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Frobenius Algebras and Comultiplication A.Ramesh Kumar Assistant Professor, Department of Mathematics, Srimed Andavan Arts & Science College, Trichy-05

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Frobenius Algebras and Comultiplication A.Ramesh Kumar Assistant Professor, Department of Mathematics, Srimed Andavan Arts & Science College, Trichy-05 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 3, 2019 (Special Issue) © Research India Publications. http://www.ripublication.com Frobenius Algebras and Comultiplication A.Ramesh Kumar Assistant Professor, Department of Mathematics, Srimed Andavan Arts & Science College, Trichy-05 R.Prabhu Assistant Professor, Department of Mathematics, M.I.E.T Engineering College, Trichy-07 Abstract theory Edward Witten, which in its axiomatisation In this article the characterize Frobenius algebras A as amounts to a precise mathematical theory. In algebras having a comultiplication which is a map of A- computer science, Frobenius algebras arise in the modules. This characterization allows a simple demonstration study of flowcharts, proof nets circuit diagrams. In any of the compatibility of Frobenius algebra structure with direct case, the reason Frobenius algebras show up is that it multiplication .The relationship between two dimensional is essentially a topological structure: it turns out the topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. Let k be a axioms for a Frobenius algebra can be given field. All vector spaces will be k-vector spaces and all algebras completely in terms of graphs or as we shall do, in will be k-algebras; all tensor products are over k.When we terms of topological surfaces Lowell Abrams . ′ write Hom (푉, 푉 ) it is always mean the space of k-linear maps. Notations: 1. Introduction Principal 2D cobordisms algebraic operation A Frobenius algebra is a finite-dimensional algebra (in a k-algebra A) equipped with a nondegenerate bilinear form compatible with the multiplication. Examples are matrix rings, group rings, the ring of characters of a representation and artinian Gorenstein rings. In algebra and representation theory such algebras have Merging multiplication AAA been studied for a century. In order to relate this to Frobenius algebras. The principal result of cobordisms and Topological Creation Ak quantum field theories establishes an alternative characterization of Frobenius algebras which goes back at least Splitting comultiplication AAA to Lawvere namely as algebras with multiplication denoted which are simultaneously coalgebras with Annihilation counit kA Comultiplication denoted subject to a certain compatibility condition between and this compatibility condition is exactly the right-hand relation drawn. In fact, the basic relations valid in 2Cob correspond precisely to 2. Algebras, Modules, and pairings: the axioms of a commutative Frobenius algebra Lowell 2.1 Definition and basic properties of Frobenius Abrams. This comparison leads to the main theorem. There is algebras: an equivalence of categories 2TQF푻한 ≃ 푨한 , given by sending 2.1.1 Definition: k-algebras a TQFT to its value on the circle. A k-algebra is a k-vector space A together with two ⨂ A Frobenius algebra is vector space that is both an k-linear maps μ: A A→A, η: k→A algebra and a coalgebra in a compatible way. This sort Satisfying the Associativity law and the unit law (μ ⨂ idA) μ = (idA ⨂ μ) μ, of compatibility is different from that involved in a ⨂ ⨂ bialgebra and Hopf algebra. More generally, Frobenius ( η idA) μ = (idA η) μ. In other words, a k-algebra is precisely a monoid in the algebras can be defined in any monoidal category and monoidal category (Vectk, ⨂, k), even in any polycategory in which case they are 2.1.1.1Definition: Frobenius algebra sometimes called Frobenius monoids Stephen Sawin. A Frobenius algebra is a k-algebra A of finite Frobenius structure. During the past decade dimension, equipped with a linear functional 훆: A → k Frobenius algebras have shown up in a variety of whose nullspace contains no nontrivial left ideals. The topological contexts, in theoretical physics and in functional 휀 ∈ 퐴∗ is called a Frobenius form. computer science. In physics, the main scenery for Frobenius algebras is that of topological quantum field Page 64 of 69 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 3, 2019 (Special Issue) © Research India Publications. http://www.ripublication.com 2.1.1.2 Definition category 2Cob. Composition of maps is pictured by A quickly leads to a couple of other joining the output holes of the first figure with the input characterizations. A nondegenerate pairing A⨂A→k, holes of the second. induces two isomorphisms 퐴 ⥲ 퐴∗, they do not in Now we can write down the axioms for algebra like this, general coincide. 2.1.1.3 Definition: Frobenius algebra is a finite dimensional k-algebra = A equipped with a left A-isomorphism to its dual. Alternatively A is equipped with a right A-isomorphism to its dual. Associativity 2.1.2 Definition: (Alternative) A Frobenius algebra is a k-algebra A of finite = = dimension, equipped with an associative nondegenerate pairing: A⨂A→k. We call this pairing the Frobenius pairing. 2.1.6 Functionals and associative pairings on A: Unit axiom Every linear functional 훆: A → k determines canonically a pairing A⨂A→k, namely 푥⨂푦 ↦ (푥푦)휀. The canonical twist map A⨂A → A⨂A, 푣 ⨂ 푤 ↦ 푤 ⨂ clearly this pairing is associative. Conversely, given an 푣 is depicted , so the property of being a 푥 commutative algebra reads associative pairing A⨂A→k, denoted 푥⨂푦 ↦ ( ), a 푦 linear functional is canonically determined, namely 1 퐴 → 푘, 푎 ↦ 〈 퐴⁄ 〉 = 〈푎⁄ 〉. = 푎 1퐴 This gives a one-to-one correspondence between linear 2.3 Pairings of vector spaces: functional on A and associative pairings. A bilinear pairing ̶ or just a pairing ̶ of two 2.2 Graphical language: vector spaces V and W is by definition a linear map The maps above are all linear maps between the 훃: V ⨂ W→k. By the universal property of the tensor tensor powers of A- note that k is naturally the 0’th tensor product, giving a pairing V ⨂ W→k is equivalent to power of A.We introduce the following graphical giving a bilinear map V ⨯ W→k. For this reason we will notation for these maps, allow ourselves to write like 훃: V ⨂ W→k, 푣 ⨂ 푤 ↦ 〈 v⁄ w 〉 2.3.1 Lemma: Given a pairing 훃: V ⨂ W→k, 푣 ⨂ 푤 ↦〈 v⁄ w 〉 , between finite-dimensional vector spaces, the following η μ 푑 퐴 are equivalent. unit identity multiplication (i) 훃 is nondegenerate. 푊 → 푉∗ These symbols are meant to have status of formal (ii) The induced linear map is an isomorphism. 푉 → 푊∗ mathematical symbols, just like the symbols → 표푟 ⨂.The (iii) The induced linear map is an isomorphism. synmbol corresponding to each k-linear map ∅: 퐴⨂푚 → If we already know for other reasons that V and W are of 퐴⨂푛 has m boundaries on the left, one for each factor of a the same dimension, then non-degeneracy can also be in the source, and ordered such that the first factor in the characterized by each of the following a priori weaker tensor product corresponds to the bottom input hole and conditions: the last factor corresponds to the top input hole. If m=0 (ii’) 〈 v⁄ w 〉 = 0 ∀푣 ∈ 푉 ⟹ 푤 = 0 we simply draw no in boundary .similarly there are n (iii’) 〈 v⁄ w 〉 = 0 ∀푤 ∈ 푊 ⟹ 푣 = 0 boundaries on the right which correspond to the target 퐴⨂푛, with the same convention for the ordering. 2.4 Pairings of A-modules: The tensor product of two maps is drawn as the union of the two symbols mimics ⊔ as monoidal operator in the Page 65 of 69 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 3, 2019 (Special Issue) © Research India Publications. http://www.ripublication.com Suppose now M is a right A-module and let N be a There exists such that left A-module. A pairing 훃: M ⨂ N→k, x ⨂ y ↦ 〈 x⁄ y 〉 is said to be associative when 〈푥푎⁄푦 〉 = 〈푥⁄푎푦 〉 for every 푥∈푀,푎∈푎,푦∈푁. 2.4.1 Lemma: = = For a pairing M ⨂ A→k as above, the following are equivalent: (i) M ⨂ N→k is associative. (ii) N→ 푀∗ is left A-linear. This is really the crucial property – we will henceforth (iii) M→ 푁∗ is right A-linear. refer to this as the snake relation. 2.5.1 Examples 2.5 Graphical expression of the Frobenius In each example, A is assumed to be a k-algebra of finite structure: dimension over k. 2.5.1.2 Algebraic field extensions: According to our principles we draw like this, Let A be a finite field extension of k.Since fields have no nontrivial ideals, any nonzero k-linear map A→ 푘 will do as Frobenius form. 휀 2.5.1.3 Division rings: Frobenius form 훽 Let A be a division ring. Since just like a field a Frobenius pairing division ring has no nontrivial left ideals, any nonzero The relations 〈푥⁄ 푦 〉 = (푥푦)휀 푎푛푑 〈1 ⁄푥 〉 = 푥휖 퐴 linear from A→ 푘 will make A into a Frobenius algebra = 〈푥⁄ 1 〉 2.1.6 then get this graphical expression. 퐴 over k. 2.5.1.4 Matrix rings: 푀푎푡 (푘) = The ring 푛 of all n-by-n matrices over k is a Frobenius algebra with the usual trace map as Frobenius form. 푇푟: 푀푎푡 (푘) → 푘, 푛 (푎푖푗) → ∑ 푎푖푖 .푖 = = 2.5.1.5 Group algebras: Let 퐺 = {푡0, … … … , 푡푛} be a finite group written multiplicatively, and with 푡0=1. The group algebra kG is It is trickier to express the axioms which 휀 and 훽 must defined as the set of formal linear combinations ∑ 푐푖푡푖 satisfy in order to be a Frobenius from and a Frobenius with multiplication given by the multiplication in G.It can pairing, respectively .The axiom for a Frobenius from be made into a Frobenius algebra by taking the Frobenius 휀: 퐴 → 푘,is not expressible in our graphical language form to be the functional 휀: 푘퐺 → 푘 , 푡0 ↦ 1, 푡푖 ↦ because we have no way to represent an ideal.
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