By Bojko Bakalov
This e-book supplies an exposition of the relatives one of the following 3 subject matters: monoidal tensor different types (such as a class of representations of a quantum group), three-dimensional topological quantum box concept, and 2-dimensional modular functors (which clearly come up in 2-dimensional conformal box theory). the next examples are mentioned intimately: the class of representations of a quantum workforce at a root of team spirit and the Wess-Zumino-Witten modular functor. the concept that those issues are similar first seemed within the physics literature within the research of quantum box idea. Pioneering works of Witten and Moore-Seiberg prompted an avalanche of papers, either actual and mathematical, exploring quite a few points of those kin. Upon getting ready to lecture at the subject at MIT, even if, the authors stumbled on that the present literature used to be tough and that there have been gaps to fill. The textual content is completely expository and finely succinct. It gathers effects, fills present gaps, and simplifies a few proofs. The publication makes a tremendous addition to the prevailing literature at the subject. it might be compatible as a direction textual content on the advanced-graduate point
Read Online or Download Lectures on Tensor Categories and Modular Functors PDF
Best linear books
Lie Groups Beyond an Introduction
This e-book takes the reader from the top of introductory Lie workforce idea to the edge of infinite-dimensional workforce representations. Merging algebra and research all through, the writer makes use of Lie-theoretic the way to improve a stunning thought having extensive purposes in arithmetic and physics. The booklet at the start stocks insights that utilize genuine matrices; it later depends on such structural good points as houses of root structures.
Lectures on Tensor Categories and Modular Functors
This publication provides an exposition of the family members one of the following 3 themes: monoidal tensor different types (such as a class of representations of a quantum group), third-dimensional topological quantum box thought, and 2-dimensional modular functors (which clearly come up in 2-dimensional conformal box theory).
We enhance the speculation of compactness of maps among toposes, including linked notions of separatedness. This thought is equipped round types of 'propriety' for topos maps, brought the following in a parallel type. the 1st, giving what we easily name 'proper' maps, is a comparatively vulnerable as a result of Johnstone.
- Mathematical Methods: For Students of Physics and Related Fields (Lecture Notes in Physics)
- Lie Groups, Lie Algebras, and Cohomology.
- Infinite-Dimensional Dynamical Systems in Mechanics and Physics
- Descriptive Topology and Functional Analysis: In Honour of Jerzy Kakol's 60th Birthday
Additional resources for Lectures on Tensor Categories and Modular Functors
Sample text
3) Vg,π := k[G]δg v = xπ, xgx−1 ∈g is an irreducible D(G)-module which depends only on the conjugacy class g and the isomorphism class of the irreducible representation π of Z(g). 4) (δf ⊗ h)(xv) = δf,hxgh−1 x−1 hxv for f, h, x ∈ G, v ∈ π. This shows that the category Repf D(G) is semisimple with simple objects Vg,π labeled by pairs (g, π), where g ∈ G is a conjugacy class in G and π ∈ Z(g) is an isomorphism class of irreducible representation of the centralizer Z(g) of some element g ∈ g (π is independent of the choice of g).
We can easily describe the Grothendieck ring of a modular tensor category. 9). 1. MODULAR TENSOR CATEGORIES 53 associative algebra with a basis xi = Vi , i ∈ I, and a unit 1 = x0 . This algebra is frequently called the fusion algebra, or Verlinde algebra. 11. Let C be an MTC, K = K(C) ⊗Z k, and let F (I) be the algebra of k-valued functions on the set I. Define a map µ : K → F (I) by the picture: V = µ(V ) (i) i i Then µ is an algebra isomorphism. Proof. 3 that µ is an algebra homomorphism. Indeed, U i U V .
When the matrix s˜ is non-singular, it is a matter of pure algebra to deduce Eq. 12). 8. In an MTC, p+ and p− are non-zero. 15) D := p+ p− , ζ := (p+ /p− )1/6 (assuming that they exist in k, otherwise we can always pass to a certain algebraic extension). 16) s := s˜/D. 17) (st)3 = p+ 2 s = ζ 3 s2 , p− s2 = c, ct = tc, c2 = 1. 18) s= , t= 1 0 0 1 52 3. MODULAR TENSOR CATEGORIES with relations (st)3 = s2 , s4 = 1, we see that the matrices s, t give a projective representation of SL2 (Z). 9. Of course, one easily sees that we can replace the matrix t by t/ζ and get a true representation of SL2 (Z) rather than a projective one.