math.QA 2008-01-17 0

Cohomology of Frobenius Algebras and the Yang-Baxter Equation

A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key examples. …

Authors: ** J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi

Cohomology of Frobenius Algebras and the Yang-Baxter Equation
Cohomolog y of F rob enius Algebras a nd the Y ang-Baxter Eq uation J. Scott Carter ∗ Univ ersit y of South Alabama Alissa S. Crans Lo y ola Marymount Univ ersit y Mohamed Elhamdadi Univ ersit y o f South Florida En v er Kara da yi Univ ersit y o f South Florida Masahico Saito † Univ ersit y o f South Florida No v em b er 21, 2018 De dic ate d to the memo ry of X iao-Song Lin Abstract A cohomolog y theory for multip lica tio ns and comultiplications o f F rob enius a lgebras is de - veloped in low dimensions in analogy with Ho chsc hild co homology of bia lgebras base d on defo r - mation theory . Concr ete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Y ang-Baxter equation using m ul- tiplications and comultiplications of F rob enius alg ebras, a nd 2-co cy cles are used to obtain de- formations of R - ma trices th us o btained. 1 In tro duc tion F r ob enius algebras are in teresting to top ologists as we ll as algebraists for n um erous r easons includ - ing the follo wing. First, 2-dimensional top ologica l quant um field theories are form ulated in terms of comm utativ e F rob enius algebras (see [13]). Second, a F rob enius algebra str u cture exists on an y finite-dimensional Hopf algebra with a left integral defi ned in th e du al space. These Hopf algebras ha v e found applications in top ology thr ough Kup erb erg’s in v arian t [14, 15], the Henn ing in v arian t [11, 17], and the theory of quant um group s from whic h the p ost-Jones in v arian ts arise. Third, there is a 2-dimensional F rob enius algebra that underlies Khov ano v’s cohomolog y theory [12]. S ee also [1]. Our interest herein is to extend the cohomology theories defined in [3, 4] to F r ob enius algebras and th ereb y co ns tr uct new solutions to the Y ang-B axter equation (YBE). W e exp ect that th ere are connections among th ese cohomology theories that extend b ey ond th eir form al definitions. F u rthermore, w e an ticipate top ological, categ orical, and/or p hysical applications b ecause of the diagrammatic nature of the theory . ∗ Supp orted in p art by NSF Grant DMS #0603926. † Supp orted in p art by NSF Grant DMS #0603876. 1 The 2-c o cycle cond itions of Hoc hschild cohomolog y of algebras and bialg ebras can b e in terp r eted via deformations of algebras [8]. In other words, a map s atisfying the asso ciativit y condition can b e deformed to obtain a new asso ciativ e map in a larger v ector space using 2-cocycles. T he same in terpretation can b e app lied to qu andle cohomology theory [2, 5 , 6]. A quandle is a s et equipp ed with a self-distributive binary op eration satisfying a few additional conditions that corresp ond to the prop erties that conjugation in a group enjo ys. Quand les ha v e b een used in knot theory extensiv ely (see [2] and r eferen ces therein for more asp ects of quandles). Quandles and related s tructures can b e used to construct set-theoretic solutions (called R -matrices) to the Y ang-Ba xter equation (see, for example, [10 ] and its references). F r om this p oint of view, com bined with the deformation 2-cocycle in terpr etation, a quandle 2-cocycle can b e regarded as giving a co cycle deformation of an R -matrix. Th us we extend this id ea to other algebraic constructions of R -matrices and construct new R -matrices f rom old via 2-cocycle deformations. In [3, 4], new R -matrices we re constru cted via 2-cocycle deformations in t wo other algebraic con texts. Sp ecifically , in [3], self-distr ibutivit y wa s revisited fr om the p oint of view of coalgebra catego ries, thereb y unify in g Lie algebras and quandles in these catego ries. Coh omology theories of Lie algebras and quandles were giv en via a single definition, and d eform ations of R -matrices w ere constructed. In [4], the adjoint map of Hopf algebras, w hic h corresp ond s to the group conju ga- tion map, was studied from the same viewp oin t. A cohomology theory was constructed based on equalities satisfied b y the adjoint map that are sufficient for it to satisfy the YBE. In this p ap er, w e pr esen t an analog for F roben iu s algebras according to the follo wing organiza- tion. After a brief review of necessary materials in Section 2, a cohomology theory for F roben ius algebras is constru cted in Section 3 via deformation theory . Then Y ang-Ba xter solutions are con- structed by s kein metho ds in S ection 4, follo w ed by d eformations of R -matrices by 2-cocycles. The reader s h ould b e a ware that the comp osition of the maps is read in the s tand ard w a y from righ t to left ( g f )( x ) = g ( f ( x )) in text and from b ottom to top in the diagrams. In this w a y , when reading from left to right one can d ra w from top to b ottom and when reading a diagram from top to b ottom, one can disp la y the maps from left to righ t. Th e argumen t of a fun ction (or input ob ject from a category) is found at the b ottom of the diagram. 2 Preliminaries A F r ob enius algebr a is an (asso ciativ e) algebra (with multiplica tion µ : A ⊗ A → A and un it η : k → A ) ov er a fi eld k with a non d egenerate asso ciativ e pairing β : A ⊗ A → k . Throughout this pap er all algebras are finite-dimensional unless sp ecifically stated otherwise. The p airing β is also expressed b y h x | y i = β ( x ⊗ y ) for x, y ∈ A , and it is asso ciative in the sense that h xy | z i = h x | y z i for an y x, y , z ∈ A . ∆ x y µ xy Multiplication η Unit ε Frobenius form β Pairing γ Copairing x y τ x y Transposition x x (2) (1) Comultiplication x Figure 1: Diagrams for F rob enius algebra m aps 2 A F r ob enius algebra A has a linear fun ctional ǫ : A → k , called the F r ob enius form , such that the k ernel con tains no nontrivial left ideal. It is defined from β by ǫ ( x ) = β ( x ⊗ 1), and con ve rsely , a F r ob enius form give s rise to a nondegenerate asso ciativ e pairing β b y β ( x ⊗ y ) = ǫ ( xy ), for x, y ∈ A . A F rob enius form has a unique copairing γ : k → A ⊗ A c haracterized by ( β ⊗ | )( | ⊗ γ ) = | = ( | ⊗ β )( γ ⊗ | ) , where | d en otes the iden tit y homomorp hism on the algebra. W e call this relation th e c anc elation of β and γ . See the mid dle en try in the b ottom r o w of Fig. 2 . T his notation will distinguish this fu nction from the iden tit y elemen t 1 = 1 A = η (1 k ) of the algebra that is the image of th e iden tit y of the ground field. A F rob enius algebra A d etermines a coalgebra structure with A -linear (coasso ciativ e) com ultiplication and the counit defin ed using the F r ob enius form. Th e com ultiplication ∆ : A → A ⊗ A is defined by ∆ = ( µ ⊗ | )( | ⊗ γ ) = ( | ⊗ µ )( γ ⊗ | ) . The multiplic ation and com ultiplication satisfy the f ollo w in g equalit y: ∆ µ = ( µ ⊗ | )( | ⊗ ∆) = ( | ⊗ µ )(∆ ⊗ | ) whic h we call the F r ob enius c omp atibility c ondition . Conversion Unit Cancelation Associativity Coassociativity Compatibility Figure 2: Equalities among F r ob enius algebra maps A F rob enius algebra is symmetric if the p airing is symm etric, meaning th at β ( x ⊗ y ) = β ( y ⊗ x ) for any x, y ∈ A . A F rob enius algebra is c ommutative if it is comm utativ e as an algebra. It is known ([13] P rop. 2.3.29) that a F r ob enius algebra is commutati ve if and only if it is cocommutativ e as a coalge br a. The map µ ∆ of a F rob enius algebra is called th e hand le op er ator, and corresp onds to multipli- cation by a cen tral elemen t called the hand le element δ h = µγ (1) ([13], page 128). An y semisimp le Hopf algebra giv es r ise to a F rob enius algebra stru cture (see, for example, [13], page 135). Let H b e a finite-dimensional Hopf algebra with m ultiplication µ and unit η . Then H is semisimple if and only if the bilinear form β a b = P µ d c d µ c a b is nondegenerate. If H is semisimple, then the ab ov e d efined β giv es rise to a F r ob enius pairing. In this case the F rob enius form (a counit of the F robeniu s algebra stru ctur e) is d efined by ǫ a = P µ d c d = T , the trace of H . Th is counit and 3 the in duced com ultiplication of the resulting F robeniu s algebra str ucture should not b e confused with the counit and the com ultiplication of the original Hopf algebra. W e th an k Y. Sommerhauss er for explaining these relationships to u s. W e compute cohomology group s and the Y ang-Baxter solutions for a v ariet y of examples, an d w e review these mostly from [13]. F r om the p oin t of view of TQFTs, th e v alue ǫη (1) corresp ond s to a sph ere S 2 , δ 0 = β γ (1) to a torus T 2 , and µ ∆ to adding an extra 1-hand le to a tub e, so we will compute these v alues and maps. Example 2.1 Complex n um b ers w ith trigonometric com ultiplication. Let A = C o v er k = R and let the basis b e denoted by 1 and i = √ − 1. Then the F rob enius form ǫ defi n ed b y ǫ (1) = 1 and ǫ ( i ) = 0 gives rise to the com ultiplication ∆, which is Swee dler’s trigonometric coalge br a with ∆(1) = 1 ⊗ 1 − i ⊗ i , and ∆( i ) = i ⊗ 1 + 1 ⊗ i . W e compute ǫη (1) = 1, µ ∆(1) = 2 η (1), and µ ∆( i ) = 2 iη (1) s o that µ ∆ is m ultiplication by 2, which is the h andle elemen t δ h . W e also ha v e δ 0 = 2. Example 2.2 P olynomial algebras. P olynomial rings k [ x ] / ( x n ) o v er a fi eld k where n is a p ositiv e in teger are F roben iu s algebras. I n particular, for n = 2, the algebra A = k [ x ] / ( x 2 ) was used in the Khov ano v homology of knots [12]. F or A = k [ x ] / ( x 2 ), th e F rob enius form ǫ : A → k is defined by ǫ ( x ) = 1 and ǫ (1) = 0. This induces the com ultiplication ∆ : A → A ⊗ A determined b y ∆(1) = 1 ⊗ x + x ⊗ 1 and ∆( x ) = x ⊗ x . T he handle elemen t is δ h = 2 x . More generally for A = k [ x ] / ( x n ) and ǫ ( x j ) = 1 for j = n − 1 and 0 otherwise, the com u ltiplica- tion is determined by ∆(1) = P n − 1 i =0 x i ⊗ x n − 1 − i . W e hav e µ ∆(1) = nx n − 1 and the hand le elemen t is δ h = nx n − 1 . Example 2.3 Group algebras. The group algebra A = k G for a finite group G o v er a field k is a F rob enius algebra with ǫ ( x ) = 0 for an y G ∋ x 6 = 1 and ǫ (1) = 1, wher e 1 is ident ified with the iden tit y elemen t. Th e induced com ultiplication is giv en b y ∆ ( x ) = P y z = x y ⊗ z . One computes ǫ η (1) = 1 and µ ∆( x ) = | G | x , where | G | is the ord er of G . In particular, n ote that µ ∆ = δ 1 | (recall th at | d enotes the identit y map), w here δ 1 = | G | (the order of the group G ), and ( µ ∆) n = δ n 1 | for any n ∈ N , so that the han d le elemen t is δ h = δ 1 = | G | , and δ 0 = | G | . There are other F r ob enius forms on the group algebra A (again from [13]). F o r example, for A = k G w here G is the sy m metric group on three letters, A = k h x, y i / ( x 2 − 1 , y 2 − 1 , xy x = y xy ), and ǫ ( xy x ) = 1 and otherwise zero, is a F roben iu s f orm and the h andle elemen t is 2( xy x + x + y ). Example 2.4 q -Commutativ e p olynomials. Let X = k h x, y i / ( x 2 , y 2 , y x − q xy ) w here q = − A − 2 for A ∈ k w ith p olynomial multiplica tion and ǫ ( xy ) = iA , and zero for other b asis elemen ts. Then γ (1)(= ∆ η (1)) = iA ( x ⊗ y ) − iA − 1 ( y ⊗ x ) − iA − 1 ( xy ⊗ 1 + 1 ⊗ xy ) . One computes ∆( x ) = − iA − 1 ( x ⊗ xy + xy ⊗ x ) , ∆( y ) = − iA − 1 ( y ⊗ xy + xy ⊗ y ) , ∆( xy ) = − iA − 1 ( xy ⊗ xy ) . The handle element is δ h = iA − 1 ( A − A − 1 ) 2 ( xy ). 4 3 Deformations and cohomology group s W e describ e the deformation theory of multiplica tion an d com ultiplication for F rob enius algebras mimic king [8], [16], and our approac h in [3, 4]. Th is appr oac h will yield the defin ition of 2-co cycles. W e w ill d efine the c hain complex for F r ob enius algebras w ith chain groups in lo w dimensions [8, 16]. W e exp ect top ological app lications in low dimensions. The differential s are defined via diagrammaticall y d efined iden tities among relations. 3.1 Deformations In [16], deformations of b ialgebras we re describ ed. W e follo w that formalism and giv e deformations of multiplica tions and comultiplica tions of F robeniu s algebras. A deformation of A = ( V , µ, ∆) is a k [[ t ]]-F rob enius algebra A t = ( V t , µ t , ∆ t ), w here V t = V ⊗ k [[ t ]] and V t / ( tV t ) ∼ = V . Deformations of µ and ∆ are giv en by µ t = µ + tµ 1 + · · · + t n µ n + · · · : V t ⊗ V t → V t and ∆ t = ∆ + t ∆ 1 + · · · + t n ∆ n + · · · : V t → V t ⊗ V t where µ i : V ⊗ V → V , ∆ i : V → V ⊗ V , i = 1 , 2 , · · · , are sequences of maps. Supp ose ¯ µ = µ + · · · + t n µ n and ¯ ∆ = ∆ + · · · + t n ∆ n satisfy the F r ob enius conditions (asso ciativit y , compatibilit y , an d coasso ciativit y) mo d t n +1 , and supp ose that there exist µ n +1 : V ⊗ V → V and ∆ n +1 : V → V ⊗ V su c h that ¯ µ + t n +1 µ n +1 and ¯ ∆ + t n +1 ∆ n +1 satisfy the F r ob enius algebra conditions mo d t n +2 . Define ξ 1 ∈ Hom( V ⊗ 3 , V ), ξ 2 , ξ ′ 2 ∈ Hom( V ⊗ 2 , V ⊗ 2 ), and ξ 3 ∈ Hom( V , V ⊗ 3 ) b y: ¯ µ ( ¯ µ ⊗ | ) − ¯ µ ( | ⊗ ¯ µ ) = t n +1 ξ 1 mo d t n +2 , ¯ ∆ ¯ µ − ( ¯ µ ⊗ | )( | ⊗ ¯ ∆) = t n +1 ξ 2 mo d t n +2 , ¯ ∆ ¯ µ − ( | ⊗ ¯ µ )( ¯ ∆ ⊗ | ) = t n +1 ξ ′ 2 mo d t n +2 , ( ¯ ∆ ⊗ | ) ¯ ∆ − ( | ⊗ ¯ ∆) ¯ ∆ = t n +1 ξ 3 mo d t n +2 . Remark 3.1 T he op erators in the qu adruple ( ξ 1 , ξ 2 , ξ ′ 2 , ξ 3 ) form the primary obstructions to f ormal deformations of multiplicat ion and com ultiplication of a F rob enius algebra [16]. F or the asso ciativit y of ¯ µ + t n +1 µ n +1 mo d t n +2 w e obtain: ( ¯ µ + t n +1 µ n +1 )(( ¯ µ + t n +1 µ n +1 ) ⊗ | ) − ( ¯ µ + t n +1 µ n +1 )( | ⊗ ( ¯ µ + t n +1 µ n +1 )) = 0 m o d t n +2 whic h is equiv alen t b y d egree calc ulations to: ( d 2 , 1 ( µ n +1 ) =) µ ( | ⊗ µ n +1 ) + µ n +1 ( | ⊗ µ ) − µ ( µ n +1 ⊗ | ) − µ n +1 ( µ ⊗ | ) = ξ 1 , (1) where d 2 , 1 is one of the d ifferen tials w e will define in the follo wing section. Similarly , f r om the F r ob enius compatibilit y condition and coassociativit y we obtain ( d 2 , 2 (1) ( µ n +1 , ∆ n +1 ) =) ∆ µ n +1 + ∆ n +1 µ − ( µ ⊗ | )( | ⊗ ∆ n +1 ) − ( µ n +1 ⊗ | )( | ⊗ ∆) = ξ 2 , (2) ( d 2 , 2 (2) ( µ n +1 , ∆ n +1 ) =) ∆ µ n +1 + ∆ n +1 µ − ( | ⊗ µ )(∆ n +1 ⊗ | ) − ( | ⊗ µ n +1 )(∆ ⊗ | ) = ξ ′ 2 , (3) ( d 2 , 3 (∆ n +1 ) =) (∆ ⊗ | )∆ n +1 + (∆ n +1 ⊗ | )∆ − ( | ⊗ ∆)∆ n +1 − ( | ⊗ ∆ n +1 )∆ = ξ 3 , (4) where there are tw o t yp es of compatibilit y conditions for d 2 , 2 . In summ ary we pro ved th e follo w ing: 5 Lemma 3.2 The maps ¯ µ + t n +1 µ n +1 and ¯ ∆ + t n +1 ∆ n +1 satisfy the asso ciativity, c o asso ciativity and F r ob enius c omp atibility c onditions mo d t n +2 if and only if the e qualities (1), (2), (3) and (4) ar e satisfie d. 3.2 Chain groups Let A b e a F roben ius algebra. W e defin e chain group s as follo ws. C n,i f ( A ; A ) = Hom( A ⊗ ( n +1 − i ) , A ⊗ i ) , C n f ( A ; A ) = ⊕ 0

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