double brackets

Double brackets ⟦ · , · ⟧ are a typographic convention that denotes a bilinear operation taking two algebraic elements and returning a tensor‑product‑valued expression. Unlike the ordinary Lie bracket ([x,y]) or the Poisson bracket ({x,y}), the double bracket records both the order of multiplication and the “output space’’ by landing in (A\otimes A) for an associative algebra (A). This extra layer makes the notation indispensable in non‑commutative geometry, deformation theory, and higher‑algebraic structures where one wishes to encode a Poisson‑type interaction without forcing commutativity.

Historical Background

The first systematic use of double brackets appears in Maxim Kontsevich’s 1994 preprint Formal (non)commutative symplectic geometry, where he introduced a bilinear map (\langle!\langle a,b\rangle!\rangle) on the free associative algebra to model non‑commutative symplectic forms. Kontsevich’s construction was later refined by Michel Van den Bergh in his 2008 paper Double Poisson algebras, which gave the operation a precise set of axioms—skew‑symmetry up to tensor flip and a Leibniz rule—thereby creating a new algebraic object now called a double Poisson algebra. The notation ⟦ · , · ⟧ quickly spread to related fields, notably to the study of Calabi–Yau algebras after the 2010 work of Ginzburg linking double brackets to cyclic (A_\infty)‑structures.

Formal Definition and Mechanism

For an associative (\mathbb{k})-algebra (A), a double bracket is a (\mathbb{k})-bilinear map
[ \llbracket a,b\rrbracket;:;A\times A\longrightarrow A\otimes A ]
satisfying (\llbracket b,a\rrbracket = -\tau\bigl(\llbracket a,b\rrbracket\bigr)), where (\tau(x\otimes y)=y\otimes x) is the tensor flip. The Leibniz property reads (\llbracket a,bc\rrbracket = \llbracket a,b\rrbracket,c + b,\llbracket a,c\rrbracket), ensuring that the bracket behaves as a derivation in each argument. When one passes to representation spaces (\operatorname{Rep}_n(A)), the induced map ({f,g} = \operatorname{Tr}\bigl(\llbracket \tilde f,\tilde g\rrbracket\big)) yields an ordinary Poisson bracket on the coordinate ring, a mechanism first proved by Van den Bergh in Theorem 3.2 of his 2008 article.

Key Applications

In deformation quantization, Kontsevich’s formality theorem (