Deformation cohomology of Schur–Weyl categories
Tóm tắt
The deformation cohomology of a tensor category controls deformations of the constraint of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur–Weyl categories). Using this description we compute the deformation cohomology of free symmetric tensor categories generated by one object with an algebra of endomorphism free of zero-divisors. We compare the answers with the exterior invariants of the general linear Lie algebra. The results make precise an intriguing connection between the combinatorics of partitions and invariants of the exterior of the general linear algebra observed by Kostant.
Tài liệu tham khảo
Batanin, M., Davydov, A.: Cosimplicial monoids and deformation theory of tensor categories. arXiv:2003.13039
Davydov, A.: Twisting of monoidal structures. Preprint of MPI, MPI/95-123. arXiv:q-alg/9703001
Davydov, A., Kong, L., Runkel, I.: Functoriality of the center of an algebra. Adv. Math. 285(5), 811–876 (2015)
Davydov, A., Molev, A.: A categorical approach to classical and quantum Schur–Weyl duality. Contemp. Math. 537, 143–171 (2011)
Drinfeld, V.G.: Quasi-Hopf algebras. Algebra Anal 1(6), 114–148 (1989)
Gelfand, S., Manin, Y.: Methods of Homological Algebra. Springer Monographs in Mathematics. Springer, Berlin (2003)
Gerstenhaber, M.: The cohomology of an associative ring. Ann. Math. 78(2), 267–288 (1963)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Itoh, M.: Invariant theory in exterior algebras and Amitsur–Levitzki type theorems. Adv. Math. 288, 679–701 (2016)
Kostant, B.: A theorem of Frobenius, a theorem of Amitsur–Levitski and cohomology theory. J. Math. Mech. 7, 237–264 (1958)
Milne, J.S.: Algebraic Geometry, p. 260. Allied Publishers, Bengaluru (2012)
Penrose, R.: Applications of negative dimensional tensors. In: Combinatorial Mathematics and Its Applications, pp. 221–244. Academic Press, Cambridge (1971)
Yetter, D.: Functorial Knot Theory: Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants. Series on Knots and Everything, vol. 26. World Scientific, Singapore (2001)