Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Frobenius Algebras

Geillan Aly

May 13, 2009 Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Outline

Background

Frobenius Algebras

Examples of Frobenius Algebras

Results and Properties Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Background

We recall the following definitions, constructions and theorems. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Theorem (1) Let K be a field, then the following are equivalent: A is a ring and a K-vector space over K such that (ka)b = k(ab) = a(kb) for all k ∈ K, a, b ∈ A. A is a K vector space with two K linear maps µ: A ⊗ A → A (multiplication) η: K → A (unit), such that the following diagrams commute:

µ AO o A ⊗ A O O

µ IA⊗µ

µ⊗IA A ⊗ A o A ⊗ A ⊗ A

µ A o A ⊗ A O jUUUU O UUU IA µ UUU UUU IA⊗η UUUU η⊗IA UU A ⊗ A o K ⊗ A  A  A ⊗ K Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Definition A structure A that satisfies the conditions of theorem 1 is an algebra. We will assume that all algebras A have a unique identity element 1A. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Examples of commutative algebras

Example A field L such that L ⊂ K. Example

Polynomial algebra K[X1,..., Xn]. Observe that a polynomial algebra satisfies the requirements of an algebra, but is an infinite dimensional vector space over K. Thus, there is no requirement that the algebra A be finite dimensional. Example K valued functions on a nonempty set S. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Examples of non-commutative algebras

Example

Matrix algebra Mn(K) with matrix multiplication being the product operation. Example

Hom K(V, V) for any vector space V with composition being the product operation.

Observe that this example and example 4 are equivalent with respect to a choice of basis. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Definition Let A be a K algebra, and M a vector space over K. Then M is a left A-module if each a, a0 ∈ A, m, m0 ∈ M and k ∈ K a product am ∈ M is defined such that a(a + m0) = am + am0 (a + a0)m = am + a0m (aa0)m = a(a0m)

1Am = m where 1A is the identity element in A (ka)m = k(am)a(km)

A right A-module is defined analogously. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Remark A is itself a left A-module, denoted AA, with canonical left multiplication. This is called the left regular A-module. Again, we can define the right regular A-module, AA as a right A-module with the canonical right multiplication. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Definition Let A be a K algebra and M a left A-module. Let M∗ be the dual space of M. Then M∗ becomes a right A-module if for ψ ∈ L ∗, a ∈ A, l ∈ L,

(ψa)(l) = ψ(al).

The right A module M∗ is the dual of M. Similarly, the dual of a right A-module is a left A-module. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Theorem (2) Let A be a finite dimensional algebra over a field K, then the following are equivalent: ∗ 1 For AA and (AA) , two left A-modules, there exists an A algebra ∗ isomorphism λ :A A → (AA) . 2 There exists a non-degenerate linear form η : A ⊗ A → K which is “associative” in the following manner

η(ab ⊗ c) = η(a ⊗ bc) for a, b, c ∈ A.

∗ 3 There exists a linear form f ∈ A whose kernel contains no non-trivial left or right ideals. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Definition An algebra A that satisfies any of the conditions of theorem 2 is a Frobenius algebra.

It is important to note that a Frobenius algebra is not a “type” of algebra, rather it is an algebra endowed with a given structure. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Proof. Part 1 (1) ⇒ (2): ∗ For λ: AA → (AA) , λ(ab) = aλ(b)∀a, b ∈ A. Thus ∀x ∈ A,

λ(ab)(x) = (aλ(b))(x) = λ(b)(xa).

Define the linear form η : A ⊗ A → K

η(x ⊗ y) B λ(y)(x).

η is non-degenerate since λ is a K isomorphism. If η(· ⊗ y) = 0 then λ(y) = 0 and thus y = 0. If η(x ⊗ ·) = 0 ⇒ x = 0.

Associativity η(xy ⊗ z) = η(x ⊗ yz) follows from λ(ab)(x) = (aλ(b))(x) = λ(b)(xa):

η(xy ⊗ z) B λ(z)(xy) = yλ(z)(x) = λ(yz)(x) = η(x ⊗ yz) Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Proof con’t.

Part 2 (2) ⇒ (1): Let η be a non-degenerate linear form on A. Define λ: AA → AA as

λ(y)x B η(x ⊗ y).

λ is a K isomorphism since η is non-degenerate and is an A algebra isomorphism by the associativity of η. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Proof con’t.

Part 3 (2) ⇒ (3): Given a linear form η(x ⊗ y), define a linear function f ∈ A∗ as

f(x) B η(x ⊗ 1A).

Then f(xA) = 0 implies η(xA ⊗ 1A) = η(x ⊗ A) = 0.

Thus x = 0 since η is non-degenerate.

Likewise, f(Ax) = 0 implies x = 0.

Thus, the kernel of f contains no non-trivial left or right ideals. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Proof con’t.

Part 4 (3) ⇒ (2): Given a linear function f ∈ A∗ with no non-trivial left or right ideals, define

η(x ⊗ y) B f(xy).

It is clear that η satisfies the conditions of (2).

f does not have a non-trivial left or right ideal. If y is in an ideal I contained in the kernel of f, then xy ∈ I implying that f(xy) = 0 = η(x ⊗ y) and η is degenerate. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Definition A symmetric Frobenius algebra is a Frobenius algebra such that the non-degenerate linear form η defined in theorem 2 is actually a trace map where η(ab) = η(ba). If A is a symmetric Frobenius algebra, the linear form will be denoted θ.

Note: The term symmetric Frobenius algebra is not the universal term used in defining this structure. Remark Clearly if A is a commutative algebra, ie. ab = ba for all a, b ∈ A then a commutative Frobenius algebra is a symmetric Frobenius algebra. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Example

Consider the algebra Mn(K) where θ(ab) = tr(ab), the trace function. θ(a · b, c) = tr((ab)c) = tr(a(bc)) = θ(a, b · c) Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Example The field C is a Frobenius algebra over R with form η(a + bi) B a.

Another form could be based on a different mapping: Consider the form η(2 + 3i) B 7; η(1 − i) B 4.[?] Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Example Let G be a finite group, R a ring and R[G] be the group ring.

θ(a · b) is the coefficient of the identity element of a · b.

This defines a Frobenius algebra: θ(a · b, c) is the coefficient of the identity element of (a · b) · c = a · b · c = a · (b · c). The coefficient of the identity element of a · (b · c) is θ(a, b · c).

θ is non-degenerate: If θ(C[G]a) = 0 for a ∈ C[G]. Then θ(g−1a) = 0∀g ∈ G. Thus, θ(g−1a) is the coefficient of g in a, implying that a = 0. Similarly, θ(aC[G]) = 0 implies a = 0. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Theorem ((3) Abrams) If A is a Frobenius algebra over a field K with form f, then every other Frobenius form on A is given by u · f for u an invertible element in A. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Proof.

Let u ∈ A be a unit. Then for any a ∈ A such that u · f(ax) = f(uax) = 0 for all a ∈ A, ua = 0 and therefore a = 0.

Thus there are no non-trivial ideals in the kernel and u · f is a Frobenius form. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Proof con’t.

Consider g ∈ A∗ a Frobenius form not equal to f.

Then by the proof in theorem 2, g = λ(u) = u · f for some u ∈ A.

Since g is a Frobenius form, the map λ0 B gβ is an isomorphism A → A∗, where β : A → End (A), the map which takes an element a ∈ A to the map “multiplication by a”.

Thus, there is a v ∈ A such that f = λ0(v) = v · g = vu · f. Then λ(1A) = f = uv · f = λ(vu) implies that 1A = vu. Since λ is an isomorphism, u is a unit in A. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Theorem (4) If A is a Frobenius algebra, then A∗ is a Frobenius algebra. Proof. Let A be a Frobenius algebra with form f ∈ A∗. By theorem 2, all elements of A∗ are of the form a · f for a ∈ A.

As A and A∗ are isomorphic, define mutiplication in A∗ by (a · f)(b · f) B (ab · f).

∗ Define τ : A → K to be “evaluation at 1A.” Then the identity τ(ax · f) = f(ax) lets A∗ with structure τ be a Frobenius algebra. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Definition A coalgebra A over a field K is a vector space A with two K-linear maps:

α: A → A ⊗ A and δ: A → K

such that the following diagrams commute

α A / A ⊗ A

α IA⊗α

 α⊗IA  A ⊗ A / A ⊗ A ⊗ A

α A / A ⊗ A UUU UUU IA UUUU α UUUU IA⊗δ UUUU  δ⊗IA U*  A ⊗ A / K ⊗ A  A  A ⊗ K

The map α is the comultiplication map, δ is the counit map with axioms coassociativity [(a ⊗ (b ⊗ c) = (a ⊗ b) ⊗ c] and the counit condition [A ⊗ K  K ⊗ A is a natural isomorphism]. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

We can use the multiplication map µ on A:

µ : A ⊗ A → A P P ai ⊗ bj 7→ aibj to define a comultiplication map β∗ on A∗:

β∗ : A∗ → A∗ ⊗ A∗  (A ⊗ A)∗ P f 7→ f ◦ µ Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

As a Frobenius algebra, A and A∗ are isomorphic, thus A is prescribed a coalgebra structure by defining α, the comultiplication map A → A ⊗ A as (λ−1 ⊗ λ−1) ◦ β∗ ◦ λ:

α N A / A A O λ λ−1⊗λ−1  β∗ N A∗ / A∗ A∗

A is coassociative and cocommutative by the definition of α. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

α can be also used to define the multiplication in A∗ since

(a · f)(b · f) = [a · f ⊗ b · f] ◦ α = ab · f.

Let g : K → A denote the unit map. The commutativity of

A∗ f ◦ β(a) O > > ∗ O PP >> g PP λ > PPP >> PPP >> PP f  ' A / K a / f(a) = f ◦ β(a)(1A) guarantees the commutativity of

∗ ∗ β N g ⊗IA∗ N A∗ / A∗ A∗ / K A∗ O O λ λ⊗λ λ−1

α N f⊗IA N A / A A / K A

Since the top row is IA∗ , the bottom row is IA. Thus f is the counit in A. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties

Theorem Every finite dimensional semi-simple algebra admits a symmetric Frobenius structure. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Proof.

Part 1 A matrix algebra over a Frobenius algebra is a Frobenius algebra: Let A be a Frobenius algebra over K with form η. Let Mn(A) be the algebra of n × n matrices over A with the standard trace. Show that the composition Tr η Mn(A) / A / K is a Frobenius form.

Recall that every finite dimensional simple algebra is a matrix algebra over a skew-field, showing that every finite-dimensional simple algebra admits a Frobenius algebra structure. Background Frobenius Algebras Examples of Frobenius Algebras Results and Properties Proof con’t.

Part 2 Every finite-dimensional simple algebra is a matrix algebra admits a symmetric Frobenius structure.

Part 3 Frobenius algebras are compatible over direct products.

Part 4 By Wedderburn, every finite dimensional semi-simple algebra is isomorphic to a direct sum of matrix algebras over skew-fields.