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Let $\mathfrak{g}$ have a nondegenerate symmetric invariant bilinear form on a Lie algebra $B$, and choose a basis $(x_i)$ with $B$-dual basis $(x^i)$. Its quadratic Casimir element isThe inverse-form tensor ${\sum }_i{x}_i\otimes x^i$ is independent of the basis and invariant under the Adjoint representation of a Lie algebra. Multiplying its factors in the universal enveloping algebra gives $[z,C_B]={\sum }_i([z,x_i]x^i+x_i[z,x^i])=0$, so $C_B$ is central. In a finite-dimensional irreducible complex representation, the Schur lemma makes its represented quadratic Casimir operator a scalar.^[ref]

With the Killing form on a complex semisimple Lie algebra, its Casimir eigenvalue on highest weight $\lambda$ is $(\lambda ,\lambda +2\rho )$, with the induced dual inner product and Weyl vector $\rho$. Scaling $B$ by $c\ne 0$ scales $C_B$ by $c^{-1}$. For the sl2 Lie algebra, the normalization $B(e,f)=1$, $B(h,h)=2$ gives $ef+fe+h^2/2$; the actual Killing form is four times this form, so its Casimir is one quarter of that expression.

For a Lie algebra $\mathfrak{g}$ over a field $k$ and a Lie algebra representation $V$, set $C^n(\mathfrak{g},V)={\mathrm{Hom}}_k({\Lambda }^n\mathfrak{g},V)$. These alternating multilinear maps form a cochain complex with differentialThe hat means that the argument is omitted. The Jacobi identity and the Lie algebra representation identity give $d^2=0$. The Lie algebra cohomology is the quotient vector space $H^n(\mathfrak{g},V)=\ker (d:C^n\to C^{n+1})/\mathrm{im}(d:C^{n-1}\to C^n)$. In degree zero it consists of the invariant vectors of a Lie algebra representation; with a trivial Lie algebra representation as coefficients, $H^1(\mathfrak{g},k)={\mathrm{Hom}}_k(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}],k)$.^[ref]

For trivial real coefficients, $H^2(\mathfrak{g},\mathbb{R})$ classifies central extensions of a Lie algebra by a fixed one-dimensional center, up to equivalences inducing the identity on the quotient and center. If $\mathfrak{g}$ is abelian and two-dimensional, every alternating bilinear form is a Lie algebra two-cocycle and every coboundary is zero; a nonzero such form produces the Heisenberg Lie algebra.

For trivial real coefficients, a Lie algebra two-cocycle is an alternating bilinear form $\kappa :\mathfrak{g}\times \mathfrak{g}\to \mathbb{R}$ satisfyingOn the vector space $\mathfrak{g}\oplus \mathbb{R}$, define $[(X,a),(Y,b)]=([X,Y],\kappa (X,Y))$. The second summand is central, and the displayed identity is exactly the remaining condition for the Jacobi identity. Thus $\kappa$ defines a central extension of a Lie algebra. Replacing the chosen linear map splitting the quotient by $X\mapsto (X,b(X))$ changes $\kappa$ by $-b([X,Y])$; this is a coboundary in Lie algebra cohomology. For the abelian plane, $\kappa ((x,y),(x^{\prime },y^{\prime }))=x{y}^{\prime }-y{x}^{\prime }$ gives the Heisenberg Lie algebra. The constant discrepancy in a moment-map equivariance obstruction is another such Lie algebra two-cocycle.^[ref]

Pavel Etingof studies representation theory, quantum groups and noncommutative algebra. His MIT research profile places this work at the intersection of algebra and mathematical physics.^[ref]

His Lie Groups and Lie Algebras by Pavel Etingof (2024) develops the bridge from continuous groups to their infinitesimal algebraic structures. A Lie algebra representation replaces a nonlinear group action with linear operators satisfying bracket identities; the theory then recovers structural information about Lie groups. His work on tensor categories broadens the study of representations by treating their tensor products and compatibility maps as part of the mathematical object.^[ref][ref]

In the general reflection convention, a root system in a finite-dimensional real inner product space $E$ is a finite spanning set $\Phi \subset E\setminus \{0\}$ such that every root reflectionpermutes $\Phi$. In particular $-\alpha \in \Phi$ whenever $\alpha \in \Phi$. A reduced root system additionally requires $\Phi \cap \mathbb{R}\alpha =\{\alpha ,-\alpha \}$. A crystallographic root system additionally requires all Cartan integers $2(\beta ,\alpha )/(\alpha ,\alpha )$ to be integers. Some authors include these extra conditions in the term root system; they are stated separately here.

For example, the A2 root system is $\{e_i-e_j:i\ne j\}$ in the plane $x_1+x_2+x_3=0$. Its six roots form a regular hexagon, and their reflections generate a Weyl group isomorphic to $S_3$. The roots of a complex semisimple Lie algebra form a reduced crystallographic root system: its root-space decomposition converts infinitesimal Lie brackets into this finite Euclidean geometry. The Dynkin diagram then records enough simple-root data to classify its simple components.^[ref]