Quaternion rings and ternary quadratic forms John Voight University - - PowerPoint PPT Presentation

Quaternion rings and ternary quadratic forms

John Voight University of Vermont RAGE Emory University 19 May 2011

Quaternion rings?

Quaternion rings?

How should one define a quaternion ring if the coefficients can come from an arbitrary commutative ring?

Quaternion rings?

How should one define a quaternion ring if the coefficients can come from an arbitrary commutative ring? i2 = j2 = k2 = ijk = −1

Quaternion rings?

How should one define a quaternion ring if the coefficients can come from an arbitrary commutative ring? i2 = j2 = k2 = ijk = −1

Sir William Rowan Hamilton (1805-1865)

Quaternion algebras over fields

Quaternion algebras over fields

Let F be a field with char F = 2.

Quaternion algebras over fields

Let F be a field with char F = 2. Then there is a functorial bijection

Quaternion algebras over fields

Let F be a field with char F = 2. Then there is a functorial bijection    Similarity classes of regular ternary quadratic forms q over F   

Quaternion algebras over fields

Let F be a field with char F = 2. Then there is a functorial bijection    Similarity classes of regular ternary quadratic forms q over F   

∼

− → Isomorphism classes of quaternion algebras B over F

Quaternion algebras over fields

Let F be a field with char F = 2. Then there is a functorial bijection    Similarity classes of regular ternary quadratic forms q over F   

∼

− → Isomorphism classes of quaternion algebras B over F

• q(x, y, z) = ax2 + by 2 + cz2

Quaternion algebras over fields

Let F be a field with char F = 2. Then there is a functorial bijection    Similarity classes of regular ternary quadratic forms q over F   

∼

− → Isomorphism classes of quaternion algebras B over F

• q(x, y, z) = ax2 + by 2 + cz2 → C 0(q)

Quaternion algebras over fields

Let F be a field with char F = 2. Then there is a functorial bijection    Similarity classes of regular ternary quadratic forms q over F   

∼

− → Isomorphism classes of quaternion algebras B over F

• q(x, y, z) = ax2 + by 2 + cz2 → C 0(q) =

−bc, −ac F

Quaternion algebras over fields

Let F be a field with char F = 2. Then there is a functorial bijection    Similarity classes of regular ternary quadratic forms q over F   

∼

− → Isomorphism classes of quaternion algebras B over F

• q(x, y, z) = ax2 + by 2 + cz2 → C 0(q) =

−bc, −ac F

• nrd : B0 → F

← B = a, b F

Quaternion algebras over fields

Let F be a field with char F = 2. Then there is a functorial bijection    Similarity classes of regular ternary quadratic forms q over F   

∼

− → Isomorphism classes of quaternion algebras B over F

• q(x, y, z) = ax2 + by 2 + cz2 → C 0(q) =

−bc, −ac F

• nrd : B0 → F

← B = a, b F

• nrd(xi + yj + zij)

= −ax2 − by 2 + abz2

Quaternion rings?

Quaternion rings?

Let R be a commutative ring.

Quaternion rings?

Let R be a commutative ring. Let B be an R-algebra

Quaternion rings?

Let R be a commutative ring. Let B be an R-algebra (an associative ring with 1 equipped with R ֒ → Z(B)).

Quaternion rings?

Let R be a commutative ring. Let B be an R-algebra (an associative ring with 1 equipped with R ֒ → Z(B)). Suppose that B is a finitely generated, locally free R-module.

Quaternion rings?

Let R be a commutative ring. Let B be an R-algebra (an associative ring with 1 equipped with R ֒ → Z(B)). Suppose that B is a finitely generated, locally free R-module. So, what does a quaternion ring over R look like?

Quaternion rings?

Let R be a commutative ring. Let B be an R-algebra (an associative ring with 1 equipped with R ֒ → Z(B)). Suppose that B is a finitely generated, locally free R-module. So, what does a quaternion ring over R look like?

• 1. Azumaya (central R-simple) algebra of rank 4 over R.

Quaternion rings?

Let R be a commutative ring. Let B be an R-algebra (an associative ring with 1 equipped with R ֒ → Z(B)). Suppose that B is a finitely generated, locally free R-module. So, what does a quaternion ring over R look like?

• 1. Azumaya (central R-simple) algebra of rank 4 over R.
• 2. Crossed products,

Quaternion rings?

Let R be a commutative ring. Let B be an R-algebra (an associative ring with 1 equipped with R ֒ → Z(B)). Suppose that B is a finitely generated, locally free R-module. So, what does a quaternion ring over R look like?

• 1. Azumaya (central R-simple) algebra of rank 4 over R.
• 2. Crossed products, e.g. B =

a, b R

• .

Quaternion rings?

Let R be a commutative ring. Let B be an R-algebra (an associative ring with 1 equipped with R ֒ → Z(B)). Suppose that B is a finitely generated, locally free R-module. So, what does a quaternion ring over R look like?

• 1. Azumaya (central R-simple) algebra of rank 4 over R.
• 2. Crossed products, e.g. B =

a, b R

• .
• 3. Quaternion orders (if R is a domain),

Quaternion rings?

Let R be a commutative ring. Let B be an R-algebra (an associative ring with 1 equipped with R ֒ → Z(B)). Suppose that B is a finitely generated, locally free R-module. So, what does a quaternion ring over R look like?

• 1. Azumaya (central R-simple) algebra of rank 4 over R.
• 2. Crossed products, e.g. B =

a, b R

• .
• 3. Quaternion orders (if R is a domain), subrings

B ⊆ B ⊗R Frac(R).

Standard involutions and exceptional rings

Standard involutions and exceptional rings

A standard involution : B → B

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B.

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B. From now on, let B have a standard involution.

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B. From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings.

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B. From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings. Let t : M → R be R-linear with rk(M) = n.

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B. From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings. Let t : M → R be R-linear with rk(M) = n. Then B = R ⊕ M

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B. From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings. Let t : M → R be R-linear with rk(M) = n. Then B = R ⊕ M with multiplication law x · y =

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B. From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings. Let t : M → R be R-linear with rk(M) = n. Then B = R ⊕ M with multiplication law x · y = t(x)y for all x, y ∈ M

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B. From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings. Let t : M → R be R-linear with rk(M) = n. Then B = R ⊕ M with multiplication law x · y = t(x)y for all x, y ∈ M is an R-algebra with standard involution x = t(x) − x for all x ∈ M.

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B. From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings. Let t : M → R be R-linear with rk(M) = n. Then B = R ⊕ M with multiplication law x · y = t(x)y for all x, y ∈ M is an R-algebra with standard involution x = t(x) − x for all x ∈ M. In particular, xx = x(t(x) − x) = 0 for all x ∈ M.

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B. From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings. Let t : M → R be R-linear with rk(M) = n. Then B = R ⊕ M with multiplication law x · y = t(x)y for all x, y ∈ M is an R-algebra with standard involution x = t(x) − x for all x ∈ M. In particular, xx = x(t(x) − x) = 0 for all x ∈ M. For an exceptional ring, we have charpoly(x; T) = T(T − t(x))n in B

Standard involutions and exceptional rings

A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B. From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings. Let t : M → R be R-linear with rk(M) = n. Then B = R ⊕ M with multiplication law x · y = t(x)y for all x, y ∈ M is an R-algebra with standard involution x = t(x) − x for all x ∈ M. In particular, xx = x(t(x) − x) = 0 for all x ∈ M. For an exceptional ring, we have charpoly(x; T) = T(T − t(x))n in B whereas in a quaternion algebra, we have charpoly(x; T) = (T 2 − trd(x)T + nrd(x))2.

Standard involutions and quaternion rings

Standard involutions and quaternion rings

Proposition (V)

Let rk(B) = 4.

Standard involutions and quaternion rings

Proposition (V)

Let rk(B) = 4. Then charpoly(x; T) = (T 2 − trd(x)T + nrd(x))2 for all x ∈ B

Standard involutions and quaternion rings

Proposition (V)

Let rk(B) = 4. Then charpoly(x; T) = (T 2 − trd(x)T + nrd(x))2 for all x ∈ B if and only if Tr(x) = 2 trd(x) for all x ∈ B.

Standard involutions and quaternion rings

Proposition (V)

Let rk(B) = 4. Then charpoly(x; T) = (T 2 − trd(x)T + nrd(x))2 for all x ∈ B if and only if Tr(x) = 2 trd(x) for all x ∈ B. B (an R-algebra with a standard involution) is a quaternion ring

Standard involutions and quaternion rings

Proposition (V)

Let rk(B) = 4. Then charpoly(x; T) = (T 2 − trd(x)T + nrd(x))2 for all x ∈ B if and only if Tr(x) = 2 trd(x) for all x ∈ B. B (an R-algebra with a standard involution) is a quaternion ring if rk(B) = 4 and (i) holds.

Quadratic modules

Quadratic modules

A quadratic module over R

Quadratic modules

A quadratic module over R is a quadratic map q : M → I

Quadratic modules

A quadratic module over R is a quadratic map q : M → I where M, I are locally free R-modules with rk(I) = 1,

Quadratic modules

A quadratic module over R is a quadratic map q : M → I where M, I are locally free R-modules with rk(I) = 1, i.e. q(rx) = r 2q(x)for all r ∈ R and x ∈ M

Quadratic modules

A quadratic module over R is a quadratic map q : M → I where M, I are locally free R-modules with rk(I) = 1, i.e. q(rx) = r 2q(x)for all r ∈ R and x ∈ M and T : M × M → I T(x, y) = q(x + y) − q(x) − q(y) is R-bilinear.

Quadratic modules

A quadratic module over R is a quadratic map q : M → I where M, I are locally free R-modules with rk(I) = 1, i.e. q(rx) = r 2q(x)for all r ∈ R and x ∈ M and T : M × M → I T(x, y) = q(x + y) − q(x) − q(y) is R-bilinear. A similarity of two quadratic modules is a commutative square: M

q

• f

≀

• I

g ≀

• M′

q′

I ′

Even Clifford algebra

Even Clifford algebra

Let q : M → I be a ternary quadratic module, so rk(M) = 3.

Even Clifford algebra

Let q : M → I be a ternary quadratic module, so rk(M) = 3. Let I ∨ = HomR(I, R).

Even Clifford algebra

Let q : M → I be a ternary quadratic module, so rk(M) = 3. Let I ∨ = HomR(I, R). We define the even Clifford algebra C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨. We have rk(C 0(q)) = 4

Even Clifford algebra

Let q : M → I be a ternary quadratic module, so rk(M) = 3. Let I ∨ = HomR(I, R). We define the even Clifford algebra C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨. We have rk(C 0(q)) = 4 and the map x ⊗ y → y ⊗ x for x, y ∈ M induces a standard involution on C 0(q).

If M = R3 and I = R then q(x, y, z) = ax2 + by 2 + cz2 + uyz + vxz + wxy with a, b, c, u, v, w ∈ R.

If M = R3 and I = R then q(x, y, z) = ax2 + by 2 + cz2 + uyz + vxz + wxy with a, b, c, u, v, w ∈ R. Then C 0(q)

If M = R3 and I = R then q(x, y, z) = ax2 + by 2 + cz2 + uyz + vxz + wxy with a, b, c, u, v, w ∈ R. Then C 0(q) = R ⊕ R(e2 ⊗ e3) ⊕ R(e3 ⊗ e1) ⊕ R(e1 ⊗ e2)

If M = R3 and I = R then q(x, y, z) = ax2 + by 2 + cz2 + uyz + vxz + wxy with a, b, c, u, v, w ∈ R. Then C 0(q) = R ⊕ R(e2 ⊗ e3) ⊕ R(e3 ⊗ e1) ⊕ R(e1 ⊗ e2) = R ⊕ Ri ⊕ Rj ⊕ Rk

If M = R3 and I = R then q(x, y, z) = ax2 + by 2 + cz2 + uyz + vxz + wxy with a, b, c, u, v, w ∈ R. Then C 0(q) = R ⊕ R(e2 ⊗ e3) ⊕ R(e3 ⊗ e1) ⊕ R(e1 ⊗ e2) = R ⊕ Ri ⊕ Rj ⊕ Rk where i2 = ui − bc jk = ai = a(u − i) j2 = vj − ac ki = bj k2 = wk − ab ij = ck.

If M = R3 and I = R then q(x, y, z) = ax2 + by 2 + cz2 + uyz + vxz + wxy with a, b, c, u, v, w ∈ R. Then C 0(q) = R ⊕ R(e2 ⊗ e3) ⊕ R(e3 ⊗ e1) ⊕ R(e1 ⊗ e2) = R ⊕ Ri ⊕ Rj ⊕ Rk where i2 = ui − bc jk = ai = a(u − i) j2 = vj − ac ki = bj k2 = wk − ab ij = ck. This defines the multiplication table since e.g. kj = jk.

If M = R3 and I = R then q(x, y, z) = ax2 + by 2 + cz2 + uyz + vxz + wxy with a, b, c, u, v, w ∈ R. Then C 0(q) = R ⊕ R(e2 ⊗ e3) ⊕ R(e3 ⊗ e1) ⊕ R(e1 ⊗ e2) = R ⊕ Ri ⊕ Rj ⊕ Rk where i2 = ui − bc jk = ai = a(u − i) j2 = vj − ac ki = bj k2 = wk − ab ij = ck. This defines the multiplication table since e.g. kj = jk. (This presentation is due to Gross and Lucianovic.)

If M = R3 and I = R then q(x, y, z) = ax2 + by 2 + cz2 + uyz + vxz + wxy with a, b, c, u, v, w ∈ R. Then C 0(q) = R ⊕ R(e2 ⊗ e3) ⊕ R(e3 ⊗ e1) ⊕ R(e1 ⊗ e2) = R ⊕ Ri ⊕ Rj ⊕ Rk where i2 = ui − bc jk = ai = a(u − i) j2 = vj − ac ki = bj k2 = wk − ab ij = ck. This defines the multiplication table since e.g. kj = jk. (This presentation is due to Gross and Lucianovic.) If q = 0, then C 0(q) ∼ = R[i, j, k]/(i, j, k)2.

Main result

Main result

Theorem (V)

There is a functorial, discriminant-preserving bijection

Main result

Theorem (V)

There is a functorial, discriminant-preserving bijection    Similarity classes of ternary quadratic modules q : M → I over R   

Main result

Theorem (V)

There is a functorial, discriminant-preserving bijection    Similarity classes of ternary quadratic modules q : M → I over R   

∼

− →        Isomorphism classes of quaternion rings B over R equipped with a parity factorization p of 4B       

Main result

Theorem (V)

There is a functorial, discriminant-preserving bijection    Similarity classes of ternary quadratic modules q : M → I over R   

∼

− →        Isomorphism classes of quaternion rings B over R equipped with a parity factorization p of 4B        q : M → I → C 0(q) with p

Parity factorization

Parity factorization

Recall C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨.

Parity factorization

Recall C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨. So, as R-modules, we have p : 4C 0(q)

Parity factorization

Recall C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨. So, as R-modules, we have p : 4C 0(q) ∼ = 3(C 0(q)/R) ∼ =

Parity factorization

Recall C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨. So, as R-modules, we have p : 4C 0(q) ∼ = 3(C 0(q)/R) ∼ = 3(2M ⊗ I ∨)

Parity factorization

Recall C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨. So, as R-modules, we have p : 4C 0(q) ∼ = 3(C 0(q)/R) ∼ = 3(2M ⊗ I ∨) ∼ = (3M)⊗2 ⊗ (I ∨)⊗3

Parity factorization

Recall C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨. So, as R-modules, we have p : 4C 0(q) ∼ = 3(C 0(q)/R) ∼ = 3(2M ⊗ I ∨) ∼ = (3M)⊗2 ⊗ (I ∨)⊗3 ∼ = (3M ⊗ I ∨)⊗2 ⊗ I ∨.

Parity factorization

Recall C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨. So, as R-modules, we have p : 4C 0(q) ∼ = 3(C 0(q)/R) ∼ = 3(2M ⊗ I ∨) ∼ = (3M)⊗2 ⊗ (I ∨)⊗3 ∼ = (3M ⊗ I ∨)⊗2 ⊗ I ∨. Let N be an invertible R-module.

Parity factorization

Recall C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨. So, as R-modules, we have p : 4C 0(q) ∼ = 3(C 0(q)/R) ∼ = 3(2M ⊗ I ∨) ∼ = (3M)⊗2 ⊗ (I ∨)⊗3 ∼ = (3M ⊗ I ∨)⊗2 ⊗ I ∨. Let N be an invertible R-module. A parity factorization of N

Parity factorization

Recall C 0(q) = R ⊕ M⊗2 ⊗ I ∨ x ⊗ x ⊗ f − f (q(x)) : x ∈ M, f ∈ I ∨. So, as R-modules, we have p : 4C 0(q) ∼ = 3(C 0(q)/R) ∼ = 3(2M ⊗ I ∨) ∼ = (3M)⊗2 ⊗ (I ∨)⊗3 ∼ = (3M ⊗ I ∨)⊗2 ⊗ I ∨. Let N be an invertible R-module. A parity factorization of N is an isomorphism p : P⊗2 ⊗ Q ∼ − → N with P, Q invertible.

The canonical exterior form

The canonical exterior form

Given a quaternion ring B,

The canonical exterior form

Given a quaternion ring B, we have the canonical exterior form

The canonical exterior form

Given a quaternion ring B, we have the canonical exterior form φB : 2(B/R) → 4B

The canonical exterior form

Given a quaternion ring B, we have the canonical exterior form φB : 2(B/R) → 4B φB(x ∧ y) = 1 ∧ x ∧ y ∧ xy.

The canonical exterior form

Given a quaternion ring B, we have the canonical exterior form φB : 2(B/R) → 4B φB(x ∧ y) = 1 ∧ x ∧ y ∧ xy. This is a quadratic form with values in 4B.

The canonical exterior form

Given a quaternion ring B, we have the canonical exterior form φB : 2(B/R) → 4B φB(x ∧ y) = 1 ∧ x ∧ y ∧ xy. This is a quadratic form with values in 4B. Using the parity factorization of 4B to adjust the coefficient module of this quadratic form, we recover the bijection.