nLab
dual gebra

Dual gebras

What is this entry about

Following Serre, gebra is a common term for associative algebras and coassociative coalgebras (also called cogebras), and sometimes more involved variants and combinations, like bialgebras (also, more properly, called bigebras) and (co)rings.

When working over a field, finite dimensional algebras are duals to finite dimensional cogebras. When the dimension is infinite, even for algebraic duals, the situation is more complicated. This entry should eventually sort out these issues (for now only the simplest cases are discussed).

Algebraic dual

For a commutative ring kk, a coassociative kk-coalgebra (C,Δ)(C,\Delta) and an associative kk-algebra (A,m)(A,m) the kk-module Hom k(C,A)Hom_k(C,A) is equipped with an associative convolution product \star given by (fg)(c)=m(fg)(Δ(c))(f\star g)(c) = m(f\otimes g)(\Delta(c)). In particular, for kk a field, the algebraic dual C *:=Hom k(C,k)C^*:= Hom_k(C,k) of a kk-coalgebra CC is an associative algebra, called its dual coalgebra whose product is also often referred to as convolution. Correspondence CC *C\mapsto C^* extends to a contravariant functor Cog kAlg kCog_k\to Alg_k, where for f:CDf:C\to D, f *:D *C *f^*:D^*\to C^* is simply the transpose, hence f *(d *)(c)=d *(f(c))f^*(d^*)(c) = d^*(f(c)).

Now for a kk-algebra (A,m)(A,m), its algebraic dual A *A^* is not necessarily a coalgebra; namely the natural candidate for the comultiplication Δ\Delta is the transpose operator m *:A *(AA) *m^*: A^*\to (A\otimes A)^* of the multiplication m:AAAm: A\otimes A\to A. There is a canonical injection A *A *(AA) *A^*\otimes A^*\to (A\otimes A)^*; over a field kk it is an isomorphism (hence taken as an identification) iff AA is finite dimensional over kk. In topological cases (e.g., if AA is filtered with filtered pieces finite-dimensional), one can replace the tensor product with some completed tensor product ^\hat\otimes and define a topological comultiplication m *:A *A *^A *m^*:A^*\to A^*\hat\otimes A^*. In algebraic situation, one usually employs so called finite dual which is the maximal subspace A A^\circ for which m *m^* factors through A A A^\circ\otimes A^\circ.

Finite dual

If kk is a field, the finite dual functor () :Alg kCog k()^\circ:Alg_k\to Cog_k is the left adjoint functor to the algebraic dual as a functor () *:Cog kAlg k()^*:Cog_k\to Alg_k. A A *=Hom k(A,k)A^\circ\subset A^*=Hom_k(A,k) as a vector spaces (actually as functors Alg kVec kAlg_k\to Vec_k, () ()^\circ is a subfunctor of () *:Alg kVec k()^*:Alg_k\to Vec_k). For a concrete construction below, the statement of adjointness is Theorem 1.5.22 in Dascalescu et al.

We say that a subspace WW of a vector space VV is of finite codimension if V/WV/W is of finite dimension. As a vector subspace of A *A^*,

A ={fA *|Ker(f)contains an ideal of finite codimension inA} A^\circ = \{ f\in A^* \, | \, Ker(f)\, \text{contains an ideal of finite codimension in}\, A\}

There are several other characterizations of the finite dual. Alternative terminologies are restricted dual and Hopf dual.

Actions on duals

We define here left actions ⇀ (or in LaTeX \rightharpoonup), ⇁ (LaTeX \rightharpoondown) and right actions ↽ (LaTeX \leftharpoondown), ↼ (LaTeX \leftharpoonup).

(Montgomery 1.6.5) If CC is a coalgebra, and C *C^* the dual algebra, then C *C^* acts from the left on CC by the transpose to the left multiplication

g,cf=fg,c \langle g, c ↼ f\rangle = \langle f g, c\rangle

or equivalently by the formula

fcf,c (1)c (2), f ↼ c \coloneqq \langle f, c_{(1)}\rangle c_{(2)},

where the Sweedler notation has been used.

Similarly for the right-hand action:

fcf,c (2)c (1), f ⇀ c \coloneqq \langle f, c_{(2)}\rangle c_{(1)},

or

g,fc=f,c (2)g,c (1)=gf,c \langle g, f ⇀ c\rangle = \langle f, c_{(2)}\rangle \langle g, c_{(1)}\rangle = \langle g f, c\rangle

According to the suggestion of Nichols, one reads ffcc as “ff hits cc” and ffcc as “ff is hit by cc”.

Similarly (cf. Montgomery 1.6.6), if AA is an algebra and A *A^* its algebraic dual, one also defines harpoon actions as transposes to left and right multiplications, for example for right multiplication

hf,k=f,kh. \langle h ↼ f, k\rangle = \langle f, k h\rangle.

Now, if fA f\in A^\circ is in finite dual, then Δ(f)\Delta(f) makes sense, hence, in Sweedler notation, hhf=f (2),hf (1)f=\langle f_{(2)}, h\rangle f_{(1)}.

We also define here left and right coadjoint actions and coactions, cf. Majid.

One should also treat rationality: a module is rational if it corresponds to a comodule of the finite dual coalgebra.

Paired bialgebras

For bigebras (and Hopf algebras n particular) one may consider the duality pairings which are compatible with their structure.

Two kk-bigebras BB and HH are paired if there is a bilinear map ,:BHk\langle,\rangle: B\otimes H\to k such that for all a,bBa,b\in B and h,gHh,g\in H the equations

ab,h=ab,Δh,1 B,h=ϵ(h), \langle a b, h\rangle = \langle a\otimes b,\Delta h\rangle, \,\,\,\,\,\,\,\,\,\,\langle 1_B,h\rangle = \epsilon(h),
Δa,hg=a,hg,ϵ(a)=a,1 H \langle \Delta a, h\otimes g\rangle = \langle a, h g\rangle, \,\,\,\,\,\,\,\,\,\,\epsilon(a) = \langle a, 1_H\rangle

They are a strictly dual pair of bigebras if the pairing is in also nondegenerate. If BB and HH are paired then one can quotient out biideals J BBJ_B\subset B, J HHJ_H\subset H of all those elements in each of them which pair as zero with all elements in the other bigebra; the quotients BB' and HH' will then be strictly paired bigebras.

Literature

Related nnLab entries: dual, Heisenberg double, gebra

Quite detailed treatment of duality of gebras is in

Other sources are

and for gebras with involution

Hit-actions are recently studied in

Cartier duality and related earlier issues on linearly compact topological spaces? due Dieudonné are in the first chapter of

Some newer applications are in

Duality of dg-algebras vs. dg-coalgebras is studied recently in great detail in

Some special cases of finite duals are treated in