##
**The Drinfel’d double and twisting in stringy orbifold theory.**
*(English)*
Zbl 1174.14048

The paper under review deals with the central role that the Drinfel’d double (and its twists) of the group ring of a finite group \(G\) has in stringy orbifold theories (and their twists) as cohomology, Chow, Grothendieck \(K_{0}\) and topological \(K\)-theory.

In the first part of the paper, the authors recall the definition of the twisted Drinfel’d double \(D^{\beta}(k[G])\) of the group ring \(k[G]\) of a finite group \(G\), where \(\beta\in Z^{3}(G,k^{*})\) is a \(3\)-cocycle. It is a quasi-triangular quasi-Hopf \(k\)-algebra which as \(k\)-vector space has a basis given by elements \(g\llcorner_{x}\) for every \(x,g\in G\) (giving a natural \(G\times G\)-grading), and whose algebra and co-algebra structures, Drinfel’d associator, \(R\) matrix and antipode are defined by means of \(\beta\) (see Definition 2.1). If \(\beta=1\), then it is the Drinfel’d double \(D(k[G])\). Moreover, as \(D^{\beta}(k[G])\) is a quasi-triangular quasi-Hopf algebra, the category \(D^{\beta}(k[G])\)-Mod of (left) \(D^{\beta}(k[G])\)-modules has a natural braided monoidal structure, and every left \(D^{\beta}(k[G])\)-module \(A\) has a \(G\)-grading.

In section 3 the authors introduce the notion of \(G\)-Frobenius algebra over a field \(k\) of characteristic 0 with character \(\chi\in \operatorname{Hom}(G,k^{*})\), which is a Frobenius algebra with a \(G\)-grading and an action of \(G\) verifying some compatibility axioms (see Definition 3.12). Any \(G\)-Frobenius algebra is shown to be a \(D(k[G])\)-module by R. M. Kaufmann [J. Algebra 282, 232–259 (2004; Zbl 1106.16040)]. In the paper the authors show that a \(G\)-Frobenius algebra with character \(\chi\) is a unital, associative, commutative algebra object in \(D(k[G])\)-Mod and defines a \(k_{\chi}\)-twisted Frobenius algebra object (see Proposition 3.15). Moreover, the main result of section 3 is Theorem 3.16, where it is shown that a \(G\)-Frobenius algebra with character \(\chi\) is a (non-degenerate) \(\mathbb{I}_{\chi}\)-twisted Frobenius algebra object in \(D(k[G])\)-Mod verifying two axioms (S) and (T), where \(\mathbb{I}_{\chi}\) is an even invertible element in \(D(k[G])\)-Mod associated to \(\chi\), i. e. it is a \(G\)-Frobenius algebra object in \(D(k[G])\)-Mod (see Definition 3.17).

In section 4 the authors relate the (twisted) Drinfel’d double to stringy cohomology theories. Consider a smooth projective variety \(X\) with an action of a finite group \(G\), and let \(I(X,G)\) be the inertia variety, i. e. \(I(X,G)=\coprod_{g\in G}X^{g}\), where \(X^{g}\) is the subset of \(X\) of the points fixed by \(g\). If \(\mathcal{F}\) is any functor among \(H^{*}\) (cohomology), \(K_{0}^{*}\) (Grothendieck \(K_{0}\)), \(A^{*}\) (Chow) and \(K^{\mathrm{top}}\) (topological \(K\)-theory) with coefficients in \(\mathbb{Q}\), then one defines \(\mathcal{F}_{\mathrm{stringy}}(X,G):=\mathcal{F}(I(X,G))\), which as vector space is isomorphic to \(\coprod_{g\in G}\mathcal{F}(X^{g})\). The stringy product giving to \(\mathcal{F}_{\mathrm{stringy}}(X,G)\) the structure of an algebra is defined by means of the obstruction bundle \(\mathcal{R}\). T. Jarvis, R. Kaufmann and T. Kimura [Invent. Math. 168, 23–81 (2007; Zbl 1132.14047)] showed that the cases where \(\mathcal{F}\) is \(H^{*}\) or \(K^{\mathrm{top}}\) yield \(G\)-Frobenius algebras. In Theorem 4.3, the authors show that if \(\mathcal{F}\) is \(A^{*}\) or \(K_{0}\), the stringy functors are \(G\)-Frobenius algebra objects.

Moreover, the authors study the stack case, namely when one considers the stack \([X/G]\). In this case we have three notions of stringy \(K\)-theory: the global stringy \(K\)-theory \(K_{\mathrm{global}}((X,G)):=K(I(X,G))\) (isomorphic as vector space to \(\bigoplus_{g\in G}K(X^{g})\)), the small stringy \(K\)-theory \(K_{\mathrm{small}}([X,G]):=K_{\mathrm{global}}((X,G))^{G}\) (isomorphic as vector space to \(\bigoplus_{[g]\in C(G)}K(X^{g})^{Z(g)}\), where \(C(G)\) is the set of conjugacy classes of \(G\) and \(Z(g)\) is the group of elements of \(G\) commuting with \(g\)) and the full stringy \(K\)-theory \(K_{\mathrm{full}}([X,G]):=K(\mathcal{I}[X/G])\), the \(K\)-theory of the inertia stack of \([X/G]\) (isomorphic as vector space to \(\bigoplus_{[g]\in C(G)}K([X^{g}/Z(g)])\)). In the case where \(X=\mathrm{pt}\) the authors show that (as rings) \(K_{\mathrm{global}}(I(\mathrm{pt},G),G)=D(k[G])^{\mathrm{comm}}\) (the subalgebra of \(D(k[G])\) generated by the elements \(g\llcorner x\) such that \([g,x]=e\)) and that \(K_{\mathrm{full}}([\mathrm{pt}/G])\simeq \mathrm{Rep}(D(k[G]))\) (see Proposition 4.10 and Theorem 4.13). This last result can be obtained, at least as vector spaces, by D. S. Freed, M. J. Hopkins and C. Teleman [Twisted \(K\)-theory and loop-group representations, arXiv:math.AT/0312155] and [Loop-groups and twisted \(K\)-theory II, arXiv:math.AT/0511232].

The last part of the paper deals with twistings by \(0\)-, \(1\)- or \(2\)-gerbes: when considering a global quotient \((X,G)\), these are simply elements in \(Z^{i}(G,k^{*})\) for \(i=1,2,3\) respectively. The choice of such an element \(\beta\), i. e. the choice of a gerbe \(\mathcal{G}\), induces a twisting of the algebra structure on \(K_{\mathrm{global}}\). If \(\beta\in Z^{3}(G,k^{*})\) one gets \(K_{\mathrm{global}}^{\beta}(I(\mathrm{pt},G),G)=D^{\beta}(k[G])^{comm}\), getting a geometrical interpretation of the twisted Drinfel’d double. When considering the stack case, in Theorem 5.8 the authors show that \(K_{\mathrm{full}}^{\beta}([\mathrm{pt}/G])=D^{\beta}(k[G])\). The final sections deal with a purely algebraic version of the twisting and equivariant \(K\)-theory using the language of modules.

In the first part of the paper, the authors recall the definition of the twisted Drinfel’d double \(D^{\beta}(k[G])\) of the group ring \(k[G]\) of a finite group \(G\), where \(\beta\in Z^{3}(G,k^{*})\) is a \(3\)-cocycle. It is a quasi-triangular quasi-Hopf \(k\)-algebra which as \(k\)-vector space has a basis given by elements \(g\llcorner_{x}\) for every \(x,g\in G\) (giving a natural \(G\times G\)-grading), and whose algebra and co-algebra structures, Drinfel’d associator, \(R\) matrix and antipode are defined by means of \(\beta\) (see Definition 2.1). If \(\beta=1\), then it is the Drinfel’d double \(D(k[G])\). Moreover, as \(D^{\beta}(k[G])\) is a quasi-triangular quasi-Hopf algebra, the category \(D^{\beta}(k[G])\)-Mod of (left) \(D^{\beta}(k[G])\)-modules has a natural braided monoidal structure, and every left \(D^{\beta}(k[G])\)-module \(A\) has a \(G\)-grading.

In section 3 the authors introduce the notion of \(G\)-Frobenius algebra over a field \(k\) of characteristic 0 with character \(\chi\in \operatorname{Hom}(G,k^{*})\), which is a Frobenius algebra with a \(G\)-grading and an action of \(G\) verifying some compatibility axioms (see Definition 3.12). Any \(G\)-Frobenius algebra is shown to be a \(D(k[G])\)-module by R. M. Kaufmann [J. Algebra 282, 232–259 (2004; Zbl 1106.16040)]. In the paper the authors show that a \(G\)-Frobenius algebra with character \(\chi\) is a unital, associative, commutative algebra object in \(D(k[G])\)-Mod and defines a \(k_{\chi}\)-twisted Frobenius algebra object (see Proposition 3.15). Moreover, the main result of section 3 is Theorem 3.16, where it is shown that a \(G\)-Frobenius algebra with character \(\chi\) is a (non-degenerate) \(\mathbb{I}_{\chi}\)-twisted Frobenius algebra object in \(D(k[G])\)-Mod verifying two axioms (S) and (T), where \(\mathbb{I}_{\chi}\) is an even invertible element in \(D(k[G])\)-Mod associated to \(\chi\), i. e. it is a \(G\)-Frobenius algebra object in \(D(k[G])\)-Mod (see Definition 3.17).

In section 4 the authors relate the (twisted) Drinfel’d double to stringy cohomology theories. Consider a smooth projective variety \(X\) with an action of a finite group \(G\), and let \(I(X,G)\) be the inertia variety, i. e. \(I(X,G)=\coprod_{g\in G}X^{g}\), where \(X^{g}\) is the subset of \(X\) of the points fixed by \(g\). If \(\mathcal{F}\) is any functor among \(H^{*}\) (cohomology), \(K_{0}^{*}\) (Grothendieck \(K_{0}\)), \(A^{*}\) (Chow) and \(K^{\mathrm{top}}\) (topological \(K\)-theory) with coefficients in \(\mathbb{Q}\), then one defines \(\mathcal{F}_{\mathrm{stringy}}(X,G):=\mathcal{F}(I(X,G))\), which as vector space is isomorphic to \(\coprod_{g\in G}\mathcal{F}(X^{g})\). The stringy product giving to \(\mathcal{F}_{\mathrm{stringy}}(X,G)\) the structure of an algebra is defined by means of the obstruction bundle \(\mathcal{R}\). T. Jarvis, R. Kaufmann and T. Kimura [Invent. Math. 168, 23–81 (2007; Zbl 1132.14047)] showed that the cases where \(\mathcal{F}\) is \(H^{*}\) or \(K^{\mathrm{top}}\) yield \(G\)-Frobenius algebras. In Theorem 4.3, the authors show that if \(\mathcal{F}\) is \(A^{*}\) or \(K_{0}\), the stringy functors are \(G\)-Frobenius algebra objects.

Moreover, the authors study the stack case, namely when one considers the stack \([X/G]\). In this case we have three notions of stringy \(K\)-theory: the global stringy \(K\)-theory \(K_{\mathrm{global}}((X,G)):=K(I(X,G))\) (isomorphic as vector space to \(\bigoplus_{g\in G}K(X^{g})\)), the small stringy \(K\)-theory \(K_{\mathrm{small}}([X,G]):=K_{\mathrm{global}}((X,G))^{G}\) (isomorphic as vector space to \(\bigoplus_{[g]\in C(G)}K(X^{g})^{Z(g)}\), where \(C(G)\) is the set of conjugacy classes of \(G\) and \(Z(g)\) is the group of elements of \(G\) commuting with \(g\)) and the full stringy \(K\)-theory \(K_{\mathrm{full}}([X,G]):=K(\mathcal{I}[X/G])\), the \(K\)-theory of the inertia stack of \([X/G]\) (isomorphic as vector space to \(\bigoplus_{[g]\in C(G)}K([X^{g}/Z(g)])\)). In the case where \(X=\mathrm{pt}\) the authors show that (as rings) \(K_{\mathrm{global}}(I(\mathrm{pt},G),G)=D(k[G])^{\mathrm{comm}}\) (the subalgebra of \(D(k[G])\) generated by the elements \(g\llcorner x\) such that \([g,x]=e\)) and that \(K_{\mathrm{full}}([\mathrm{pt}/G])\simeq \mathrm{Rep}(D(k[G]))\) (see Proposition 4.10 and Theorem 4.13). This last result can be obtained, at least as vector spaces, by D. S. Freed, M. J. Hopkins and C. Teleman [Twisted \(K\)-theory and loop-group representations, arXiv:math.AT/0312155] and [Loop-groups and twisted \(K\)-theory II, arXiv:math.AT/0511232].

The last part of the paper deals with twistings by \(0\)-, \(1\)- or \(2\)-gerbes: when considering a global quotient \((X,G)\), these are simply elements in \(Z^{i}(G,k^{*})\) for \(i=1,2,3\) respectively. The choice of such an element \(\beta\), i. e. the choice of a gerbe \(\mathcal{G}\), induces a twisting of the algebra structure on \(K_{\mathrm{global}}\). If \(\beta\in Z^{3}(G,k^{*})\) one gets \(K_{\mathrm{global}}^{\beta}(I(\mathrm{pt},G),G)=D^{\beta}(k[G])^{comm}\), getting a geometrical interpretation of the twisted Drinfel’d double. When considering the stack case, in Theorem 5.8 the authors show that \(K_{\mathrm{full}}^{\beta}([\mathrm{pt}/G])=D^{\beta}(k[G])\). The final sections deal with a purely algebraic version of the twisting and equivariant \(K\)-theory using the language of modules.

Reviewer: Arvid Perego (Mainz)

### MSC:

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

19L47 | Equivariant \(K\)-theory |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

### Keywords:

quantum cohomology; Hopf algebras; orbifolds; stringy \(K\)-theory; twisted \(K\)-theory; gerbes; fusion ring
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\textit{R. M. Kaufmann} and \textit{D. Pham}, Int. J. Math. 20, No. 5, 623--657 (2009; Zbl 1174.14048)

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