rewritten as

̺U̺U⊥ = q1E, which shows that q1E πU 1E.

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Indeed, the ̺U are just the operators of this interval: {̺U | U ∈ L(E)} = [q1E, 1E]. For U, V ∈ L(E), we have ̺U̺V ∈ [q1E, 1E] ⇐ ⇒ U ⊥ ⊂ V ⇐ ⇒ U⊤V. If ̺U̺V ∈ [q1E, 1E], then ̺U̺V = ̺U∩V . The following theorem was proved by Carsten Dietzel: Theorem 10. PPUq(E) is the structure group

• f the discrete L-algebra L1(E). The L-algebra

L(E) is the ∧-closure C(L1(E)), and PPU −

q (E) is

the self-similar closure S(L1(E)) of L1(E). L1(E) ֒ → L(E) ֒ → PPU −

q (E) ֒

→ PPUq(E) ≀ ≀ ≀ C(L1(E)) ֒ → S(L1(E)) ֒ → G(L1(E)) What is the L-algebra structure of PPU −

q (E)?

For any right ℓ-group G, the negative cone has two L-algebra structures: x → y := yx−1 ∧ 1, x y := (y−1x ∨ 1)−1 which give rise to two lattice structures: x y :⇐ ⇒ x → y = 1, x y :⇐ ⇒ x y = 1.

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They are related as follows: x y ⇐ ⇒ y−1 x−1. Thus, for a (two-sided) ℓ-group, both lattice orders

• coincide. In the above example, as well as for Garside

groups, the two lattice orders are distinct. Now if U, V ∈ E, then ̺U → ̺V = ̺U·V , where U · V = (U ∩ V ) + U ⊥, as previously defined. We did not mention the second, distinct L-algebra

• peration of PPU −

q (E). For subspaces U, V of E,

̺U → ̺V = ̺U ̺V . Notice that the right ℓ-group PPUq(E) is a com- plete invariant of L1(E). Since L(E) ∼ = [q1E, 1E], this means that q1E is a distinguished element of PPUq(E). For example, q−11E can be characterized as the smallest strong order unit in PPUq(E). Final remark. The generators ̺U of PPUq(E) which represent the subspaces of E satisfy quadratic relations, like those in Iwahori-Hecke algebras: (̺U − 1)(̺U − q) = 0, with a decisive difference: For q = 1, the ̺U turn into the identity; for q = 0, they degenerate into a projection πU. What kind of ring do they generate?

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