On Jaeger's HOMFLY-PT expansions, branching rules and link homology: a progress report
aa r X i v : . [ m a t h . QA ] S e p ON JAEGER’S HOMFLY-PT EXPANSIONS, BRANCHING RULES AND LINKHOMOLOGY: A PROGRESS REPORT
PEDRO VAZA
BSTRACT . We describe Jaeger’s HOMFLY-PT expansion of the Kauffman polynomial and howto generalize it to other quantum invariants using the so-called “branching rules” for Lie algebrarepresentations. We present a program which aims to construct Jaeger expansions for link homo-logy theories. This note is an updated write-up of a talk given by the author at the Meeting of theSociedade Portuguesa de Matemática in July 2012.
1. L
INK POLYNOMIALS AND J AEGER EXPANSIONS
This story starts with two celebrated invariant polynomials of links.
Definition 1.1.
The
Kauffman polynomial F “ F p a, q q is the unique invariant of framed uno-riented links satisfying F ´ ¯ ´ F ´ ¯ “ p q ´ q ´ q ˆ F ´ ¯ ´ F ´ ¯ ˙ F ˆ ˙ “ a ´ q F ˆ ˙ and F ´ ¯ “ a q ´ ´ a ´ qq ´ q ´ ` . Definition 1.2.
The
HOMFLY-PT polynomial P “ P p a, q q is the unique invariant of framedoriented links satisfying aP ´ ¯ ´ a ´ P ´ ¯ “ p q ´ q ´ q P ´ ¯ P ˆ ˙ “ a ´ P ˆ ˙ , P ˆ ˙ “ aP ˆ ˙ and P ´ ¯ “ a ´ a ´ q ´ q ´ . Either the Kauffman and the HOMFLY-PT polynomials can be normalized to give ambientisotopy invariants of (oriented) links, but we will not proceed in this direction. In 1989 FrançoisJaeger showed that the Kauffman polynomial of a link L can be obtained as a weighted sum ofHOMFLY-PT polynomials on certain links associated to L . Consider the following formalism(1) ” ı “ p q ´ q ´ q ˆ” ı ´ ” ı˙ ` ” ı ` ” ı ` ” ı ` ” ı” ı “ ” ı ` ” ı Key words and phrases.
Quantum invariant, branching rules, link homology. Pedro Vaz where the r.h.s. is evaluated to HOMFLY-PT polynomials completed with information about rotation numbers : r ~D s “ p a ´ q q rot D P p ~D q (of course we only take the diagrams which areglobally coherently oriented). The expression in (1) can be seen as a state expansion with thecoefficients given as vertex weights. For a state σ we denote its weight by w p σ q . The proof of thefollowing can be found in [1]. Theorem 1.3 (F. Jaeger, 1989) . Let D be a diagram of a link L and Σ p D q denote the set of statesof D . The sum (2) ÿ σ P Σ p D q w p σ qr σ s “ F p D q is a HOMFLY-PT expansion of the Kauffman polynomial of L . It is not hard to see how to extend this expansion to tangles (see [5]). To explain this expansionwe look into the representation theory of quantum enveloping algebras of the simple Lie algebras(QEAs). It is known that the HOMFLY-PT polynomial is related to the representation theory ofthe QEAs of type A n ´ (e.g. sl n ) and that in turn the Kauffman polynomial is related to theQEAs of types B n , C n and D n ( so n ` , sp n and so n q respectively). Taking a “ q n in F p a, q q and P p a, q q we obtain the so n and the sl n polynomials respectively.Following N. Reshetikhin and V. Turaev there is a functor from the category of tangles whosearcs are colored by irreducible finite dimensional (f.d.) representations of a QEA g to the (tensor)category of f.d. representations of g (see [3, 4]). In other words, for each of these tangles there is a g -invariant map which depends only on the (regular) isotopy class of the tangle which gives a fullisotopy invariant in the cases we are interested in. So what we really have in Theorem 1.3 is an sl n -expansion of the so n -polynomial (the case where all strands are colored by the fundamentalrepresentation) !There are other (2-variable) HOMFLY-PT expansions of F p a, q q resulting in sl n -expansions ofthe so n ` and sp n polynomials after the specialization a “ q n . For example the assignment (thiswas found by the author together with E. Wagner and will appear somewhere in the literature [6]) ” ı B n “p q ´ q ´ q ˆ” ı B n ` ” ı B n ` ” ı B n ´ ” ı B n ´ ” ı B n ´ ” ı B n ˙ ` ” ı B n ` ” ı B n ` ” ı B n ` ” ı B n ` ” ı B n ´ ” ı B n ´ ” ı B n ´ ” ı B n ´ ” ı B n ” ı B n “ ” ı B n ` ” ı B n ` ” ı B n where r ~D s B n “ a ´ rot D P p ~D q and a dashed line means the corresponding strand is to be erased,gives an sl n -expansion of the so n ` -polynomial. aeger’s HOMFLY-PT expansions, branching rules and link homology
2. B
RANCHING RULES , LINK HOMOLOGY AND CATEGORIFICATION
Let us give an explanation for this phenomenon. An inclusion l ã Ñ g of Lie algebras (resp.QEAs) gives rise to functors ( Ind and
Res ) between their categories of representations. In gene-ral
Res does not send an irreducible M over g to an irreducible over l but if we restrict ouselvesto finite-dimensional representations we know that Res p M q decomposes as a direct sum of irre-ducibles for l . The branching rule tells us how to obtain such a decomposition i.e. how to expressan irreducible for g as a direct sum of irreducibles for l .This is what we had before ! For example, the expression “ ‰ D n “ “ ‰ D n ` “ ‰ D n , which canbe obtained from the extension of Theorem 1.3 to tangles, is a diagrammatic interpretation of theisomorphism V fund p so n q – V fund p sl n q ‘ V ˚ fund p sl n q for so n Ą sl n , and “ ‰ B n “ “ ‰ B n ` “ ‰ B n ` “ ‰ B n corresponds to V fund p so n ` q – V fund p sl n q ‘ V ˚ fund p sl n q ‘ V triv p sl n q for so n ` Ą sl n .The general picture of categorification of quantum link invariants , pioneered by M. Khova-nov [2] lifts the representations W appearing in the RT picture to categories C g p W q (which arerequired to satisfy certain properties) and the RT map f RT to a (derived) functor F RT between(the derived categories of) these categories. Again, the isomorphism class of this functor dependsonly on the isotopy class of the tangle. The categorification of f RT for general f.d. irreduciblerepresentations of QEAs was constructed by B. Webster in [7, 8]. W ˚ b W b W W ˚ b W b W Inv g p W, W q P f RT C g p W ˚ , W , W q C g p W ˚ , W , W q Fun g p C g p W q , C g p W qq P F RT We can try to use Webster’s work to construct categorical l -expansions for the categorified g -RT invariants. The categories C g appearing in [7] extend to linear combinations of (arbitrary)f.d. irreducibles of g which means that Webster’s functors extend to (formal) linear combinationsof tangles. Definition 2.1. A categorical Jaeger expansion consists of (i) categorified branching rules i.e.a functor C g p V g q Ñ C l p‘ i V l i q for l Ă g , which is full, bijective on objects and descends toa map between the respective Grothendieck groups giving an isomorphism of l -representations V g – ‘ i V l i , and (ii) its extension to corresponding decompositions of the “tangle functors”.Here V g and each of the V l i are irreducible f.d. representations of g and l respectively (resp.tensor products of such representations).Although (ii) seems desirable from the topological point of view (work still in progress), thefulfillment of (i) (see [9, 10]) is already very interesting, due to the potential applications to areaslike representation theory and physics. Pedro Vaz
Theorem 2.2.
There are functors C g p V g q Ñ C l p‘ i V l i q categorifying the branching rules for sl n ` Ą sl n , so n ` Ą so n ´ , sp n Ą sp n ´ , so n Ą so n ´ , (for all finite dimensional repre-sentations and tensor products of minuscule representations) , so n , so n ` Ą sl n (for funda-mental representations and their tensor products). R ÉFÉRENCES [1] L. Kauffman,
Knots and Physics , World Scientific, Singapore, 1991.[2] M. Khovanov, “A categorification of the Jones polynomial”,
Duke. Math. J. , Vol. 101, No. 3 (2000), pp. 359-426.[3] N. Reshetikhin e V. Turaev, “Ribbon graphs and their invariants derived from quantum groups”,
Commun.Math. Phys. , Vol. 127, No. 1 (1990), pp. 1-26.[4] V. Turaev, “The Yang-Baxter equation and invariants of links”,
Invent. Math. , Vol. 92, No. 3 (1988), pp. 527-553.[5] P. Vaz e E. Wagner, “A remark on BMW algebra, q -Schur algebras and categorification”, arXiv :1203.4628v1[math.QA] (2012).[6] P. Vaz e E. Wagner, “(work in progress)”, (2012).[7] B. Webster, “Knot invariants and higher representation theory I : diagrammatic and geometric categorificationof tensor products”, arXiv :1001.2020v7 [math.GT] (2011).[8] B. Webster, “Knot invariants and higher representation theory II : the categorification of quantum knot inva-riants”, arXiv :1005.4559v5 [math.GT] (2011).[9] P. Vaz, “KLR algebras and the branching rule I : The Gelfand-Tsetlin basis in type A n ”, arXiv :1309.0330v1[math.RT] (2013).[10] P. Vaz, KLR algebras and the branching rule II : The Gelfand-Tsetlin basis in types B , C and D ,(2013) (in preparation).I NSTITUT DE R ECHERCHE EN M ATHÉMATIQUE ET P HYSIQUE , U
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