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Algebras of functions on quantum groups: Part I. (English) Zbl 0923.17017
Mathematical Surveys and Monographs. 56. Providence, RI: American Mathematical Society (AMS). ix, 150 p. (1998).
Let us first recall the definition of a Poisson Lie group. It is a Lie group \(G\) with a Poisson structure, compatible with the Lie group structure. A Poisson structure is given by a bilinear map from \(C^\infty(G)\times C^\infty(G)\) to \(C^\infty(G)\), denoted by \(f,g \to \{f,g\}\), called the Poisson bracket, making \(C^\infty(G)\) into a Lie algebra and satisfying the Leibniz rule \(\{fg,h\}=f\{g,h\}+\{f,h\}g\). The compatibility is expressed by taking the natural Poisson structure on \(G\times G\) and by requiring that the product on \(G\) preserves these Poisson structures.
Now, let \(G\) be a Lie group and denote by \(\mathfrak g\) its Lie algebra. Consider a Poisson structure on \(G\). It is natural to ask what this induces on the Lie algebra. It turns out that it makes the Lie algebra into a Lie bialgebra in the following sense. The Poisson structure induces in a natural way a linear map \(\varphi :\mathfrak g \to \mathfrak g \wedge \mathfrak g\) such that the adjoint of \(\varphi\) makes the dual space \(\mathfrak g^*\) again into a Lie algebra. The Leibniz rule yields the relation \(\varphi([a,b])=a.\varphi(b)-b.\varphi(a)\) where the action of \(\mathfrak g\) on \(\mathfrak g\wedge\mathfrak g\) is given by \(a.(b\otimes c)=[a,b]\otimes c+ b\otimes [a,c]\).
It can be shown that for a given Lie group, there is a one-to-one correspondence between the Poisson structures on the group and the Lie bialgebra structures on the corresponding Lie algebra.
This describes the general setting for the Drinfel’d approach to quantum groups as it is treated in this book. These preliminaries are explained in the first Chapter of the book. In the next chapter comes the notion of a quantization of a Lie bialgebra. So, start with a Lie bialgebra \((\mathfrak g,\varphi)\) as defined above. Consider the enveloping algebra \(U_0\). This enveloping algebra carries a natural Hopf algebra structure. The comultiplication \(\Delta_0\) is defined by \(\Delta_0(a)=a\otimes 1 + 1\otimes a\) when \(a\in \mathfrak g\). There is a unique extension of the map \(\varphi\) to a map \(\tilde\varphi\) satisfying \(\tilde\varphi(ab)=\Delta_0(a)\tilde\varphi(b)+\tilde\varphi(a)\Delta(b)\) whenever \(a,b\in \mathfrak g\). This makes the pair \((U_0,\tilde\varphi)\) into a so-called co-Poisson Hopf algebra.
This co-Poisson structure is used to deform the comultiplication \(\Delta_0\) on \(U_0\). Roughly speaking, a quantization of a co-Poisson algebra \((U_0,\tilde\varphi)\) is a Hopf algebra \(U\) over the formal power series (in a variable \(h)\) over the number field such that \(U/hU\) is isomorphic with \(U_0\) as a Hopf algebra and such that \(h^{-1}(\Delta(a)-\Delta'(a)) \bmod h=\tilde\varphi(a\bmod h)\). Here \(\Delta\) denotes the comultiplication on \(U\) and \(\Delta'\) is obtained from \(\Delta\) by composing it with the flip \(b\otimes c\to c\otimes b\). So, in some sense, the co-Poisson structure determines in what direction the original (trivial) comultiplication \(\Delta_0\) has to be changed.
An important result is that for any finite-dimensional Lie bialgebra (defined by a Cartan matrix), there exists a quantization and it is essentially unique. Moreover, any finite dimensional simple complex Lie algebra can be given a standard Lie bialgebra structure and the corresponding quantization gives the famous Drinfel’d examples of quantized universal enveloping algebras (QUE algebras). These QUE algebras are studied in Chapter 2 of the book. Also the relation with \(R\)-matrices and many other topics are investigated.
Now, let us go back to the original Poisson Lie groups. We have seen that there is a one-to-one correspondence between the Poisson structures on a Lie group \(G\) and the Lie bialgebra structures on the corresponding Lie algebra. The algebra \(C^\infty(G)\) is in a sense dual to the enveloping algebra \(U_0\) of the Lie algebra of the Lie group. The triviality of the coproduct on \(U_0\) is related with the commutativity of the product in \(C^\infty(G)\). So it is no surprise that the deformation of the coproduct yields a deformation of the algebra \(C^\infty(G)\) into a non-abelian algebra. These are the quantized function algebras. They are studied in the third Chapter of the book.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)