Center of quantum affine algebras

Question

Are there some references about the center of quantum affine algebras? I searched on google and only find the paper. In particular, what is the center of $U_q(\widehat{\mathfrak{sl}_2})$. Thank you very much.

Answer 1

For affine (quantum or not) algebras the center appears only for special value of element "c" - the so-called "critical level". The second order central element related to "Sugavara construction" is well-known for a long time. Somewhat nice explicit formulas for the center of affine (not quantum) algebras for type A_n first appeared in our paper, sorry for self-advertisement. For the generalization to quantum case see paper. It is somewhat surprising that in quantum case there is an alternative approach to describe the center - it is presented in the paper you cited, which appeared before cited above developments.

Let me briefly sketch the ideas behind these works. Let us start with gl_n. Basic linear algebra tells that characteristic polynomial of a matrix det(L-t) gives invariant polynomials. The algebra U(gl_n) is somehow a quantization/deformation of commutative Poisson algebra S(gl), so we may hope that "det(L-t)" can be somehow deformed to give generators of the center of U(gl_n), that indeed can be done - there exists Capelli determinat which does the job. We need to deal with the matrix with non-commuting elements, but nevertheless it works. It appears that in affine case somewhat similar formula "det(d/dz - L(z))" does the job.

Going further to quantum case, it is well-known that quantum algebras can be described in the so-called RLL = LLR formalism, it is actually not one line to describe U_q(gl_n) (or affine) version since one has L^+, L^- operators, however this is standard. Roughly speaking the formula $det(1-q^{d/dz} L(z) )$ will do the job for U_q(gl_n) both affine or not. For non-affine case one should take only the first term of L(z) in "z" expansion. Sorry for being too sketchy.