Tag - Lie algebras

Let n be a positive integer and let Q = (q_ij) be a multipicatively antisymmetric n × n matrix over a field 𝕂, that is q_ii = 1 for 1 ≤ i ≤ n and, for 1 ≤ i,j ≤ n, q_ij ≠ 0 and q_ji=q_ij^-1. The quantized (co-ordinate ring of) quantum n-space R=𝒪_Q(𝕂ⁿ) is the 𝕂-algebra generated by x₁,x₂, . . .,x_n subject to the relations x_ix_j = q_ijx_jx_i for 1 ≤ i < j ≤ n.

Although the space of derivations Der(R) is well understood through work of Alev and Chamarie in 1982, less is known about the space Der_σ(R) of σ-derivations of R. The only case in the literature where the σ-derivations of R are determined appears to be when n = 2 and, for some λ ∈ 𝕂*, σ(x₁)=λx₁ and σ(x_2)=λ^-1 x₂. This case appears in a 2018 paper by Almulhem and Brzeziński that was motivated by differential geometry. This talk will discuss the classification of the σ-derivations of R for all n when σ is toric, that is each x_i is an eigenvector for σ, with a view to applications to iterated Ore extensions of 𝕂. Any such classification must include the inner σ-derivations of R, that is those for which there exists a ∈ R such that δ_a(r)=ar-σ(r)a for all r ∈ R.

The methods are based on two of the classical methods of non-commutative algebra, namely localization and grading, in this case by ℤⁿ. Localization at the set {x₁^d₁x₂^d₂ . . . x_n^d_n} yields the quantum n-torus T=𝒪_Q((𝕂*)ⁿ) to which σ and all σ-derivations extend. A σ-derivation δ of T is homogeneous, of weight (d₁,d₂, . . ., d_n), if δ(x_i) ∈ 𝕂x₁^d₁x₂^d₂ . . .,x_i^d_i+1. . . x_n^d_n for 1 ≤ i ≤ n and every σ-derivation of T is a unique linear combination of homogeneous σ-derivations. It turns out that if δ is a homogeneous σ-derivation of T then either the automorphism σ is inner or the σ-derivation δ is inner and the Ore extension T[x ; σ,δ] can, by a change of variables, be expressed as an Ore extension of either automorphism type or derivation type. This dichotomy influences the space Der_σ(R) which can be identified with { δ ∈ Der_σ(T) : δ(R) ⊆ R }. The most obvious σ-derivations included here are the homogeneous σ-derivations of weight (d₁,d₂, . . ., d_n) where each d_i ≥ 0, but more interesting are those for which one d_i = -1. There are two types of these, depending on whether σ or δ is inner on T. In the latter case we are in a common situation where a σ-derivation of a ring R is not inner on R but becomes inner on the localization of R at the powers of a normal element of R, giving rise to a distinguished normal or central element of the Ore extension R[x ; σ,δ].

The history of Lie bialgebras began with the paper where the Lie bialgebras were defined: V. G. Drinfeld, "Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations".

The aim of my talk is to celebrate 40 years of Lie bialgebras in mathematics and to explain how these important algebraic structures can be classified. This classification goes "hand in hand" with the classification of the so-called Manin triples and Drinfeld doubles also introduced in Drinfeld's paper cited above.

The ingenious idea how to classify Drinfeld doubles associated with Lie algebras possessing a root system is due to F. Montaner and E. Zelmanov. In particular, using their approach the speaker classified Lie bialgeras, Manin triples and Drinfeld doubles associated with a simple finite-dimensional Lie algebra 𝔤 (the paper was based on a private communication by E. Zelmanov and it was published in Comm. Alg. in 1999).

Further, in 2010, F. Montaner, E. Zelmanov and the speaker published a paper in Selecta Math., where they classified Drinfeld doubles on the Lie algebra of the formal Taylor power series 𝔤[[u]] and all Lie bialgebra structures on the polynomial Lie algebra 𝔤[u].

Finally, in March 2022 S. Maximov, E. Zelmanov and the speaker published an arXiv preprint, where they made a crucial progress towards a complete classification of Manin triples and Lie bialgebra structures on 𝔤[[u]].

Of course, it is impossible to compress a 40 years history of the subject in one talk but the speaker will try his best to do this.