Definition
A Lie bialgebra is a vector space equipped with two compatible algebraic structures: a Lie algebra bracket and a Lie coalgebra cobracket. Formally, a Lie bialgebra $(\mathfrak{g},[\cdot,\cdot],\delta)$ consists of a Lie algebra $(\mathfrak{g},[\cdot,\cdot])$ together with a linear map $\delta:\mathfrak{g}\to\mathfrak{g}\wedge\mathfrak{g}$ (the cobracket) such that $\delta$ is a 1‑cocycle for the adjoint representation and the dual map $\delta^{}:\mathfrak{g}^{}\wedge\mathfrak{g}^{}\to\mathfrak{g}^{}$ endows $\mathfrak{g}^{}$ with a Lie algebra structure. The compatibility condition, often called the Drinfel’d compatibility, requires that $\delta$ be a Lie algebra 1‑cocycle and that the induced bracket on $\mathfrak{g}^{}$ satisfies the Jacobi identity.
Overview
Lie bialgebras arise in the study of Poisson–Lie groups, quantum groups, and integrable systems. They provide the infinitesimal counterpart to Poisson–Lie group structures: the tangent Lie algebra of a Poisson–Lie group carries a canonical Lie bialgebra structure, and conversely, under suitable integrability conditions, a Lie bialgebra integrates to a Poisson–Lie group. The theory was introduced by Vladimir Drinfel’d in the mid‑1980s as part of his work on quantum groups, leading to the construction of quantum universal enveloping algebras as deformations of the Hopf algebra $U(\mathfrak{g})$.
Etymology / Origin
The term combines “Lie,” after the Norwegian mathematician Sophus Lie, whose work founded the theory of Lie algebras and Lie groups, with “bialgebra,” a hybrid of “bi‑” (two) and “algebra,” indicating the presence of both algebraic (bracket) and coalgebraic (cobracket) structures. The notion was formalized by Drinfel’d in his 1983 papers on quantum groups and later elaborated in the works of Kosmann‑Schwarzbach and others.
Characteristics
| Aspect | Description |
|---|---|
| Underlying vector space | A finite‑dimensional (or, more generally, a locally finite) vector space $\mathfrak{g}$ over a field of characteristic 0 (commonly $\mathbb{R}$ or $\mathbb{C}$). |
| Lie bracket | A bilinear map $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ that is antisymmetric and satisfies the Jacobi identity. |
| Cobracket | A linear map $\delta:\mathfrak{g}\to\mathfrak{g}\wedge\mathfrak{g}$ that is co‑antisymmetric ($\delta(x) = -\tau\delta(x)$ where $\tau$ swaps the tensor factors) and satisfies the co‑Jacobi identity dual to the Lie algebra Jacobi identity. |
| Compatibility (Drinfel’d condition) | $\delta([x,y]) = (\operatorname{ad}_x\otimes 1 + 1\otimes\operatorname{ad}_x)\delta(y) - (\operatorname{ad}_y\otimes 1 + 1\otimes\operatorname{ad}_y)\delta(x)$ for all $x,y\in\mathfrak{g}$. This says that $\delta$ is a 1‑cocycle for the adjoint representation. |
| Dual Lie algebra | The dual space $\mathfrak{g}^{}$ inherits a Lie bracket $[\xi,\eta]_{} = \delta^{*}(\xi\wedge\eta)$. The original bracket and the cobracket are mutually dual. |
| Morphisms | A Lie bialgebra morphism $f:\mathfrak{g}\to\mathfrak{h}$ is a linear map that is simultaneously a Lie algebra homomorphism and a Lie coalgebra homomorphism (i.e., $\delta_{\mathfrak{h}}\circ f = (f\otimes f)\circ\delta_{\mathfrak{g}}$). |
| Examples |
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| Classification | For simple Lie algebras, Lie bialgebra structures are classified by solutions of the modified classical Yang‑Baxter equation (Belavin–Drinfel’d classification). |
Related Topics
- Poisson–Lie group – a Lie group equipped with a multiplicative Poisson structure; its infinitesimal object is a Lie bialgebra.
- Quantum group – a non‑commutative, non‑cocommutative Hopf algebra deformation of $U(\mathfrak{g})$ guided by a Lie bialgebra structure.
- Classical Yang‑Baxter equation – an equation for an element $r\in\mathfrak{g}\otimes\mathfrak{g}$ whose solutions generate coboundary Lie bialgebras.
- Manin triple – a triple $(\mathfrak{d},\mathfrak{g},\mathfrak{g}^{})$ of a Lie algebra $\mathfrak{d}$ equipped with a non‑degenerate invariant bilinear form, where $\mathfrak{g}$ and $\mathfrak{g}^{}$ are isotropic subalgebras; Lie bialgebras correspond bijectively to Manin triples.
- Drinfel’d double – the Lie algebra $\mathfrak{d}$ of a Manin triple; it plays a central role in the construction of quantum doubles of Hopf algebras.
- Cobracket cohomology – the cohomological framework describing deformations of Lie bialgebra structures.
References
- V. G. Drinfel’d, Quantum groups, Proceedings of the International Congress of Mathematicians, Berkeley, 1986.
- A. Y. Kosmann‑Schwarzbach, Lie bialgebras, Poisson‑Lie groups and dressing transformations, in Integrability of Nonlinear Systems, 1995.
- M. E. Semenov‑Tian‑Shansky, What is a classical r‑matrix?, Functional Analysis and Its Applications, 1985.