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165 results on '"KHOVANOV, MIKHAIL"'

155. The Odd nilHecke Algebra and its Diagrammatics.

Author

Ellis, Alexander P., Khovanov, Mikhail, and Lauda, Aaron D.

Subjects
*PICARD number, *CALCULUS, *COHOMOLOGY theory, *QUANTUM groups, *GEOMETRIC quantization
Abstract

We introduce an odd version of the nilHecke algebra and develop an odd analog of the thick diagrammatic calculus for nilHecke algebras. We graphically describe idempotents which give a Morita equivalence between odd nilHecke algebras and the rings of odd symmetric functions in finitely many variables. Cyclotomic quotients of odd nilHecke algebras are Morita equivalent to rings which are odd analogs of the cohomology rings of Grassmannians. Like their even counterparts, odd nilHecke algebras categorify the positive half of quantum sl(2). [ABSTRACT FROM AUTHOR]

Published
2014
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162. A categorification of finite-dimensional irreducible representations of quantum <img src="/fulltext-image.asp?format=htmlnonpaginated&src=7172Q7UT0M2Q6487_html\29_2007_Article_31_TeX2GIFIEq1.gif" border="0" alt=" $${\mathfrak{sl}_2}$$ " /> and their tensor products

Author

Frenkel, Igor, Khovanov, Mikhail, and Stroppel, Catharina

Abstract

Abstract.: The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for $${\mathfrak{sl}_2}$$ . The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra $${\mathfrak{gl}_n}$$ . For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur–Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.

Published
2006
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163. Matrix factorizations and link homology II

Author

Khovanov, Mikhail and Rozansky, Lev

Subjects
Mathematics::K-Theory and Homology, Mathematics::Quantum Algebra, 16. Peace & justice, Mathematics::Geometric Topology, Mathematics::Algebraic Topology
Abstract

To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.

164. HOMAGES TO LOUIS H. KAUFFMAN.

Author

Spencer, Joel, Carter, Scott, Morton, Hugh, and Khovanov, Mikhail

Subjects
MATHEMATICS
Abstract

The article presents homages of various people to Louis H. Kauffman. The first homage came from Joel Spencer who praises the mathematical talents of Kauffman. Another homage came from Scott Carter who remembered the time he met Kauffman. One Homage is from Hugh Morton who remembered his first encounter with Kauffman in Bangor, Northern Wales.

Published
2006
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165. Lectures on Knot Homology and Quantum Curves

Author

Ingmar Saberi, Sergei Gukov, Gukov, Sergei, Khovanov, Mikhail, and Walcher, Johannes

Subjects
High Energy Physics - Theory, Khovanov homology, Pure mathematics, FOS: Physical sciences, Volume conjecture, 01 natural sciences, Planar algebra, Mathematics - Algebraic Geometry, Mathematics - Geometric Topology, Mathematics::K-Theory and Homology, Mathematics - Quantum Algebra, 0103 physical sciences, FOS: Mathematics, Quantum Algebra (math.QA), 0101 mathematics, Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, Physics, 010102 general mathematics, Geometric Topology (math.GT), Knot polynomial, 16. Peace & justice, Mathematics::Geometric Topology, Knot theory, High Energy Physics - Theory (hep-th), Floer homology, Knot invariant, 010307 mathematical physics, Knot (mathematics)
Abstract

Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this interpretation allows one to pose questions that would not have been asked otherwise, such as, "Is there a direct relation between Khovanov homology and the A-polynomial of a knot?" We will explain that the answer to this question is "yes," and introduce a certain deformation of the planar algebraic curve defined by the zero locus of the A-polynomial. This novel deformation leads to a categorified version of the Generalized Volume Conjecture that completely describes the "color behavior" of the colored sl(2) knot homology, and eventually to a similar version for the colored HOMFLY homology. Furthermore, this deformation is strong enough to distinguish mutants, and its most interesting properties include relations to knot contact homology and knot Floer homology., Comment: 48 pages, 9 figures. Comments are welcome

Published
2014

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