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matrix Hopf algebra

Let BB be a bialgebra, possibly noncommutative, over a field kk and G=(g j i) j=1,,n i=1,,nG = (g^i_j)^{i = 1,\ldots, n}_{j = 1,\ldots, n} an n×nn\times n-matrix over BB. BB is a matrix bialgebra with basis GG if

  • the set of entries of GG generates BB and

  • the comultiplication Δ\Delta and counit ϵ\epsilon satisfy the matrix equations ΔG=GG\Delta G = G \otimes G (i.e. in components Δg j i= k=1 ng k ig j k\Delta g^i_j = \sum_{k = 1}^n g^i_k \otimes g^k_j) and ϵG=1\epsilon G = 1 (reading in components ϵ(g j i)=δ j i\epsilon (g^i_j) = \delta^i_j).

According to a result of Redford every finite-dimensional Hopf algebra over a field is a matrix Hopf algebra with respect to some basis.

The free (noncommutative) associative algebra FF on n 2n^2 generators f j if^i_j has a unique coalgebra structure making it a matrix bialgebra with basis (f j i) j=1,,n i=1,,n(f^i_j)^{i = 1,\ldots, n}_{j = 1,\ldots, n}. We call it the free matrix bialgebra of rank n 2n^2. Every bialgebra quotient of that bialgebra is a matrix bialgebra.

A matrix Hopf algebra \mathcal{H} with basis T=(t j i)T = (t^i_j) is a Hopf algebra which possess a matrix subbialgebra BB with basis TT such that the map H(id B):H(B)H(id_B):H(B)\rightarrow\mathcal{H} is onto (where H(B)H(B) denotes the Hopf envelope of BB and HH is understood as a functor).

A matrix Hopf algebra \mathcal{H} with basis TT is often not a matrix bialgebra with basis TT: e.g. the commutative coordinate ring of GL(n,k)GL(n,k) is not a matrix bialgebra with respect to the obvious basis TT; in this example this can be repaired by enlarging the basis by one group-like element: the inverse of the determinant. On the other hand, the coordinate algebra of the special linear group O(SL(n,k))O(SL(n,k)) is a matrix bialgebra and a matrix Hopf algebra with the same standard basis TT.

Revised on March 8, 2010 15:44:56 by Zoran Škoda (161.53.130.104)