crossed G-algebra

Crossed G-algebra


(Here GG will be a group (a discrete one for the moment).)

A crossed GG-algebra is a type of GG-graded algebra, with an inner product and a ‘crossed’ or ‘twisted’ multiplication. They arise as the analogues of Frobenius algebras for 2d-HQFTs, equivariant TQFTs? and in slight generality in the study of symmetries of singularities. They were introduced by Turaev in 1998. The generalised structures have been studied by R. Kaufmann.


Moore and Segal (see references below) do not like the term ‘crossed algebra’ and suggest the alternative name ‘Turaev algebra’.


We will lead up to the definition of crossed GG-algebra through various stages.

We need a particular form of graded algebra in which the summands are projective modules, so we give that form first.

Graded GG-algebra

A graded GG-algebra or GG-algebra over a field (or more generally a commutative ring), 𝕜\mathbb{k} is an associative algebra, LL, over 𝕜\mathbb{k} with a decomposition,

L= gGL g,L = \bigoplus_{g\in G} L_g,

as a direct sum of projective 𝕜\mathbb{k}-modules of finite type such that

(i) L gL hL ghL_g L_h \subseteq L_{gh} for any g,hGg,h \in G (so, if 1\ell_1 is graded gg, and 2\ell_2 is graded hh, then 1 2\ell_1\ell_2 is graded ghgh),


(ii) LL has a unit 1=1 LL 11 = 1_L\in L_1 for 1, the identity element of GG.


(i) The group algebra, 𝕜[G]\mathbb{k}[G], has an obvious GG-algebra structure in which each summand of the decomposition is free of dimension 1.

(ii) For any associatve 𝕜\mathbb{k}-algebra, AA, the algebra, A[G]=A 𝕜𝕜[G]A[G]= A\otimes_\mathbb{k}\mathbb{k}[G], is also GG-algebra. Multiplication in A[G]A[G] is given by (ag)(bh)=(ab)(gh)(ag)(bh) = (ab)(gh) for a,bAa,b \in A, g,hGg,h \in G, in the obvious notation.

(iii) If GG is the trivial group, then a GG-graded algebra is just an algebra (of finite type), of course.

Frobenius GG-algebra

A Frobenius GG-algebra is a GG-algebra, LL, together with a symmetric 𝕜\mathbb{k}-bilinear form,

ρ:LL𝕜\rho : L\otimes L \to \mathbb{k}

such that

(i) ρ(L gL h)=0\rho(L_g\otimes L_h) = 0 if hg 1h \neq g^{-1};

(ii) the restriction of ρ\rho to L gL g 1L_g \otimes L_{g^{-1}} is non-degenerate for each gGg\in G;


(iii) ρ(ab,c)=ρ(a,bc)\rho(ab,c) = \rho(a,bc) for any a,b,cLa,b,c \in L.


We note that (ii) implies that L g 1L g *L_{g^{-1}} \cong L_g^*, the dual of L gL_g.

Examples continued:

(i) The group algebra, L=𝕜[G]L = \mathbb{k}[G], is a Frobenius GG-algebra with ρ(g,h)=1\rho(g,h) = 1 if gh=1gh = 1, and 0 otherwise, and then extending linearly. (Here we write gg both for the element of GG labelling the summand L gL_g, and the basis element generating that summand.)

(iii) For GG trivial, a Frobenius 1-algebra is a Frobenius algebra.

Crossed GG-algebra

Finally the notion of crossed GG-algebra combines the above with an action of GG on LL, explicitly:

A crossed GG-algebra over 𝕜\mathbb{k} is a Frobenius GG-algebra, LL, over 𝕜\mathbb{k} together with a group homomorphism,

φ:GAut(L)\varphi: G \to Aut(L)


(i) if gGg\in G and we write φ g=φ(g)\varphi_g = \varphi(g) for the corresponding automorphism of LL, then φ g\varphi_g preserves ρ\rho, (i.e., ρ(φ ga,φ gb)=ρ(a,b)\rho(\varphi_ga,\varphi_gb) = \rho(a,b)) and

φ g(L h)L ghg 1\varphi_g(L_h) \subseteq L_{ghg^{-1}}

for all hGh\in G;

(ii) φ g| L g=id\varphi_g|_{L_g} = id for all gGg\in G;

(iii) (twisted or crossed commutativity) for any g,hGg,h \in G, aL ga\in L_g, bL hb\in L_h, φ h(a)b=ba\varphi_h(a)b = ba;

(iv) for any g,hGg,h \in G and cL ghg 1h 1c \in L_{ghg^{-1}h^{-1}},

Tr(cφ h:L gL g)=Tr(φ g 1c:L hL h),Tr(c\varphi_h : L_g \to L_g) = Tr(\varphi_{g^{-1}}c : L_h \to L_h),

where TrTr denotes the 𝕜\mathbb{k}-valued trace of the endomorphism. (The homomorphism cφ hc\varphi_h sends aL ga\in L_g to cφ h(a)L gc\varphi_h(a) \in L_g, whilst (φ g 1c)(b)=φ g 1(cb)(\varphi_{g^{-1}}c)(b) = \varphi_{g^{-1}}(cb) for cL hc \in L_h. This is sometimes called the ‘torus condition’.)


a) We note that the usage of terms differs between Turaev’s book (2010) and here, as we have taken ‘crossed GG-algebra’ to include the Frobenius condition. We thus follow Turaev’s original convention (preprint 1999) in this.

b) The useful terminology ‘twisted sector’ in a crossed GG-algebra refers to a summand, L gL_g, for and index, gg, which is not the identity element of GG, of course, then L 1L_1 is called the ‘untwisted sector’

Operations on crossed GG-algebras

(To be added)

twisted G-algebra?


  • V. Turaev, Homotopy Quantum Field Theory , EMS Tracts in Math.10, European Math. Soc. Publ. House, Zurich 2010. (for the detailed development and the links with low dimensional topology and HQFTs).

  • Greg Moore, Graeme Segal, D-branes and K-theory in 2D topological field theory (arXiv hep-th 0609042)

Last revised on May 26, 2011 at 06:15:03. See the history of this page for a list of all contributions to it.