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Braiding を持つもの達

Yang-Baxter トポロジ での 不変 などの , なもの “braiding” つようにな ている

なのは braided monoidal category である

なのは monoidal category object x morphism c : x x x x braid relation つものである のときには , Lebed [ Lebb ] はそのようなも のを braided set んでいる monoidal category では braided object べきだろう

  • braided object

Lebed [ Leba ] , いくつかの object まりで braiding つものを braided system , 調 べている

  • braided system

これ にも braided いたものや braid relation をみたすものには 以下 ようなものがある

Braided group とは , Majid [ Maj93 ] により 導入 された , linear braided monoidal category での Hopf algebra object (group object) のことである Majid [ Maj94 ] Lyubashenko Majid [ LM94 ] による がある

Braided vector space braided Hopf algebra などについては , Andruskiewitsch Schneider pointed Hopf algebra する survey [ AS02 ] をみるとよいだろう Braided vector space Hashimoto Hayashi [ HH92 ] では Yang-Baxter pair ばれている , ベクトル から algebra braided vector space ている えば , symmetric algebra, exterior algebra, divided power algebra など

Hopf algebra つとして group ring があるが , braided Hopf algebra から ることができる ただし matched pair of groups という からである Andruskiewitsch Natale [ AN03 ] など

Triangulated category exceptional collection への braid などにつ いては , Gorodentsev Kuleshov “Helix theory” [ GK04 ] がよくまとま ている ,

には , braided differential algebra [ GPS11 ] というものもある

Lebed [ Leb13 ] , (co)homology pre-braiding というもので , pre-braiding ベクトル (co)homology している , にそのような (co)homology えたのは , Carter, Elhamdadi, Saito [ CES04 ] である

  • braided (co)homology

References

[AN03]     Nicolás Andruskiewitsch and Sonia Natale. Braided Hopf algebras arising from matched pairs of groups. J. Pure Appl. Algebra , 182(2-3):119–149, 2003, arXiv:math/0209210 .

[AS02]     Nicolás Andruskiewitsch and Hans-Jürgen Schneider. Pointed Hopf algebras. In New directions in Hopf algebras , volume 43 of Math. Sci. Res. Inst. Publ. , pages 1–68. Cambridge Univ. Press, Cambridge, 2002, arXiv:math/0110136 .

[ATL]     Andrea Appel and Valerio Toledano-Laredo. Quasi-Coxeter categories and quantum groups, arXiv:1610.09741 .

[Bon89]     A. I. Bondal. Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. , 53(1):25–44, 1989.

[Bre98]     Lawrence Breen. Braided n -categories and Σ-structures. In Higher category theory (Evanston, IL, 1997) , volume 230 of Contemp. Math. , pages 59–81. Amer. Math. Soc., Providence, RI, 1998, arXiv:math/9810045 .

[BZ08]     Arkady Berenstein and Sebastian Zwicknagl. Braided symmetric and exterior algebras. Trans. Amer. Math. Soc. , 360(7):3429–3472, 2008, arXiv:math/0504155 .

[CES04]     J. Scott Carter, Mohamed Elhamdadi, and Masahico Saito. Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of quandles. Fund. Math. , 184:31–54, 2004, arXiv:math/0206255 .

[GK04]     A. L. Gorodentsev and S. A. Kuleshov. Helix theory. Mosc. Math. J. , 4(2):377–440, 535, 2004.

[Gor88]     A. L. Gorodentsev. Surgeries of exceptional bundles on n . Izv. Akad. Nauk SSSR Ser. Mat. , 52(1):3–15, 240, 1988.

[GPS11]     Dimitri Gurevich, Pavel Pyatov, and Pavel Saponov. Braided differential operators on quantum algebras. J. Geom. Phys. , 61(8):1485–1501, 2011, arXiv:1004.4721 .

[GR87]     A. L. Gorodentsev and A. N. Rudakov. Exceptional vector bundles on projective spaces. Duke Math. J. , 54(1):115–130, 1987, http://dx.doi.org/10.1215/S0012-7094-87-05409-3 .

[GS09]     Dimitri Gurevich and Pavel Saponov. Braided affine geometry and q -analogs of wave operators. J. Phys. A , 42(31):313001, 51, 2009, arXiv:0906.1057 .

[GS13]     Dimitri Gurevich and Pavel Saponov. Braided algebras and their applications to noncommutative geometry. Adv. in Appl. Math. , 51(2):228–253, 2013, arXiv:1211.5506 .

[HH92]     Mitsuyasu Hashimoto and Takahiro Hayashi. Quantum multilinear algebra. Tohoku Math. J. (2) , 44(4):471–521, 1992, http://dx.doi.org/10.2748/tmj/1178227246 .

[HLV12]     I. Heckenberger, A. Lochmann, and L. Vendramin. Braided racks, Hurwitz actions and Nichols algebras with many cubic relations. Transform. Groups , 17(1):157–194, 2012, arXiv:1103.4526 .

[Leba]     Victoria Lebed. Braided Systems: a Unified Treatment of Algebraic Structures with Several Operations, arXiv:1305.0944 .

[Lebb]     Victoria Lebed. Cohomology of idempotent braidings, with applications to factorizable monoids, arXiv:1607.08081 .

[Leb13]     Victoria Lebed. Homologies of algebraic structures via braidings and quantum shuffles. J. Algebra , 391:152–192, 2013, arXiv:1204.3312 .

[LM94]     Volodimir Lyubashenko and Shahn Majid. Braided groups and quantum Fourier transform. J. Algebra , 166(3):506–528, 1994, http://dx.doi.org/10.1006/jabr.1994.1165 .

[Maj93]     Shahn Majid. Braided groups. J. Pure Appl. Algebra , 86(2):187–221, 1993, http://dx.doi.org/10.1016/0022-4049(93)90103-Z .

[Maj94]     Shahn Majid. Algebras and Hopf algebras in braided categories. In Advances in Hopf algebras (Chicago, IL, 1992) , volume 158 of Lecture Notes in Pure and Appl. Math. , pages 55–105. Dekker, New York, 1994, arXiv:q-alg/9509023 .

[Roy]     Sutanu Roy. Braided C * -quantum groups, arXiv:1601.00169 .