dual gebra

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).

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

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

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

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

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

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

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

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

$\langle g, c ↼ f\rangle = \langle f g, c\rangle$

or equivalently by the formula

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

where the Sweedler notation has been used.

Similarly for the right-hand action:

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

or

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

According to the suggestion of Nichols, one reads $f$⇀$c$ as “$f$ hits $c$” and $f$↼$c$ as “$f$ is hit by $c$”.

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

$\langle h ↼ f, k\rangle = \langle f, k h\rangle.$

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

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.

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

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

$\langle a b, h\rangle = \langle a\otimes b,\Delta h\rangle,
\,\,\,\,\,\,\,\,\,\,\langle 1_B,h\rangle = \epsilon(h),$

$\langle \Delta a, h\otimes g\rangle = \langle a, h g\rangle,
\,\,\,\,\,\,\,\,\,\,\epsilon(a) = \langle a, 1_H\rangle$

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

Related $n$Lab entries: dual, Heisenberg double, gebra

Quite detailed treatment of duality of gebras is in

- Sorin Dăscălescu, Constantin Năstăsescu, Serban Raianu,
*Hopf algebras: an introduction*, Marcel & Dekker 2000

Other sources are

- Susan Montgomery,
*Hopf algebras and their actions on rings*, AMS 1994, 240 pp. - Shahn Majid,
*Foundations of quantum group theory*, Cambridge Univ. Press - Eiichi Abe,
*Hopf algebras*, Cambridge Tracts in Mathematics 74, 1977

and for gebras with involution

- A. van Daele,
*Dual pairs of Hopf $\ast$-algebras*, Bull. London Math. Soc. (1993) 25 (3): 209-230 doi

Hit-actions are recently studied in

- M. Cohen, S. Westreich,
*Hit-actions and commutators for Hopf algebras*, Bull. Math. Soc. Sci. Math. Roumanie**56**(104) No. 3, 2013, 299–313 pdf

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

- Jean Dieudonné,
*Introduction to the theory of formal groups*, Marcel Dekker, New York 1973.

Some newer applications are in

- Lowell Abrams, Charles Weibel,
*Cotensor products of modules*, math.RA/9912211

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

- Matthieu Anel, André Joyal,
*Sweedler Theory for (co)algebras and the bar-cobar constructions*, 260 pp. arxiv/1309.6952; cf. also Boston 2012 slides

Some special cases of finite duals are treated in

- Stephen Donkin,
*On the Hopf algebra dual of an enveloping algebra*, Math. Proc. Camb. Phil. Soc. (1982), 91, 215-224, doi - Jahn Astrid,
*The finite dual of crossed products*, thesis, pdf - MathOverflow: Hopf algebra duality and algebraic groups