Overview Algebraic Background MOY Graphs and Their Matrix Factorizations Colored Link Homology A colored sl ( N )-homology for links in S 3 Hao Wu The George Washington University A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background MOY Graphs and Their Matrix Factorizations Colored Link Homology Overview Algebraic Background Symmetric Polynomials Matrix Factorizations MOY Graphs and Their Matrix Factorizations Definition Decompositions Colored Link Homology Definition Invariance Open Problems and More A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background MOY Graphs and Their Matrix Factorizations Colored Link Homology ◮ I will introduce an sl ( N )-homology associated to links colored by integers, which generalizes the Khovanov-Rozansky sl ( N )-homology. ◮ The construction of this colored sl ( N )-homology uses matrix factorizations over rings of symmetric polynomials. ◮ I conjecture that this colored sl ( N )-homology decategorifies to the quantum sl ( N )-polynomial of links colored by exterior powers of the defining representation. A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Rings of Symmetric and Partially Symmetric Polynomials ◮ An alphabet is a set X = { x 1 , . . . , x m } of finitely many indeterminants. Denote by Sym ( X ) the ring of symmetric polynomials in X with complex coefficients. The grading on Sym ( X ) is given by deg x j = 2. A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Rings of Symmetric and Partially Symmetric Polynomials ◮ An alphabet is a set X = { x 1 , . . . , x m } of finitely many indeterminants. Denote by Sym ( X ) the ring of symmetric polynomials in X with complex coefficients. The grading on Sym ( X ) is given by deg x j = 2. ◮ Let X 1 , . . . , X l be a collection of pairwise disjoint alphabets. Denote by Sym ( X 1 | · · · | X l ) the ring of polynomials in X 1 ∪ · · · ∪ X l over C that are symmetric in each X i , which is naturally a Sym ( X 1 ∪ · · · ∪ X l )-module. This is a free module whose structure is known. A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Simple Symmetric Polynomials For an alphabet X = { x 1 , . . . , x m } , ◮ � elementary: X k := x i 1 x i 1 · · · x i k , 1 ≤ i 1 < i 2 < ··· < i k ≤ m � complete: h k ( X ) := x i 1 x i 1 · · · x i k , 1 ≤ i 1 ≤ i 2 ≤···≤ i k ≤ m m � x k power sum: p k ( X ) := i . i =1 A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Simple Symmetric Polynomials For an alphabet X = { x 1 , . . . , x m } , ◮ � elementary: X k := x i 1 x i 1 · · · x i k , 1 ≤ i 1 < i 2 < ··· < i k ≤ m � complete: h k ( X ) := x i 1 x i 1 · · · x i k , 1 ≤ i 1 ≤ i 2 ≤···≤ i k ≤ m m � x k power sum: p k ( X ) := i . i =1 ◮ Sym ( X ) = C [ X 1 , . . . , X m ] = C [ h 1 ( X ) , . . . , h m ( X )] = C [ p 1 ( X ) , . . . , p m ( X )] A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Cohomology of Complex Grassmannian Denote by G m , N the complex ( m , N ) Grassmannian. Let X and Y be alphabets of m and N − m indeterminants. ◮ Usual cohomology: H ∗ ( G m , N ; C ) ∼ = Sym ( X ) / ( h N +1 − m ( X ) , h N +2 − m ( X ) , . . . , h N ( X )) as graded C -algebras. A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Cohomology of Complex Grassmannian Denote by G m , N the complex ( m , N ) Grassmannian. Let X and Y be alphabets of m and N − m indeterminants. ◮ Usual cohomology: H ∗ ( G m , N ; C ) ∼ = Sym ( X ) / ( h N +1 − m ( X ) , h N +2 − m ( X ) , . . . , h N ( X )) as graded C -algebras. ◮ GL ( N ; C )-equivariant cohomology: H ∗ GL ( N ; C ) ( G m , N ; C ) ∼ = Sym ( X | Y ) as graded Sym ( X ∪ Y )-algebras. A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Grading Shifts Let M be a graded vector space. For j ∈ Z , define M { q j } to be M with grading shifted by j , i.e. M { q j } = M as ungraded R -modules and, for every homogeneous element m ∈ M , deg M { q j } m = j + deg M m . More generally, let f ( q ) = � l j = k a j q j be a Laurent polynomial whose coefficients are non-negative integers. Define l � ( M { q j } ⊕ · · · ⊕ M { q j } M { f ( q ) } = ) . � �� � j = k a j − fold A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Quantum Integers Quantum integers are a particular class of such Laurent polynomials. We use the following definitions: [ j ] := q j − q − j q − q − 1 , [ j ]! := [1] · [2] · · · [ j ] , � j � [ j ]! := [ k ]! · [ j − k ]! . k It is well known that � m + n � � = q − mn q 2 | λ | . n λ =( λ 1 ≥···≥ λ m ): l ( λ ) ≤ m , λ 1 ≤ n A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Graded Matrix Factorizations Fix an integer N > 0. Let R be a graded commutative unital C -algebra, and w a homogeneous element of R with deg w = 2 N + 2. A graded matrix factorization M over R with potential w is a collection of two graded free R -modules M 0 , M 1 and two homogeneous R -module homomorphisms d 0 : M 0 → M 1 , d 1 : M 1 → M 0 of degree N + 1, called differential maps, s.t. d 1 ◦ d 0 = w · id M 0 , d 0 ◦ d 1 = w · id M 1 . We usually write M as d 0 d 1 − → M 1 − → M 0 . M 0 A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Koszul Matrix Factorizations If a 0 , a 1 ∈ R are homogeneous s.t. deg a 0 + deg a 1 = 2 N + 2, then denote by ( a 0 , a 1 ) R the graded matrix factorization a 0 → R { q N +1 − deg a 0 } a 1 R − − → R , which has potential a 0 a 1 . A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology Koszul Matrix Factorizations (cont’d) More generally, if a 1 , 0 , a 1 , 1 , . . . , a k , 0 , a k , 1 ∈ R are homogeneous with deg a j , 0 + deg a j , 1 = 2 N + 2, then define a 1 , 0 , a 1 , 1 a 2 , 0 , a 2 , 1 . . . . . . a k , 0 , a k , 1 R to be the tenser product ( a 1 , 0 , a 1 , 1 ) R ⊗ R ( a 2 , 0 , a 2 , 1 ) R ⊗ R · · · ⊗ R ( a k , 0 , a k , 1 ) R , which is a graded matrix factorization with potential � k j =1 a j , 0 · a j , 1 . A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology The Categories HMF R , w and hmf R , w If M , M ′ are both graded matrix factorizations over R with potential w , then Hom R ( M , M ′ ) is a graded Z 2 -chain complex of R -modules. Its homology, Hom HMF ( M , M ′ ), is the R -module of homotopy classes of morphisms of matrix factorizations from M to M ′ . Denote by Hom hmf the C -subspace of Hom HMF ( M , M ′ ) of homogenous elements of bi-degree (0 , 0). Category Objects Morphisms all homotopically finite graded matrix HMF R , w Hom HMF factorizations over R of potential w with quantum gradings bounded below hmf R , w same as above Hom hmf A colored sl ( N ) -homology for links in S 3 Hao Wu
Overview Algebraic Background Symmetric Polynomials MOY Graphs and Their Matrix Factorizations Matrix Factorizations Colored Link Homology The Krull-Schmidt Property An additive category C is called Krull-Schmidt if ◮ every object of C is isomorphic to a finite direct sum A 1 ⊕ · · · ⊕ A n of indecomposible objects of C ; ◮ and, if A 1 ⊕ · · · ⊕ A n ∼ = A ′ 1 ⊕ · · · ⊕ A ′ l , where A 1 , . . . A n , A ′ 1 , . . . , A ′ l are indecomposible objects of C , then n = l and there is a permutation σ of { 1 , . . . , n } such that A i ∼ = A ′ σ ( i ) for i = 1 , . . . , n . Theorem (Khovanov-Rozansky) If R is a polynomial ring with homogeneous indeterminants of positive gradings and w is a homogeneous element of R with deg w = 2 N + 2 , then hmf R , w and hCh b ( hmf R , w ) are both Krull-Schmidt, where hCh b ( hmf R , w ) is the homotopy category of bounded chain complexes over hmf R , w . A colored sl ( N ) -homology for links in S 3 Hao Wu
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