semi-simple lie algebra is a non-associative frobenius algebra (all two-dimensional frobenius...

Semi-simple Lie algebra isa non-associative Frobenius algebra

(All two-dimensional Frobenius algebras)

Zbigniew Oziewicz and William S. Page

Abstract. An algebra Y is said to be a solvable Frobenius alge-bra if it possesses a non-zero one-sided Y -module morphism withnon-trivial radical. It is said to be a Frobenius algebra if thereexists a Y -module isomorphism. We determined the necessaryand sufficient condition on an algebra to be a solvable Frobeniusalgebra. The notion of a solvable Frobenius algebra makes it possi-ble to find all commutative non-associative Frobenius algebras (atleast two-dimensional, Theorem 8.1), and all Frobenius structuresfor commutative associative Frobenius algebras. Frobenius alge-bra is formulated within the abelian monoidal category of operadof graphs.

Frobenius algebra allows S2-permuted opposite algebra to beextended to S3-permuted algebras.

Contents

1. Non associative Frobenius algebra 22. Duality and solvable Frobenius algebra 33. Operad of graphs 64. Clifford algebra of one-dimensional space 125. Digression on Lie algebras 176. What it is a Y -associative scalar product? 187. Frobenius ideal 208. Commutative Frobenius algebras 229. Commutative associative Frobenius algebras 24

2000 Mathematics Subject Classification. 16W99,17D99.Key words and phrases. Monoidal category, Abelian category, Frobenius Alge-

bra, Nonassociative Algebra, Clifford Algebra, S3-permutation.October 22, 2010.

1

2 ZBIGNIEW OZIEWICZ AND WILLIAM S. PAGE

10. Non commutative Frobenius algebras 2511. S3-permuted Frobenius algebra 2712. Conclusion 30References 30WEB pages 31

1. Non associative Frobenius algebra

Duality with evaluations and co-evaluations allows one to regardevery algebra Y as a right Y -module and as a left Y -module, in twoways, on both dual objects. An associative algebra Y is a two-sided Y -module, known as Y -bimodule, and this Y -bimodule structure is againdoubled on both dual objects.

Frobenius algebra is usually defined to be associative and unital al-gebra Y possessing a (left or right) Y -module isomorphism, or, equiv-alently, possessing a Y -associative invertible scalar product, denotedhere by ∪, called the Frobenius structure or the Frobenius pairing, seee.g. [Eilenberg and Nakayama 1955; Caenepeel, Militaru, Zhu 2002,Definition 3 on page 32; Kock 2003, pages 95–97, Definition 2.2.5].

Exactly this last property, admission of associative invertible scalarproduct, permits many interesting properties for such algebras, and inthe present paper we are interested in Frobenius algebras that are notnecessarily associative nor unital.

In the present paper by Frobenius algebra Y we mean an algebraY possessing an Y -associative and invertible scalar product. We dropthe extra conditions of associativity and unitality of Y.

This permits one to see the analogy between Frobenius algebrasand Lie algebras. Namely, Elie Cartan in his These in 1894, introducedwhat is nowadays baptized as the Killing trace form for Lie algebra,and we will refer in the sequel to this form (or metric) as to the Cartan-Killing trace form. We note that the Cartan-Killing trace form for Liealgebra satisfies exactly the same relation as the Y -associative form∪ for the Frobenius algebra. Cartan proved that the invertibility ofCartan-Killing trace form, the ∪-radical Y ⊥ = 0, is equivalent to semi-simplicity of the corresponding Lie algebra Y. Moreover Cartan foundalso the characterization of solvable and nilpotent Lie algebras in termsof the Cartan-Killing form (also called Y -invariant form).

We decide to introduce the notion of a solvable Frobenius algebraY in an analogy to solvable Lie algebra. An algebra Y is said tobe a solvable ∪-Frobenius if ∪-radical Y ⊥ 6= 0. Equivalently, if thereexists a non-zero (left or right) Y -module morphism, that is not an

FROBENIUS ALGEBRA 3

isomorphism. An algebra is ∪-Frobenius if ∪-radical Y ⊥ = 0, i.e. ifthere exists a Y -module isomorphism.

We determined the necessary and sufficient condition on an algebrato be a solvable Frobenius algebra. The notion of a solvable Frobe-nius algebra makes it possible to explicitly find all two-dimensionalcommutative Frobenius algebras, Theorem 8.1 and Section 9.

Our next observation is that an arbitrary algebra Y possesses apriori five different Y -forms U = U(Y ). Among these five, two of themare the Cartan-Killing trace-forms, trace of the composition of one-sided (left or right) regular representations, and they are equal for aLie algebra. Let a → aR and a → La, denote the right and the leftregular representations of Y. These five Y -forms are as follows

trace(aR ◦ bR), trace(La ◦ Lb), trace(aR ◦ Lb),(1)

(trace aR) ◦ Y, (trace La) ◦ Y.(2)

In general an associative Y does not require these Y -forms to beY -associative, as can be seen from an example of the Clifford algebraof complex numbers C. We are interested in relations among these fiveY -forms assigned by every algebra Y.

The Frobenius algebra is formulated within the abelian monoidalcategory of the operad of graphs.

2. Duality and solvable Frobenius algebra

A monoidal category with duality is a two-color monoidal categorygenerated by a single object and its two-sided dual. We will visualize adual generating object by a dotted outer or inner edge (another color).

In this section we wish to return to the original concept of a Frobe-nius algebra as introduced by Georg Frobenius in 1903. The left andthe right regular representation of an algebra Y, is the same as the regu-lar left- and right- Y -modules, and they gives rise to adjoint (pull-back)representations on the dual space. An algebra is said to be a Frobeniusalgebra if there is an isomorphism among one-sided Y -modules. Ouraim is to introduce a pre-Frobenius algebra in the same spirit when anisomorphism is weakened to non-trivial Y -module morphism.

A monoidal category with two-sided dual (for every object) is saidto be closed if it possesses both left and right morphisms of evaluations,evL ∈ cat(1∗ ⊗ 1, 0) and, evR ∈ cat(1⊗ 1∗, 0), as illustrated on Figure1. A monoidal category is said to be pivotal if it possesses left andright coevaluation morphisms co-evL ∈ cat(0, 1∗ ⊗ 1), etc.


left ev right ev

left co-ev right co-ev

= = dimension ∈ N

Figure 1. Evaluations and coevaluations: alphabet andone equational ‘spherical’ axiom

A monoidal category is said to be bi-closed, or autonomous, orcompact-closed, if every object possess the left and the right dual ob-ject(s), both reflexive, together with morphisms of evaluation and co-evaluations, such that the equational axioms depicted on Figure 2 hold.

= =

= =

Figure 2. Bi-closed category: axioms

For compact closed (bi-closed) monoidal categories we refer to [Kelly& Laplaza 1980].

Duality with evaluation assigns to every left (right) Y -module, adual right (left) Y -module [Eilenberg and Nakayama 1955], and thisis visualized in Figure 3. Analogously to an algebra Y, a coalgebra isdenoted by . Thanks to co-evaluation the left and the right -co-modules are also always together.

FROBENIUS ALGEBRA 5

=

ev ev

Figure 3. The left and the right Y -modules always together

At this point we are ready to graft the letters from Figure 8 with theduality letters from Figure 1. The results are morphisms ∈ cat(1, 1∗)and ∈ cat(1∗, 1), see Figure 4.

The isomorphisms, cat(2, 0) ' cat(1, 1∗), and cat(0, 2) ' cat(1∗, 1),allows, by abuse of notation, to use the same letter ∪, for ∪ ∈ cat(2, 0),and as well for, ∪ ∈ cat(1, 1∗). Analogously, ∩ ∈ cat(0, 2) ' cat(1∗, 1).

•

•

•

= =∪

∪ ∪

co-ev

•

•

•

•

•

= =∩

Figure 4. A morphism, cat(1, 1∗) ' cat(1⊗ 1, 0)

Let a morphism ∪ ∈ cat(1, 1∗) be a homomorphism among rightY -modules as shown on Figure 5. This equality within cat(2, 1∗) isequivalent to the equality within cat(3, 0) as shown on Figure 6. Thegraphical proof of this equivalence uses the axioms of Figures 3 - 4 - 2.

The crucial observation about the equivalence of Figure 5 with Fig-ure 6, is that this equivalence does not need to assume that a Y -modulemorphism, ∪ ∈ cat(1, 1∗) ' cat(2, 0) be an isomorphism. In fact thisequivalence holds for every non-trivial Y -module morphism, where tobe non-trivial means to be not zero within an abelian spherical mono-idal category considered in the next section 6 .

A morphism Y ∈ cat(2, 1) is said to be Frobenius when the tworight Y -representations in Figure 5 are equivalent, i.e. are isomorphic.


••

=∪

∪ ∈ cat(1, 1∗)

Figure 5. A right Y -module morphism is an equation ∈ cat(2, 1∗)

=

Figure 6. A Y -associative scalar product ∪, as equalityof parallel arrows in cat(3, 0).

2.1. Definition (Frobenius algebra). Consider an abelian monoi-dal category. Let 0 6= Y ∈ cat(2, 1) and a right Y -module morphism,0 6= ∪ ∈ cat(1, 1∗) ' cat(2, 0), be such that the parallel arrows1 areequal as shown in Figure 5, or equivalently in Figure 6. Then a non-zeromorphism 0 6= ∪ ∈ cat(2, 0) is said to be Y -associative.

Let Y ⊥ ≡ ker ∪ ⊂ Y, denotes a ∪-radical of Y. We shall say

• If Y ⊥ 6= 0, then Y is said to be a solvable ∪-Frobenius.• If Y ⊥ = 0, then Y is said to be a ∪-Frobenius.

The following statements are equivalent,{Y is solvable ∪-Frobenius

}⇐⇒

{∪ is Y -associative

and Y ⊥ 6= 0

}{Y is ∪-Frobenius

}⇐⇒

{∪ is Y -associative

and invertible

}(3)

Moreover solvable ∪-Frobenius algebra is said to be ∪-symmetric if theFrobenius structure is symmetric, ∪T = ∪.

Note that the Cartan-Killing trace Y -form, a turtle ∪ = ∪(Y ), isonly the Y -associative form for a Lie algebra Y, ∪ is Y -invariant metric.

3. Operad of graphs

We work within a bi-closed monoidal category (another name atensor category) generated by a single object and its dual, two colors.In the present section we disregard duality. In this case the set of allobjects of the single generated (one-color) monoidal category coincides

1Arrows (= morphisms) are parallel if they possesses the same arity-in and thesame arity-out

FROBENIUS ALGEBRA 7

with the set of non-negative integers (with the set of natural numbers)N. Thus the set of all objects is, obj cat = N, and 1 ∈ N is a generatingobject.

Therefore each morphism (an arrow of a category) within this fixedcolor, is characterized by a pair of non-negative integers, morphims arebi-graded, and we refer to this pair {input, output} = {entrance, exit},as to the type or arity of the morphism = {arity-in, arity-out},

N 3 m morphisms of (m→ n)-arity−−−−−−−−−−−−−−−−−−−→ n ∈ N.(4)

Garrett Birkhoff in his Lattice Theory in 1940, within Universalalgebra considered an n-ary operation symbol to be carefully distin-guished from a model of the operation. Every such operation is consi-dered here to be a morphism in a monoidal category. For example thetraditional 0-ary (or null-ary) operation in our terminology must bebi-graded. It is a (0 → 1)-operation/morphism. Similarly traditionaln-ary operations are seen here as morphisms ∈ cat(n, 1).

We need to introduce the graphical notation we are using. Everymorphism ∈ cat(m,n) is visualized as a node with a number of outerleaves (representing the source and target objects): on top the inputobject of m-arity-in, and at bottom the output object of n-arity-out asillustrated in Figure 7. Throughout this paper the graphs are directedand we read them from the top to the bottom.

For example, a morphism ∈ cat(2, 1) in Figure 7 is traditionallycalled a binary algebra. Each morphism is a graph with no outer nodes.Every node is inner, and every edge is an outer leave or an inner edge,which represents an object of a monoidal category.

•

. . .

. . .

︷︸︸︷input: m

output: n︸︷︷︸•

︷︸︸︷input: 2

output: 1

Figure 7. A (m→ n)-morphism is an arrow from m ton, and (2→ 1)-morphism ∈ cat(2, 1)

Two morphisms with the same arity-in and arity-out are said to beparalell, they are parallel arrows, m⇒ n.


In abelian monoidal category cat, we postulate a set, k ≡ cat(0, 0),to be a ring, and, in particular, to be an associative commutative fieldwith unit 1 6= 0.

Morphisms can be composed in many different ways, like gardenplants are grafted. Composition of morphisms generalizes the con-struction of words from an alphabet, however words are constructedby concatenations only, whereas our bigraded morphisms allow diverse‘non-linear’ compositions.

Figure 8. Alphabet of three letters: Y ∈ cat(2, 1), a∪ ∈ cat(2, 0), and, ∩ ∈ cat(0, 2).

We refer to each composition as a grafting. The concatenation ofarrows is considered also as a special grafting. For example a concate-nation of an arrow from cat(2, 1) with another arrow from cat(2, 1),gives an arrow from cat(4, 2), see Figure 9.

Figure 9. The concatenation of Y ∈ cat(2, 1) with Ygives an arrow in cat(4, 2).

Except for concatenations that will be not used in what follows,each grafting is also bigraded, and (i → j)-graft means joining of thei-th output of the first (m→ n 6= 0)-morphism, i ∈ {1, 2, . . . , n}, withj-th input of the second (0 6= k → l)-morphism, j ∈ {1, 2, . . . , k}, asillustrated in several examples in Figures 19, 11, 16, 17.

Of special importance for what follows are morphisms from ∪ ∈cat(2, 0), and ∩ ∈ cat(0, 2). We say that a pair of such morphisms ismutually inverse to each other, or that they are invertible (nondegen-erate), if the condition on Figure 19 holds.

∼∼

Figure 10. The Reidemeister moves zero 0. Cancella-tion of maxima and minima [Kauffman 2001, 2002].

FROBENIUS ALGEBRA 9

Reidemeister in 1932 introduced three moves, I, II, III, each oneinvolving 1-, 2-, 3-, crossings. Figure 19 shows the move zero, no cross-ings, see, e.g. [Kauffman 2001].

The double grafting, (1 ← 1)-grafting & (2 ← 2)-grafting, of thecoevaluation with evaluation gives the graphical definition of the di-mension of the generating object as shown on Figure 11.{

The dimension ofthe generating object

}≡ trace id ≡ ∈ N ⊂ cat(0, 0).

Figure 11. The dimension of the generating object:dim = trace |= evaluation ◦ coevaluation.

From alphabet in Figure 1 we can graft two chiral three-letter wordsfrom cat(1, 0). This word is a one-sided trace of an algebra Y as shownon Figure 18.

≡

Figure 12. Two compound chiral words ∈ cat(1, 0).Each |Y | ≡|-form grafted from three letters: Y and co-evaluation and evaluation.

The self-grafted trees from a morphisms Y ∈ cat(2, 1) define ‘thelower central series’ of ideals, in analogy to the lower central seriesof a group and series of ideals of a Lie algebra. Every k-algebra Y ∈cat(2, 1), self-grafted, Y ◦Y, gives two chiral (the mirror image) parallelmorphisms, Y ◦ Y ∈ cat(3, 1), shown on Figure 13. A k-algebra Y issaid to be associative if these parallel morphisms are equal.

Figure 13. Two different graftings of Y with Y, Y ◦ Y ∈ cat(3, 1).

Applying three different traces to each of two chiral ‘Y ◦ Y ’ onFigure 13 gives the five Y -forms, all in cat(2, 0), defined in Figures14-15.

3.1. Exercise. The Cartan-Killing trace forms (turtles) are sym-metric. Let Y be either commutative or anti-commutative, then theright-turtle is equal to the left-turtle.


Figure 14. Definition: the two Cartan-Killing traceforms baptized as chiral turtles ∈ cat(2, 0), right andleft. Each grafted from four letters.

≡

Figure 15. Definition: the deer, and right - left chiralsnails ∈ cat(2, 0). Each grafted from four letters.

These five Y -forms play a fundamental role in the present paper,they will be used frequently in what follows, therefore we found itconvenient to baptize them as (right and left) turtle, snail and deercorrespondingly. In fact all these five Y -forms carry important infor-mation about an algebra Y, and to be more precise one must refer tothem as Y -turtle, Y -snail and Y -deer.

3.2. Exercise. Let an algebra Y be associative. Show the followingequalities hold

= =

3.3. Exercise. Let a morphism Y be either commutative or anti-commutative, Y T = ±Y. Then Y -snails are equal up to sign, and Y -deer is symmetric.

= ± =

{ }T

3.4. Exercise. Consider anti-commutative morphism Y T = −Y ∈cat(2, 1), that does not need to be Lie. Show

• dim = 2. The snails vanishes. The Cartan-Kiling trace-form(any turtle) is the only one independent Y -form. This Cartan-Killing trace form is Y -associative and degenerate. Y is solv-able but does not need to be Lie.• dim = 3. Let Y be Lie or associative. Then the Cartan-

Killing trace Y -form (the right turtle) is non-degenerate andY -associative. In other words anti-commutative associative Yis Frobenius (semi-simple).

FROBENIUS ALGEBRA 11

In Figure 16 more sophisticated graftings are illustrated. Considertwo morphisms, one binary and another ternary,

C ∈ cat(2, 1) and A ∈ cat(3, 1).(5)

They can be grafted as shown in Figure 16, giving two different mor-phims ∈ cat(3, 0). We use the alphabet from Figure 8.

A

C

A C

Figure 16. On the left a ternary A ∈ cat(3, 1) graftedwith the right trace of C ∈ cat(2, 1) gives a ‘grasshopper’F ≡ (traceC) ◦A ∈ cat(3, 0). On the right the entangleddouble grafting gives another 3-arity-in ‘animal’, denoted

by F ∈ cat(3, 0), among many other alternative grafting.

We show in Section 7 that both 3-arity-in compound morphismsfrom Figure 16 define the Frobenius ideal at least in dimension two.

There are two different graftings of Y ∈ cat(2, 1) with ∪ ∈ cat(2, 0),see Figure 17.

1 = 2 =

Figure 17. (1← 1)-grafting and (2← 1)-grafting.

As usual, every composition of morphisms gives rise to enrichedmorphisms ‘among hom-objects’, i.e. two functors, the push-forwardand pull-back,

∪ ◦ Y =

{∪∗ Y push-forward

Y ∗ ∪ pull-back(6)


The two different compositions of Figure 17, i , lead to two differentpush-forwards and pullbacks,

∪ i Y = ∪∗i Y = Y ∗i ∪,(7)

cat(2, 1)∪∗i−−−−−→ cat(3, 0)(8)

cat(2, 0)Y ∗i

−−−−−→ cat(3, 0).(9)

In this way the one-color monoidal category with hom-sets {cat(m,n)}has been enriched in (8)–(9) to hom-objects cat(m,n) of an enrichedcategory.

4. Clifford algebra of one-dimensional space

The concept of the Frobenius algebra was introduced by GeorgFrobenius in 1903. Every matrix ring is an example of the Frobeniusalgebra with the trace form as a Frobenius form. However in generala given algebra allows that exists a manifold of inequivalent Frobeniusforms.

Clifford algebra C` possess a manifold of Frobenius forms {ε =ε(C`)}, dim{ε} = dim C`. Therefore C` is a Frobenius algebra {C`, ε}in a many inequivalent ways. One example of different Frobenius formson Clifford algebra of complex numbers C is considered in [Kock 2003,page 99, Example 2.2.14].

We believe that the investigation of the manifold of all Frobeniusforms allowed by a given Clifford algebra C`, and therefore also a ma-nifold of all Frobenius structures given by scalar products, ∪ ≡ ε◦C` =∪(C`), compatible with C`, that study of these up to now hidden Frobe-nius structures will enrich the understanding of what is a Clifford al-gebra, anyway? It is a Frobenius algebra in many different ways. Itis also a manifold of coalgebras, identified with the graphical camels,≡ C` ◦ ∩.

If C` denotes a Clifford algebra, then |C`| denotes an underlyingvector space.

A scalar product on a vector space |C`|, |C`| ⊗ |C`| ∪−−−−→ R, C`-compatible, that is baptized in literature (missleadingly) to be C`-associative, ∪ = ∪(C`), must be not confused with a scalar producton the generating vector space of the Clifford algebra! In particular, afixed uniquely scalar product on a generating space, determine a mani-fold of Frobenius scalar products on C`. Our main question is what isthe relevance of a manifold of Frobenius scalar products and a manifoldof coalgebras (seen as graphical camels) for mathematics, physics, andengineering computing of Clifford algebras.


We are going to elucidate a (two-dimensional) manifold of all Frobe-nius forms for two-dimensional Clifford algebra of one-dimensional vec-tor space of arbitrary signature. This seems to be a trivial case, how-ever still it is showing the richness of yet ignored manifold of Frobeniusstructures of the Clifford algebra. This will leads to a manifold ofall Frobenius structures, i.e. all scalar products on a vector space |C`|compatible with an algebra C`, denoted here as Y ≡ C`. Moreover weare showing entire manifold of coalgebras defined as graphical camels.In particular we are showing that only one specific Frobenius form,exactly the one-sided partial trace of an algebra, i.e. only one specificand not expected a priori coalgebra leads to a bialgebra structure of C.This last result rise a conjecture that the Clifford coalgebra that fits tobialgebra must involve a factor 1/ dim C`.

Before elucidating a manifold of all Frobenius forms of a given bi-nary algebra, we must stress that every algebra possess two naturalchiral (left and right) forms given by partial trace (left and right). Apriori these forms need not to be a Frobenius forms. Namely fromalphabet in Figure 1 we can graft two chiral three-letter words fromcat(1, 0). This word is a one-sided trace of an algebra Y as shown onFigure 18.

≡

Figure 18. Two compound chiral words ∈ cat(1, 0).Each |Y | ≡|-form grafted from three letters: Y and co-evaluation and evaluation.

Let’s calculate these forms as a trace of a Clifford algebra.Let σ ∈ {−1, 0,+1} be signature of the (necessarily symmetric)

metric tensor g of the one dimensional vector space

σ =

−1 for complex numbers C0 for Grassmann algebra of ‘dual numbers’

+1 for double numbers

(10)

The basis of |C`(σ)| is 1, i ∈ |C`|. where σ = i2 ≡ g(i⊗ i).The dual basis in |C`|∗ is as follow. A form, < ≡ 1∗ ∈ |C`|∗, take

the real part, and, = ≡ i∗ ∈ |C`∗|, take the ‘imaginary’ part,

1∗1 = 1, i∗1 = 0

1∗i = 0, i∗i = 1.(11)


The mixed tensor C` = C = Y of multiplication is as follows

Y = 1⊗ (1∗ ⊗ 1∗ + σ i∗ ⊗ i∗) + i⊗ (i∗ ⊗ 1∗ + 1∗ ⊗ i∗).(12)

The trace of Clifford Y is = 2 · 1∗.

4.1. Theorem (All Frobenius forms). The following form

ε = r1∗ + si∗ ∈ |C`|∗, r ≡ ε1 and s ≡ εi,(13)

is a Frobenius form for C`(σ) if and only if

r2σ 6= s2.(14)

Proof.

∪ ≡ ε ◦ Y = r(1∗ ⊗ 1∗ + σ i∗ ⊗ i∗) + s(1∗ ⊗ i∗ + i∗ ⊗ 1∗),(15)

∩ =1

r2σ − s2{r(σ 1⊗ 1 + i⊗ i)− s(1⊗ i+ i⊗ 1)}. �(16)

One can check that the above morphisms, ∪ ∈ cat(2, 0), and ∩ ∈cat(0, 2), are mutually inverse to each other, or that they are invertible(nondegenerate), because the condition on Figure 19 holds.

∼∼

Figure 19. The Reidemeister moves zero 0.

Reidemeister in 1932 introduced three moves, I, II, III, each oneinvolving 1-, 2-, 3-, crossings. Figure 19 shows the move zero, no cross-ings.

The Cartan-Killing form for C` is within a manifold (15), for

s ≡ εi = 0 and r ≡ ε1 = dim C` = 2,(17)

Cartan = 2 · (1∗ ⊗ 1∗ + σ i∗ ⊗ i∗).(18)

4.1. Clifford coalgebra as camel. Every C`-associative and in-vertible scalar product (15)-(14) give rise to a Clifford coalgebra, de-noted here by a more suggestive symbol, ≡ Y ◦∩, instead of Sweedler’s4.

=

Figure 20. Two compound chiral camels ∈ cat(1, 2).Each camel is grafted from two letters: Y and ∩ (16).


We are ready to show a manifold of all Clifford coalgebras. Ourconvention for evaluation of tensors is ‘vertical’ as follows. Let α, β becovectors, and v, w be vectors. Then

ev{(α⊗ β)(v ⊗ w)} ≡ ev(αv) · ev(βw).(19)

(20) (r2σ − s2) = (σ 1⊗ 1 + i⊗ i)⊗ (r1∗ − si∗)+ (1⊗ i+ i⊗ 1)⊗ (rσi∗ − s1∗).

Frobenius form ε (14) is counit for Clifford coalgebra,

(ε⊗ id) ◦ = id = (id⊗ ε) ◦ .(21)

Analogously to a trace of Y what is shown on Figure 18, there isalso a trace of as a word in cat(0, 1), that does not need a priori tobe a unit for Y,

trace =2

r2σ − s2(σr · 1− s · i).(22)

Indeed for σ 6= 0 and for s = 0, r = 2, this is unit for Y.

4.2. Notation. By abuse of notation we use the same letter i fora vector i ∈ |C`|, and for a Clifford multiplication by i ∈ End C`,

C` 3 z 7−→ iz ∈ C`,(23)

i ≡ i⊗ 1∗ + σ 1⊗ i∗ ∈ End |C`|, i2 = σ · id .(24)

In (24), i on the left stand for endomorphism, z 7−→ iz, whereas on theright i stand for a vector, i ∈ |C`|.

The handle endomorphism is defined as the composition Y ◦ ,

Y ◦ (ε) =2

r2σ − s2(rσ· | −s · i) ∈ End |C`|.(25)

4.2. Frobenius algebra. For each Frobenius form, the Cliffordalgebra and the Clifford coalgebra as a camel, satisfy the followingFrobenius conditions,

(26) (Y⊗ |) ◦ (| ⊗ ) = ◦ Y = (| ⊗Y ) ◦ ( ⊗ |)

=r(σ | ⊗ | + i⊗ i)− s( | ⊗ i+ i⊗ | )

r2σ − s2.


∼

Figure 21. Identity (27) for Clifford algebra of one-dimensional space.

4.3. Bialgebra. Let X denotes a permutation X ◦X ≡ | ⊗ | . ForClifford algebra of one-dimensional vector space the following identityholds

(Y ⊗ Y ) ◦ (| ⊗X⊗ |)( ⊗ ) = ( ◦ Y )2.(27)

4.3. Theorem (Frobenius algebra as bialgebra). Consider the Clif-ford algebra of one-dimensional space with signature σ and with Frobe-nius form ε. Then this Frobenius algebra {C`, ε} is a bilagebra if andonly if

ε i = s = 0, and ε 1 = r = 2.(28)

Proof. The conditions (28) are equivalent that ◦ Y is idempo-tent. The following two conditions are necessary and sufficient thatthis Frobenius algebra is a bialgebra,

s(r2σ − 4rσ − s2) = 0,

(r − 2)r2σ = s2(r + 2)

}⇐⇒ s = 0 and r = 2. �(29)

In particular by these conditions a Grassmann algebra, σ = 0,can not be a bialgebra, but it is a Frobenius algebra for the followingcoalgebra,

1 = 1⊗ i+ i⊗ 1 + r i⊗ i,i = i⊗ i.(30)

If σ 6= 0, then Frobenius algebra is a bialgebra for s = 0 and r = 2only,

1 = 12(1⊗ 1 + σ i⊗ i),

i = 12(1⊗ i+ i⊗ 1).

(31)

A factor 12

is not expected a priori.At the first sight seems that a Frobenius form is defined up to non-

zero scalar. Therefore we have in fact two one-parameter families ofFrobenius forms (covectors) ε ∈ C`∗ ' cat(1, 0), linear functionals in


[Kock 2.2.3 on pages 94–95]. Kock’s formulas one must read from theleft to right, we write as in old times, from the right to left,

ε =

{1∗ + s i∗ if s2 6= σ,

i∗ + r1∗ if r2σ 6= 1.(32)

What is the nullspace (the front) of ε (32)?

ε(s− ri) = 0, front of {ε = r1∗ + si∗} ' s1− ri.(33)

5. Digression on Lie algebras

Elie Cartan in his Doctoral Thesis in 1894, introduced theCartan-Killing trace form ∈ cat(2, 0) for a Lie algebra Y (often calledthe Killing form, but this concept was not introduced by Killing). Wewill visualize this form in the operad of graphs within the bi-closed(two-color) monoidal category with duality in section 2, adleft Y = ,

= trace(◦

)One can show (exercise) that the Cartan-Killing trace Y -form is the

same as the Y -turtle in Figure 14. We know that turtles are the samefor anti-commutative Y T = ±Y. Therefore one can ask where are deerand snails within Lie algebras? We must look for the consequencesof the Jacobi non-associativity (equivalent to the derivation, and alsocalled as the Jacobi associativity or Jacobi identity).

Within an abelian monoidal category, see section 6, the Jacobi non-associativity (equivalent to the Leibniz derivation) is as follows

− =

Figure 22. The Jacobi non-associativity or the Leibnizderivation is an identity ∈ cat(3, 1).

A generalization of the the Laibniz derivation to pair of binaryoperation is said to be the Leibniz or the Loday algebra.

5.1. Exercise. Let Y be Jacobi non-associative. Then

• Both Y -snails vanishes• Moreover we have the following relation among Y -deer andY -turtle,


+ = 0

5.2. Corollary (Lie algebra). Let Y be a Lie algebra, i.e.Y beJacobi non-associative and anti-commutative. Then Y -snails vanishes,the left and right Cartan-Kiling trace-forms (turtles) are equal, and aturtle is equal minus deer.

Therefore the Lie algebra Y possess only one independent Y -form,exactly the Cartan-Killing trace Y -form (Y -turtle). The Cartan-Killingtrace Y -form is Y -invariant with respect to adjoint representation ofthe Lie algebra. This last condition is identical with the condition tobe Y -associative (exactly the same as for the Frobenius algebra Y ).Therefore for a form ∈ cat(2, 0) to be Y -associative (for a Frobeniusalgebra) is exactly the same as to be Y -invariant (for a Lie algebra).

Jacobson, following Dieudonne, considered a Y -associative symmet-ric form for arbitrary non-associative algebra, generalizing the traceform for a Lie algebra, [Jacobson 1962, Chapter III §5, page 70-71].On the other hand the associative Frobenius algebra Y possesses aY -associative form. Therefore we consider that it is worthwhile to con-sider Y -associative forms for arbitrary algebra, i.e. for arbitrary non-associative and associative algebra. An arbitrary algebra Y admittingY -associative invertible form we shall call a Frobenius algebra.

Elie Cartan proved in his These in 1894 that a Lie algebra Y is semi-simple (no non-zero commutative ideal) if and only if its Cartan-Kllingtrace Y -form (a Y -turtle) is invertible. Therefore we can conclude thatthe semi-simple Lie algebra is an example of the Jacobi non-associativeFrobenius algebra.

6. What it is a Y -associative scalar product?

In what follows we assume that the enriched monoidal categoryabove is abelian. This is the case in particular if each hom-set ofmonoidal category cat(m,n), is an additive abelian group. This willbe specified later on to be an abelian category of k-modules for thecommutative ring k, or even to the category of k-bimodules.

Within an Abelian category the Frobenius condition of Figure 6can be recast as the following ‘enriched’ and entangled system,

cat(2, 1)∪∗1−∪∗2−−−−−−−−→ cat(3, 0)

Y ∗1−Y ∗2←−−−−−−−−− cat(2, 0)(34)

(∪∗1 − ∪∗2)Y (∪) = 0 ∈ cat(3, 0),(35)

(Y ∗1 − Y ∗2) ∪ (Y ) = 0 ∈ cat(3, 0).(36)


Invertibility of the enriched morphism (∪∗1 − ∪∗2) implies Y = 0,therefore we must look for necessary and sufficient conditions on ∪such that the enriched morphism (∪∗1−∪∗2) is not invertible, and suchrestricted ‘subobject’ ∪ must be understood in both equations (35) and(36).

This implies that in fact the enriched system (35)-(36) is entangledin the meaning that both morphisms, ∪ ∈ cat(2, 0) and Y ∈ cat(2, 1),must be restricted to sub-morphisms such that both enriched mor-phisms are not invertible,

det (∪∗1 − ∪∗2) = 0 and ∀ det (Y ∗1 − Y ∗2) = 0.(37)

In (37) ‘∀ det’ means for all determinants of square matrix of maximalrank.

Not all co-authors of the present paper agree with the above entan-glement (35)-(36)-(37). There is also the opinion that the equations(35) and (36) must be considered as totally equivalent (redundant),but we are not yet able to come to consensus.

Our aim is to solve the above entangled system (35)-(36)-(37), inthe case dim = 2.

6.1. Proposition. We found the following necessary or sufficientcondition on invertible scalar product ∪ that the enriched morphism(∪∗1 − ∪∗2) is not invertible.

• If dim = 1, 2, ∪ must be symmetric.• If dim > 2, it is sufficient, but not necessary that ∪ is sym-

metric, but must not be skew-symmetric.

Our result allows a scalar product ∪ that is not necessarily symmet-ric in dim > 2, but from another side it does not allow skew-symmetricscalar products.

We found the possibility of an Y -associative skew-symmetric scalarproduct (bilinear form) mentioned only in the following theorem byOsborn.

6.2. Theorem (Osborn 1972,Theorem 8.2, page 246). Let A be afinite-dimensional algebra over a field K, let A have a nondegenerateassociative bilinear form which is either symmetric or skew-symmetric,and let A contain no nonzero ideals C such that CC = 0. Then A isa direct sum of ideals which are simple algebras, and the restriction ofthe bilinear form to each summand is nondegenerate.

Our aim is to find the necessary and sufficient conditions on analgebra, Y ∈ cat(2, 1), such that the enriched map (Y ∗1 − Y ∗2) is notinvertible. In the case of dim = 2 this enriched pull-back must be


restricted necessarily to symmetric scalar products (36) only. This willensure the existence of non-trivial Y -associative scalar products.

7. Frobenius ideal

Let for simplicity k ≡ cat(0, 0) be an associative and commutativering with unit 1 6= 0. For every binary k-algebra Y ∈ cat(2, 1), binarymeans arity-in = 2, we denote by k[Y ] the commutative ring of allpolynomials in all components of algebra Y with coefficients in thering k ≡ cat(0, 0).

There are two important finitely generated ideals in the above ringk[Y ]. The commutative ideal we denote by C ⊂ k[Y ], is finitely gene-rated by the following polynomials of degree 1,

C ≡ Y − Y opp, Ckij ≡ Y k

ij − Y kji ∈ cat(0, 0),(38)

C ≡ (Y kij − Y k

ji) ≡ gen{Y kij − Y k

ji} ⊂ k[Y ].(39)

Another finitely generated ideal is the associative ideal A genera-ted by all n4 polynomials, each of degree two, originating from theassociators. This ideal is denoted as follows

A = − ⊂ k[Y ].(40)

The main subject of the present paper is another finitely genera-ted ideal, denoted by F ⊂ k[Y ], and referred to in the sequel as theFrobenius ideal. The condition F (Y ) = 0 ensures that Y has a non-zero Y -associative ∪ 6= 0, however this ∪ need not be invertible. Ifin addition this Y -associative ∪ happens to be invertible, than Y is a∪-Frobenius algebra.

The Frobenius ideal is defined in terms of a system of equations(35)-(36) as follows. Consider equation (36) for

0 6= ∪ ∈ cat(1, 1∗) ' cat(2, 0).(41)

This equation in dim = n, is a system of n3-scalar linear equations forindependent scalar components of ∪. A non-trivial ∪ 6= 0, prevents thevanishing of the ideal generated by determinants of all

(3× 3)− submatrices for dim = 2,

(9× 9)− submatrices for dim = 3.(42)

Each such determinant is polynomial in Y of a degree 3 and 9correspondingly.


7.1. Definition (Frobenius ideal). The Frobenius ideal, F ⊂ k[Y ],is finitely generated by a set of(

8

3

)− polynomials in Y for dim = 2,(

27

9

)− polynomials in Y for dim = 3.

(43)

Each generating polynomial of Frobenius ideal F is the determinant ofsome submatrix (42).

7.2. Definition (Solvable Frobenius algebra). A k-algebra Y issaid to be solvable Frobenius if F (Y ) = 0. A solvable Frobenius k-algebra Y possesses a non-zero Y -associative scalar product that neednot necessarily be invertible.

7.3. Theorem. Let the morphism F ∈ cat(3, 0) be defined as inFigure 16. At least in dimension two, the Frobenius ideal F is gene-rated by the components of F , F = gen{F}. The algebra Y is solvableFrobenius if and only if F (Y ) = 0.

7.4. Corollary. In dim = 2, every commutative algebra (but notnecessarily associative), and every associative algebra (but not nec-essarily commutative), possesses non-zero associative scalar products,i.e. is a solvable Frobenius algebra. In other words the Frobenius idealis a proper subideal of the intersection

F ⊂ (C ∩ A).(44)

Proof. For dim = 2, the Frobenius ideal is expected to be genera-ted quite generally by

(83

)= 56 polynomials of degree 3 in Y. However

‘accidentally’ this number of generators is reduced to only(43

)= 4.

Each determinant of (3 × 3)-sub-matrix of the following (3 × 4)-matrix, is a generating polynomial,

Y 121 − Y 1

12 Y 221 − Y 2

12 00 Y 1

21 − Y 112 Y 2

21 − Y 212

−Y 112 Y 1

11 − Y 212 Y 2

11

−Y 122 Y 1

12 − Y 222 Y 2

12

�(45)

7.5. Conjecture. It is a challenge to extend the Corollary (44) forevery finite-dimensional pre-Frobenius algebra. However our prelimi-nary Singular package symbolic calculations do not leave much hopefor this. Moreover we believe that Theorem 7.3 can be generalizedappropriately for every finite-dimensional pre-Frobenius algebra.


8. Commutative Frobenius algebras

8.1. Theorem. At least in dimension two, the following basis-freescalar product, turtle - snail, is the only Y -associative scalar productfor the commutative non-associative Y,

∪(Y ) = −

In other words every commutative non-associative Y is ∪-Frobenius ina unique way.

Proof. In dim = 2 a Y -associative ∪ must necessarily be symmet-ric, i.e. all ∪-Frobenius algebras must be ∪-symmetric. We look for theunknown Y -associative ∪ = ∪(Y ), which is the solution of the tensorequation, Figure 6. If Y is commutative we arrive at the followingsystem of two equations,(

−Y 112 Y 1

11 − Y 212 Y 2

11

−Y 122 Y 1

12 − Y 222 Y 2

12

) ∪11∪12 = ∪21∪22

=

(00

)(46)

8.2. Lemma. The system of two equations (46) is linearly indepen-dent if and only if Y is non-associative.

Proof of Lemma 8.2. The determinants of (2× 2)-sub-matricesare as follows,

det

(−Y 1

12 Y 111 − Y 2

12

−Y 122 Y 1

12 − Y 222

)= +A1

122 = −A1221,

det

(−Y 1

12 Y 211

−Y 122 Y 2

12

)= −A1

112 = A2122 = A1

211 = −A2221,

det

(Y 111 − Y 2

12 Y 211

Y 112 − Y 2

22 Y 212

)= −A2

112 = +A2211.

�(47)

Proof of Theorem 8.1. For the explicit expression for the Y -associative scalar product ∪ ∈ cat(2, 0), one needs to consider threecases, each given by an inequality.

8.1. ∪22 = −A1122 = A1

221 6= 0.(∪11

∪12

)=

(Y 112 − Y 2

22 Y 212 − Y 1

11

Y 122 −Y 1

12

)(Y 211

Y 212

)∪11 = A2

112, ∪12 = ∪21 = −A1112.

(48)


8.2. ∪12 = ∪21 = −A1112 = A2

122 6= 0.(∪11∪22

)=

(Y 212 −Y 2

11

Y 122 −Y 1

12

)(Y 212 − Y 1

11

Y 222 − Y 1

12

)∪11 = A2

112, ∪22 = −A1122.

(49)

8.3. ∪11 = A2112 6= 0.(∪12∪22

)=

(−Y 2

12 Y 211

Y 112 − Y 2

22 Y 212 − Y 1

11

)(Y 112

Y 122

)∪12 = ∪21 = −A1

112, ∪22 = −A1122.

�(50)

8.3. Example. The following non-associative and non-unital k-algebra Y,

Y (e1 ⊗ e1) = e2,

Y (e1 ⊗ e2) = e1 + e2,

Y (e2 ⊗ e2) = e1,

(51)

is(1 00 1

)-Frobenius algebra. Here, A2

112 = 2, A1112 = 0, and A1

122 = −2.

8.4. Conjecture. Theorem 8.1 suggest the plausible hypothesisthat every finite-dimensional non-associative commutative Frobeniusalgebra Y possesses a Y -associative scalar product given as a somelinear combinations of the Y -turtle, Y -snail and Y -deer.

8.5. Exercise. Find that in dim = 2 and for Y T = Y the Y -deeris symmetric.

8.6. Comment. The following inverse is the result of the bruteforce, and it is awkward.

(det ∪)∪−1 =∑

kAikkj ei ⊗ ej.(52)

Better expression needs some right inverse of associator, A ◦A−1 ≡ id,or, a co-associator of a co-algebra, {Apqri } ∈ cat(1, 3).

8.7. Comment (Kock page 98). Joachim Kock made an importantRemark [Kock, Remark 2.2.10 on page 98]. If the particular algebraY admits a Frobenius Y -associative and invertible scalar product ∪(called also the Frobenius pairing or the Frobenius structure), this notimply that the Frobenius structure ∪ must necessarily be unique upto some overall factor. The Frobenius algebra must be seen as a pair{Y,∪}, an algebra Y is ∪-Frobenius, and different Frobenius structuresdetermine different ∪-Frobenius algebras. Theorem 8.1 shows that thisis not the case for non-associative commutative and symmetric Frobe-nius algebra at least in two-dimension. Namely the non-associtivity


of algebra Y necessarily implies the unique (up to an overall factor)Frobenius structure, Y -associative ∪ ∈ cat(2, 0) if det ∪ 6= 0.

However the associative Frobenius algebras can admit a parameter-dependent family of different Frobenius structures. In the next Sectionwe will determine, in particular, all Frobenius structures for the algebraof complex numbers C. One can see C in many different ways as ∪-Frobenius algebra.

8.8. Exercise. In dim = 2, there are the following relations

A2121 + A1

111 = 0

A2122 + A1

112 = 0

A2221 + A1

211 = 0

A2222 + A1

212 = 0

(53)

8.9. Exercise. In dim = 2, dozen of associators are inside of thecommutative ideal,

Ai111, Ai121 Ai212, Ai222 ∈ C ∩ A,Ai221 + Ai122, Ai211 + Ai112 ∈ C ∩ A.

(54)

9. Commutative associative Frobenius algebras

9.1. Theorem. All two-dimensional commutative associative ∪-Frobenius algebras possesses the continuous three- two- or one-parameterfamily of Y -associative ∪’s.

In dim = 2, the other alternative is that the system (46) is linearlydependent, and this implies that Y must be necessarily associative.The linear dependence of the system (46) means the following twoalternatives for arbitrary scalars, for λ or for µ,

Y 122 = λY 1

12, Y 212 = λY 2

11, Y 112 = Y 2

22 + λY 111 − λ2Y 2

11,

or

Y 211 = µY 2

12, Y 112 = µY 1

22, Y 212 = Y 1

11 − µ2Y 122 + µY 2

22.

(55)

The above relations, together with (54), imply (and are equivalent)that Y must be necessarily associative.

9.1. Proof: all two-dimensional commutative associativeFrobenius algebras. If a commutative algebra Y is associative then


the Frobenius equation (46), using (55), is reduced to one of the fol-lowing two alternative forms,

(−Y 222 − λY 1

11 + λ2Y 211) ∪11 +(Y 1

11 − λY 211) ∪12 +Y 2

11∪22 = 0,

−Y 122 ∪11 +(µY 1

22 − Y 222) ∪12 +Y 2

12∪22 = 0.(56)

This implies to consider six separate cases of Y -associative ∪’s. Inwhat fallows r, s, µ, λ ∈ k are arbitrary scalars.

9.1.1. Y 122 6= 0.

∪ =

(r(Y 2

22 − µY 122)− sY 2

12 −rY 122

−rY 122 −sY 1

22

)9.1.2. Y 2

12 6= 0.

∪ =

(rY 2

12 sY 212

sY 212 rY 1

22 + s(Y 222 − µY 1

22)

)9.1.3. Y 2

22 6= µY 122.

∪ =

(r(µY 1

22 − Y 222) Y 1

22r − Y 212s

Y 122r − Y 2

12s s(µY 122 − Y 2

22)

)9.1.4. Y 2

22 6= λ2Y 211 − λY 1

11.9.1.5. Y 1

11 6= λY 211.

9.1.6. Y 211 6= 0.

10. Non commutative Frobenius algebras

10.1. Anti-commutative algebras.

10.1. Exercise. Let Y 6= 0 be anti-commutative and associative.Show that every three-dimensional such Y is Frobenius with respect tothe Cartan-Killing form, with respect to the right Y -turtle.

10.2. Two-dimensional algebras. In dim = 2 the reduced Grobnerbasis of the Frobenius ideal F = gen{F} (45) consists of a set four po-lynomials.

10.2. Proposition. Let k be associative and commutative field withunit 1 6= 0, and V be k-space of dimk V = 2. The Frobenius idealF = gen{F}, generated by all eight components of three-hand F asdisplayed on Figure 16, possesses the Grobner basis of four followingpolynomials

Fpqr ≡∑

AipqrCjij, Fpqr ≡

∑AipjqC

jir ∈ k,(57)

F221 = F212, F122 = F122,

F112 = F121, F121 = F112.(58)


In dim = 2 we must determine Y -dependent determinants of all(2× 2)-sub-matrices of Frobenius matrix (45),(

3

2

)×(

4

2

)= 18.(59)

These determinants will allow to determine the Y -associative symmet-ric scalar products, ∪ = ∪(Y ), not yet necessarily invertible.

10.3. Problem. A solvable Frobenius k-algebra is said to be notcommutative if C(Y ) 6= 0, i.e. if (C1

21)2 + (C2

21)2 6= 0. Solve (58) for not

commutative two dimensional algebras.

10.4. Example. A two-dimansional Frobenius k-algebra that is noncommutative, C1

21 6= 0, and non associative. We put in (58), C221 = 0

and

A2212 = Y 1

12Y221 − Y 2

12Y121 − Y 2

22C221 = 0

A1111 = −Y 2

11C121 = 0

A2112 = Y 2

12(Y212 − Y 1

11) + Y 211(Y

112 − Y 2

22) = 0

A2122 = Y 2

11Y122 − Y 1

12Y212 = 0

(60)

This gives

Y 211 = 0, Y 1

12Y212 = 0, . . . Y 2

11 = Y 212 = Y 2

21 = 0.(61)

Then F (Y ) = 0. We will find the Y -associative scalar product forthe above algebra.

Frobenius matrix (45) specified for the above algebra givesY 121 − Y 1

12 0 00 Y 1

21 − Y 112 0

−Y 112 Y 1

11 0Y 122 Y 2

22 − Y 121 0

(62)

The choice of the first and the the last rows gives the followingequation for Y -associative scalar product

if Y 222 = Y 1

12 6= Y 121,(63) (

C121 0 0Y 122 Y 2

22 − Y 112 0

) ∪11∪21 = ∪12 = 1

∪22

=

(00

)(64)

Therefore we have the following Y -associative scalar product

∪11 = 0, Y 122∪11 = (Y 1

12 − Y 222)∪21, ∪22 = 1.(65)

Another choices of the set of two another rows in the matrix (62) cangives another Y -associative scalar product for the same non-associative


algebra Y. This illustrate that the given algebra can allows a lot of dif-ferent Frobenius structures, with different Y -associative scalar prod-ucts. The list of all Y -associative scalar products is given by the listof all 2× 2 -sub-matrix of the matrix (62), with no zero determinants.These not zero determinants do not assure yet that the resulting Y -associative scalar product is invertible.

Looks like that all solutions of the Frobenius ideal (45), for noncommutative algebra - the following hypothesis:

(C121)

2 + (C221)

2 6= 0 =⇒ ∪ is not invertible.(66)

In another word, a ∪-radical Y ⊥ 6= 0.

11. S3-permuted Frobenius algebra

Greg Wene in lecture in Nayarit in Janury 2010, proposed to extendS2-opposed algebra (S2-permuted) to S3-permuted algebra, where S3

is symmetric group of order 3! = 6 of permutations of a set of threeelements. Such S3-permutation to be unambiguous needs a priori theFrobenius algebra.

The unipotent permutation (a transposition, a trivial braid ∈ S2)is commonly called a switch. The opposite algebra is the compositionof the switch with the multiplication

Y opp ≡ Y ◦ switch ∈ cat(2, 1).(67)

We denote by (∪−1)⊗ ≡ ∪−1⊗∪−1, and analogously by ∪⊗ ≡ ∪⊗∪.

(68) Y ∈ cat(2, 1) & ∪ ∈ cat(1, 1∗)

=⇒ ∪ ◦ Y ∈ cat(2, 1∗) ' cat(3, 0),

∩⊗ ∈ cat(2∗, 2) =⇒ Y ◦ ∩⊗ ∈ cat(2∗, 1) ' cat(3∗, 0).

Let σ ∈ S3, then

σ(∪ ◦ Y ) ∈ cat(3, 0) ' cat(2, 1∗),

σ(Y ◦ (∩ ⊗ ∩)) ∈ cat(3∗, 0).(69)

One can get from an algebra/coalgebra Y the new algebras/coalge-bras by transforming pure tensors (69) ∈ cat(3, 0), into mixed tensors∈ cat(2, 1). There are four ways to get new algebras

(∩ ⊗ id⊗) ◦ σ(∪ ◦ Y ) ∈ cat(2, 1) ' cat(1∗, 2∗)

(id⊗⊗∩) ◦ σ(∪ ◦ Y ) ∈ cat(1∗, 2∗) ' cat(2, 1)

(∪⊗ ⊗ id) ◦ σ(Y ⊗ ∩⊗) ∈ cat(2, 1)

(id⊗∩⊗) ◦ σ(Y ⊗ ∩⊗) ∈ cat(2, 1).

(70)


The S3-permuted algebra, extending the concept of the S2-oppositealgebra, consists of three following steps

(1) Invert the mixed tensor of multiplication m, into purely co-variant or purely contra-variant tensor (68)

(2) Apply a permutation σ ∈ S3 to purely co- or contra-varianttensor

(3) Invert pure tensor into new mixed multiplication tensor.

cat(2, 1) cat(2, 1)

cat(0, 3)cat(0, 3)

New algebra

∪ ⊗ ∪

S3 or B3

∩ ⊗ ∩

Figure 23. S3-permuted algebra

cat(2, 1) cat(2, 1)

cat(3, 0)cat(3, 0)

New algebra

∩

S3 or B3

∪

Figure 24. S3-permuted algebra

11.1. S3-permuted Frobenius C. Now we can determine the S3-permuted Frobenius C using Figure 23, or one can go an alternativeway using Figure 24 as follows

∪ ◦ Y = 1∗ ⊗ 1∗ ⊗ 1∗ − i∗ ⊗ (i∗ ⊗ 1∗ + 1∗ ⊗ i∗)− 1∗ ⊗ i∗ ⊗ i∗,(71)

(72) (σ ⊗ id)(∪ ◦ Y )

= 1∗ ⊗ 1∗ ⊗ 1∗ − i∗ ⊗ i∗ ⊗ 1∗ − 1∗ ⊗ i∗ ⊗ i∗ − i∗ ⊗ 1∗ ⊗ i∗.

(73) Y σ ≡ ∩(σ ⊗ id)(∪ ◦ Y )

= 1⊗ 1∗ ⊗ 1∗ + i⊗ i∗ ⊗ 1∗ − 1⊗ i∗ ⊗ i∗ + i⊗ 1∗ ⊗ i∗.Therefore in this particular case, Y σ = Y.


Do we get the same k-algebra C by S3-permuting Frobenius C?

(74) ∪ ◦Y= (a1∗ + bi∗)⊗ (1∗ ⊗ 1∗ − i∗ ⊗ i∗) + (bi∗ − ai∗)⊗ (i∗ ⊗ 1∗ + 1∗ ⊗ i∗)

Permutation of the first two vectors with σ ⊗ id ∈ S3, gives

(σ ⊗ id)(∪ ◦ Y ) =

+ a 1∗ ⊗ 1∗ ⊗ 1∗

− a i∗ ⊗ 1∗ ⊗ i∗

− a i∗ ⊗ i∗ ⊗ 1∗

− a 1∗ ⊗ i∗ ⊗ i∗

+ b 1∗ ⊗ i∗ ⊗ 1∗

− b i∗ ⊗ i∗ ⊗ i∗

+ b i∗ ⊗ i∗ ⊗ 1∗

+ b 1∗ ⊗ i∗ ⊗ i∗.

(75)

Now we must come back to Y σ with ∩,

(a2 + b2) ∩ ◦(σ ⊗ id) ◦ ∪ ◦ Y =

+ a (a1 + bi)⊗ 1∗ ⊗ 1∗

− a (b1− ai)⊗ 1∗ ⊗ i∗

− a (b1− ai)⊗ i∗ ⊗ 1∗

+ b (a1 + bi)⊗ i∗ ⊗ 1∗

+ b (b1− ai)⊗ i∗ ⊗ 1∗

− a (a1 + bi)⊗ i∗ ⊗ i∗

− b (b1− ai)⊗ i∗ ⊗ i∗

+ b (a1 + bi)⊗ i∗ ⊗ i∗.

(76)

The Cayley multiplication table looks like

(a2 + b2)Y σ(1⊗ 1) = a2 + abi

(a2 + b2)Y σ(i⊗ 1) = −ab+ a2i

(a2 + b2)Y σ(1⊗ i) = b2 + (a2 + b2 − ab)i(a2 + b2)Y σ(i⊗ i) = ab− a2 − b2 + b2i

(77)

We are getting some ‘non associative’ Y σ from associative Y.


12. Conclusion

The innovative concept of this paper we propose is the variety of thesolvable Frobenius algebras, and an equational theory for such algebrasthat is basis-free identity. To be solvable ∪-Frobenius Y means thatexists no zero ∪-radical Y ⊥ 6= 0 for Y -associative ∪ 6= 0.

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WEB pages

http: //www.singular.uni-kl.de/ A free package for Grobner basis calculations. Sin-gular is faster than Maple for some calculations.

http: //wxmaxima.sourceforge.net

Universidad Nacional Autonoma de Mexico, Facultdad de EstudiosSuperiores de Cuautitlan, C.P. 54714 Cuautitlan Izcalli, ApartadoPostal #25, Estado de Mexico, Mexico

E-mail address: [email protected]

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semi-simple lie algebra is a non-associative frobenius algebra (all two-dimensional frobenius...

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