Displaying similar documents to “A colored Khovanov bicomplex”

New categorifications of the chromatic and dichromatic polynomials for graphs

Marko Stošić (2006)

Fundamenta Mathematicae

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For each graph G, we define a chain complex of graded modules over the ring of polynomials whose graded Euler characteristic is equal to the chromatic polynomial of G. Furthermore, we define a chain complex of doubly-graded modules whose (doubly) graded Euler characteristic is equal to the dichromatic polynomial of G. Both constructions use Koszul complexes, and are similar to the new Khovanov-Rozansky categorifications of the HOMFLYPT polynomial. We also give a simplified definition...

On Ordinary and Z2-graded Polynomial Identities of the Grassmann Algebra

Ribeiro Tomaz da Silva, Viviane (2012)

Serdica Mathematical Journal

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2010 Mathematics Subject Classification: Primary: 16R10, Secondary: 16W55. The main purpose of this paper is to provide a survey of results concerning the ordinary and Z2-graded polynomial identities of the infinite dimensional Grassmann algebra over a field of characteristic zero, as well as of its sequences of ordinary and Z2-graded codimensions and cocharacters. We also intend to describe briefly the techniques used by the authors in order to illustrate some important...

Matrix factorizations and link homology

Mikhail Khovanov, Lev Rozansky (2008)

Fundamenta Mathematicae

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For each positive integer n the HOMFLYPT polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities. ...

A colored 𝔰𝔩(N) homology for links in S³

Hao Wu

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Fix an integer N ≥ 2. To each diagram of a link colored by 1,...,N we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain complex defined by Khovanov and Rozansky. The homology of this chain complex decategorifies to the Reshetikhin-Turaev 𝔰𝔩(N) polynomial of links colored by exterior powers...

Gradings and Graded Identities for the Matrix Algebra of Order Two in Characteristic 2

Koshlukov, Plamen, César dos Reis, Júlio (2012)

Serdica Mathematical Journal

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2010 Mathematics Subject Classification: 16R10, 16R99, 16W50. Let K be an infinite field and let M2(K) be the matrix algebra of order two over K. The polynomial identities of M2(K) are known whenever the characteristic of K is different from 2. The algebra M2(K) admits a natural grading by the cyclic group of order 2; the graded identities for this grading are known as well. But M2(K) admits other gradings that depend on the field and on its characteristic. Here we describe...

Z2-Graded Polynomial Identities for Superalgebras of Block-Triangular Matrices

Di Vincenzo, Onofrio (2004)

Serdica Mathematical Journal

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000 Mathematics Subject Classification: Primary 16R50, Secondary 16W55. We present some results about the Z2-graded polynomial identities of block-triangular matrix superalgebras R[[A M],[0 B]]. In particular, we describe conditions for the T2-ideal of a such superalgebra to be factorable as the product T2(A)T2(B). Moreover, we give formulas for computing the sequence of the graded cocharacters of R in some interesting case. Partially supported by MURST COFIN...

The ℤ₂-graded sticky shuffle product Hopf algebra

Robin L. Hudson (2006)

Banach Center Publications

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By abstracting the multiplication rule for ℤ₂-graded quantum stochastic integrals, we construct a ℤ₂-graded version of the Itô Hopf algebra, based on the space of tensors over a ℤ₂-graded associative algebra. Grouplike elements of the corresponding algebra of formal power series are characterised.

The Hochschild cohomology of a closed manifold

Yves Felix, Jean-Claude Thomas, Micheline Vigué-Poirrier (2004)

Publications Mathématiques de l'IHÉS

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Let M be a closed orientable manifold of dimension and 𝒞 * ( M ) be the usual cochain algebra on M with coefficients in a field. The Hochschild cohomology of M, H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) is a graded commutative and associative algebra. The augmentation map ε : 𝒞 * ( M ) → 𝑘 induces a morphism of algebras I : H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) → H H * ( 𝒞 * ( M ) ; 𝑘 ) . In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of H H * ( 𝒞 * ( M ) ; 𝑘 ) , which is in general quite small. The algebra H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) is expected to...