By Igor Frenkel, Mikhail Khovanov, Catharina Stroppel

The aim of this paper is to review categorifications of tensor items of finite-dimensional modules for the quantum staff for sl2. the most categorification is got utilizing yes Harish-Chandra bimodules for the advanced Lie algebra gln. For the targeted case of straightforward modules we obviously deduce a categorification through modules over the cohomology ring of sure flag kinds. extra geometric categorifications and the relation to Steinberg forms are discussed.We additionally provide a specific model of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) common bases when it comes to projective, tilting, regular and easy Harish-Chandra bimodules.

Show description

Read Online or Download A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products PDF

Similar quantum physics books

Quantum Field Theory of Many-body Systems by Wen X.G. PDF

For many of the final century, condensed subject physics has been ruled by means of band conception and Landau's symmetry breaking conception. within the final 20 years, although, there was the emergence of a brand new paradigm linked to fractionalization, topological order, emergent gauge bosons andfermions, and string condensation.

Read e-book online Contextual logic for quantum systems PDF

During this paintings we construct a quantum good judgment that permits us to consult actual magnitudespertaining to diverse contexts from a hard and fast one with no the contradictionswith quantum mechanics expressed in no-go theorems. This common sense arises from consideringa sheaf over a topological area linked to the Boolean sublattices ofthe ortholattice of closed subspaces of the Hilbert house of the actual approach.

Read e-book online Quantum Coherence: From Quarks to Solids PDF

Quantum coherence performs a vital function in a number of different types of topic. The thriving box of quantum details in addition to unconventional techniques to utilizing mesoscopic structures in destiny optoelectronic units give you the fascinating history for this set of lectures. The lectures originate from the Schladming iciness faculties and are edited to handle a wide readership starting from the graduate pupil as much as the senior scientist.

Extra resources for A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products

Sample text

4 we get isomorphisms of graded C i -modules Fi+1 Ei (C) ∼ = C i,i+1 ⊗C i+1 C i,i+1 ⊗C i C −n + 1 i n−i−1 ∼ = C 2k + 2l −n + 1 , l=0 k=0 i,i−1 Ei−1 Fi (C) ∼ =C ⊗C i−1 C i,i−1 ⊗C i C −n + 1 n−i i−1 ∼ = C 2l + 2k −n + 1 . l=0 k=0 If n − 2i ≥ 0, then n−i−1 G G FiG = EiG − Ei−1 Fi+1 G (2k + 2i) (1 − n) k=0 i−1 − G (2(n − i) + 2k) 1 − n k=0 422 I. Frenkel, M. Khovanov and C. Stroppel   0 =   Sel. , New ser. if n − 2i = 0, n−2i−1 G (2i + 2k) 1 − n if n − 2i > 0, k=0 = [n − 2i] Id . If n − 2i < 0, then i−1 G G G EiG ∼ FiG − Fi+1 Ei−1 = (2(n − i) + 2k) (−n + 1) k=0 n−i−1 − G (2k + 2i) 1 − n k=0 2i−n−1 = G (n − 2i + 1 + 2k) k=0 = [2i − n] Id .

J. Amer. Math. Soc. 3 (1990), 421–445. [Soe92] W. Soergel. The combinatorics of Harish-Chandra bimodules. J. Reine Angew. Math. 429 (1992), 49–74. Vol. 12 (2006) [Soe97] [Str03] [Str05] [Str06] [Sus05] [Zhu04] Categorification of representations of quantum sl2 431 W. Soergel. Kazhdan–Lusztig polynomials and a combinatorics for tilting modules. Represent. Theory 1 (1997), 83–114. C. Stroppel. Category O: Gradings and translation functors. J. Algebra 268 (2003), 301–326. C. Stroppel. Categorification of the Temperley–Lieb category, tangles, and cobordisms via projective functors.

4). We believe that this is a necessary ingredient of a more substantial and general construction which will be considered in the next subsection. Let W = Sn be the symmetric group of order n! with subgroup Wi as above. For any 0 ≤ i ≤ n let B i = Func(W/Wi ) be the algebra of complex-valued functions on the (finite) set W/Wi . Similarly, for 0 ≤ i, i + 1 ≤ n let B i,i+1 = Func(W/Wi,i+1 ) be the algebra of functions on W/Wi,i+1 . e. ew (x) = (i) δw,x . In fact, the ew , w ∈ W/Wi , form a complete set of primitive, pairwise orthogonal, idempotents.

Download PDF sample

A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products by Igor Frenkel, Mikhail Khovanov, Catharina Stroppel

by Brian

Rated 4.64 of 5 – based on 4 votes