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If $D:\mathfrak{b}\to \mathrm{Der}(\mathfrak{a})$ is a Lie algebra homomorphism into the derivations of a Lie algebra, the semidirect product has underlying vector space $\mathfrak{a}\oplus \mathfrak{b}$ and bracket $[(a,b),(a^{\prime },b^{\prime })]=([a,a^{\prime }]+D_b{a}^{\prime }-D_{b^{\prime }}a,[b,b^{\prime }])$. The homomorphism and derivation identities ensure the Jacobi identity. The first factor is an ideal; the second is a subalgebra whose action records the twisting. The diamond Lie algebra arises by letting a one-dimensional algebra act on a Heisenberg Lie algebra.

// Target: linear-algebra.bigb

The Lie algebra over a commutative ring $R$ generated by a set $X$ subject only to bilinearity, alternation and the Jacobi identity. Every map from $X$ to the underlying set of an $R$-Lie algebra extends uniquely to a Lie algebra homomorphism. It can be constructed from bracketed words by quotienting by those identities; elements are represented by Lie polynomials.

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra, with a chosen normalized invariant bilinear form on a Lie algebra $B$. Its affine current algebra is the central extension of a Lie algebra $\widehat{L\mathfrak{g}}=(\mathfrak{g}\otimes \mathbb{C}[t,t^{-1}])\oplus \mathbb{C}K$ of the loop algebra, withInvariance of $B$ makes the extra term a Lie algebra two-cocycle, hence preserves the Jacobi identity. In a representation where $K=kI$, the scalar $k$ is the level. The central term vanishes for zero modes, which recover the original Lie algebra, but modifies the commutator of opposite nonzero modes.^[ref]

A restricted module has $J_{n}v=0$ for all sufficiently large $n$ for each vector $v$; this makes the Sugawara construction finite on each vector after normal ordering. Nontrivial unitary integrable positive-energy highest-weight representations have positive integer levels in the basic normalization. The zero-level integrable highest-weight representation is trivial. Adjoining the degree derivation gives the further extension with $[d,x\otimes t^n]=nx\otimes t^n$.

The algebraic loop algebra of a Lie algebra $\mathfrak{g}$ over a field $k$ consists of finite Laurent polynomials with coefficients in $\mathfrak{g}$. Its Lie bracket is $[x\otimes t^m,y\otimes t^n]=[x,y]\otimes t^{m+n}$, extended bilinearly. Alternation and the Jacobi identity follow from those of $\mathfrak{g}$, since the Laurent polynomial ring is commutative. Appropriate smooth or formal completions give other loop-algebra variants; the affine current algebra adds a central extension to these Fourier modes. For the affine extension, see Kac and Wakimoto, Section 0.2.

For a finite-dimensional Lie algebra over a field of characteristic zero, Cartan's criterion characterizes a solvable Lie algebra by vanishing of its Killing form on $\mathfrak{g}\times [\mathfrak{g},\mathfrak{g}]$, and a semisimple Lie algebra by nondegeneracy of that form. The Cartan solvability criterion and Cartan criterion for semisimplicity state the two distinct tests.

Equip the Boolean hypercube with uniform probability measure and graph distance divided by its graph diameter. These spaces form a Lévy family of graphs. The vertex-isoperimetric inequality in the discrete cube reduces the complement of a positive-density set's radius-$\epsilon n$ graph neighbourhood to a binomial distribution tail. For fixed density $\delta >0$, Chebyshev's inequality puts a lower density quantile at least $n/2-C_{\delta }\sqrt{n}-O(1)$. After enlarging by $\epsilon n$, the omitted mass is at most $n/[4(\epsilon n-C_{\delta }\sqrt{n}-O(1){)}^2]$, which tends to zero.

Equip $[n{]}^d$ with uniform probability measure and its nearest-neighbour graph distance divided by $d(n-1)$, for fixed positive $d$. The set $\{x:x_1\le \lceil n/2\rceil \}$ has at least half the vertices. Its radius-$\epsilon d(n-1)$ graph neighbourhood still omits a proportion tending to $1/2-\epsilon d$ when $0<\epsilon <1/(2d)$. These grids therefore do not form a Lévy family of graphs.

For a probability measure on the real line with characteristic function $\phi$, averaging $1-\cos (tx)$ over $[-h,h]$ gives $1-\sin (hx)/(hx)$. This is at least $1/2$ when $|x|\ge 2/h$. Integrating gives the displayed bound. Consequently pointwise convergence of characteristic functions to a function continuous at zero implies tightness of probability measures, by the dominated convergence theorem on $[-h,h]$ and a separate tail bound for the finitely many initial measures.

On a trivialized fiber bundle over $U$, assume the fiber has finite-dimensional total cohomology over a field. The Künneth theorem identifies the total-space cohomology with a finite free graded module over $H^{\ast }(U)$. Classes restricting to a basis on every fiber give an invertible change of generators: its degree-zero coefficient blocks are invertible on each path component, while its positive-base-degree terms strictly decrease fiber degree and are therefore nilpotent. Inverting the diagonal blocks and then using a finite geometric series proves that multiplication by the chosen classes is an isomorphism over $U$.

For global cohomology classes satisfying the local basis criterion for Leray-Hirsch classes, multiplication commutes with the Mayer–Vietoris sequence, including its connecting homomorphism: multiply cochains on the right by global cocycle representatives. Induct over a finite trivializing open cover. If $U$ is a union of $r-1$ patches and $V$ is the last patch, $U\cap V$ has a cover by at most $r-1$ trivialized intersections. The induction hypothesis on $U$, $V$ and $U\cap V$, followed by the Five lemma, proves the theorem on $U\cup V$. A compact base supplies the finite cover.

The tau is the charged lepton of the third fermion generation. In the massless approximation its charged-current W boson partial decay width equals those of the electron and muon, reflecting lepton universality.

The tau lepton has electric charge $-1$ in units of the positive elementary electric charge; ${\tau }^{+}$ is its antiparticle. Its weak charged current decays make the tau lepton useful for testing lepton universality. They also illustrate how particle detection can establish a short-lived parent through visible daughters and a consistent invisible recoil.

For the opposite internal torques on a black hole and its accretion disk, the total angular momentum ${\mathbf{J}}_t$ and the black-hole spin magnitude $J_h$ are constant. If $A=K_2|{\mathbf{J}}_h\times {\mathbf{J}}_d{|}^2$, then $d({\mathbf{J}}_h\cdot {\mathbf{J}}_t)/dt=A$ and $d{J}_d^2/dt=-2A$. The $K_1$ term produces precession; the positive $K_2$ term increases the alignment of the hole with the total vector while decreasing the disk angular-momentum magnitude. The exact direction equation is $\dot{c}=K_2{J}_h{J}_t(1-c^2)$ for $c=\cos \angle ({\mathbf{J}}_h,{\mathbf{J}}_t)$.

For a weakly gravitating thin spherical shell of mass $M$, radius $R$ and uniform angular velocity $\Omega$, its interior linearized gravity metric has constant scalar potentials $A=-GM/R$, $C=GM/R$ and $g_{0i}=\omega (\hat{\mathbf{z}}\times \mathbf{x}{)}_i$. The scalar mass shell is a monopole. Its mass current is a dipole: regular interior and decaying exterior radial solutions are proportional to $r$ and $r^{-2}$, and their derivative jump fixes the factor $4/3$. Local inertial axes co-rotate with the shell at $\omega$ relative to asymptotic nonrotating axes. This is the leading interior Lense-Thirring precession, neglecting quadratic rotation and higher gravitational orders.

For closed orientable hyperbolic surfaces of curvature $-1$, the spectrum of the Laplace-Beltrami operator, including multiplicities, determines the unmarked length spectrum of closed geodesics, and conversely. Count either oriented geodesics throughout or unoriented ones throughout. Primitive lengths with multiplicities determine the lengths of all iterates, and these determine the primitive lengths recursively.^[ref]

The Selberg trace formula connects the two lists: the spectral side sums functions of eigenvalues, while the geometric side sums explicit contributions from closed-geodesic lengths. Equality of either complete list forces equality of the other. The theorem concerns these spectral data; distinct surfaces can still have equal spectra.

For nonorientable hyperbolic surfaces, the trace contributions distinguish orientation-preserving and orientation-reversing geodesics. Their separate counts, rather than an unspecified undifferentiated convention, belong in the extended statement.^[ref]

For a closed hyperbolic surface with a specified marking of its fundamental group, assign to each nontrivial conjugacy class the length of its unique closed geodesic. Retaining the group label is stronger than retaining only the unmarked length multiset: it also records which lengths belong to intersecting curves or powers of particular curves.

A Hausdorff topological group admits a compatible left-invariant group metric exactly when it has a countable neighbourhood basis at the identity element. Choose symmetric neighbourhoods of the identity element $U_n$ with $U_{n+1}^3\subset U_n$ and trivial intersection. Assign a step in $U_n$ cost $2^{-n}$, and take the infimum of total costs of chains joining two points. The dyadic product estimate for group neighbourhoods makes this a genuine compatible metric. The Hausdorff assumption separates distinct points; without it the same construction can give only a pseudometric.

A group distance is bi-invariant when both left and right translations are isometries. Conjugation then fixes the identity element and is an isometry, so each metric ball there is invariant under every inner automorphism. A compatible bi-invariant group metric therefore requires arbitrarily small conjugation-invariant neighbourhoods. This concerns a distance function, not merely a bi-invariant Riemannian metric tensor.

A maximal left ideal is a proper left ideal contained in no larger proper left ideal. In a unital Banach algebra it is norm closed: its closure is a left ideal, and if that closure were the entire algebra it would contain elements of $L$ within distance less than one of $1$, hence invertible by the Neumann series. A proper left ideal contains no invertible element. The quotient $A/L$ is consequently a quotient Banach space and a simple module.

On a dyadic interval $Q<p\le 2Q$, every such exceptional prime makes all $Q^{0.01}$-smooth integers quadratic residues. By positive density of power-smooth integers, there are $\gg Q^2$ of these integers up to $Q^2$. Apply the variance form of the large sieve or its forbidden-residue consequence with interval length $Q^2$ and prime cutoff $2Q$. Each exceptional prime excludes $(p-1)/2$ residue classes and contributes at least a fixed positive amount to the sieve denominator. Hence each dyadic interval contains $O(1)$ exceptional primes. Summing the intervals gives the logarithmic bound.

// Target: analytic-number-theory.bigb

The porous medium equation $h_t=\lambda (h{h}_x{)}_x-\Gamma h$ with constant $\Gamma >0$ becomes $H_{\tau }=\lambda (H{H}_x{)}_x$ under the displayed transformation. Substitution gives $h_t+\Gamma h=e^{-\Gamma t}{\tau }^{\prime }{H}_{\tau }$ and $\lambda (h{h}_x{)}_x=\lambda e^{-2\Gamma t}(H{H}_x{)}_x$, so ${\tau }^{\prime }=e^{-\Gamma t}$. This transformation applies to every solution with appropriately transformed initial and boundary conditions, not just a similarity solution. A one-sided constant-volume porous gravity current therefore has remaining physical volume $V{e}^{-\Gamma t}$ and a limiting front $[9\lambda V/(\varphi \Gamma ){]}^{1/3}$. The finite transformed time $\tau \to 1/\Gamma$ means that arbitrary initial profiles need not reach the universal large-time source shape before drainage removes them.