Relative to $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$, a symplectic Lie algebra matrix has blocks $\left(\begin{array}{cc}P& Q\\ R& -P^{\mathsf{T}}\end{array}\right)$ with $Q,R$ symmetric. Its diagonal Cartan subalgebra has basis $E_{ii}-E_{l+i,l+i}$. Write ${\epsilon }_i(h)={\lambda }_i$. A root-space decomposition is obtained fromThe matrix unit rule $[h,E_{ab}]=(h_{aa}-h_{bb})E_{ab}$ proves $[h,e_r]=r(h)e_r$. These root vectors and the diagonal basis span all the permitted blocks and are independent. There are $2l^2$ roots and $l$ diagonal basis elements, giving dimension $l(2l+1)$.

In the jump chain of the symmetric stochastic attrition process, the two possible increments of $D$ are $-2A+1$ and $2B-1$. Their mean is the displayed quantity, bounded by one in absolute value. Subtracting these conditional means produces a martingale. If both initial populations are at most $N$, it has at most $2N$ nonzero increments, each bounded by $4N$. Thus Azuma-Hoeffding inequality controls changes of order $N^2$ with exponentially small probability of order $e^{-cN}$.

The three-dimensional irreducible representation $V_2$ of SU(2) has weights $2,0,-2$. Thus the character of its $n$th symmetric power is ${\sum }_{a+b+c=n}{t}^{2(a-c)}$. Its weight $2\ell$ has multiplicity $\lfloor (n-|\ell |)/2\rfloor +1$ for $|\ell |\le n$. Subtracting adjacent weight multiplicities gives one copy of each $V_{2n},V_{2n-4},\dots$. The invariant subspace is one-dimensional for even $n$ and zero for odd $n$.

In the symmetric power ${\mathrm{Sym}}^m{\mathbb{C}}^n$ of the defining unitary group representation, diagonal torus weights are indexed by nonnegative integer tuples summing to $m$, with one monomial per weight space. Torus averaging extracts a monomial from any nonzero invariant complex subspace. The complexified Lie algebra acts by $E_{ij}=x_i{\partial }_{x_j}$ on the symmetric algebra. Whenever $a_j>0$, the displayed action moves one unit of degree from $j$ to $i$ with nonzero coefficient. Finite transfers connect every degree-$m$ monomial, so the subspace is the entire representation. The case $m=0$ is the one-dimensional trivial representation. This proof needs neither the Weyl character formula nor the Weyl dimension formula.

When the drift bound is a deterministic summable sequence ${\delta }_n\ge 0$, the displayed correction is a nonnegative supermartingale. Its added tail tends to zero, so the almost sure supermartingale convergence theorem directly implies finite almost sure convergence of $X_n$. This simpler tail correction requires deterministic bounds; an arbitrary future random drift tail is not generally measurable at the current time.

In the leading radiation domination limit, neglect $k^2{\delta }_r$ and the matter contribution to gravity. The radiation equation gives $S^{\prime \prime }=0$. Substitution into ${\delta }_c^{\prime \prime }+{\delta }_c^{\prime }/\tau =3{\delta }_r/{\tau }^2$ givesThus adiabatic initial conditions select both $S_0=0$ and $S_1=0$, leaving the growing and decaying power-law modes. The four constants describe the leading supplied coupled system; regularity and additional initial Einstein field equations constraints can further restrict their physical interpretation.

For RNS string matter normalized by $[{\dot{X}}^{\mu }(\sigma ),X^{\nu }({\sigma }^{\prime })]=-i\pi {\eta }^{\mu \nu }\delta (\sigma -{\sigma }^{\prime })$ and $\{{\psi }_1^{\mu }(\sigma ),{\psi }_1^{\nu }({\sigma }^{\prime })\}=\pi {\eta }^{\mu \nu }\delta (\sigma -{\sigma }^{\prime })$, put $A={\partial }_{+}X$ and $J={\psi }_1\cdot A$. Then $[A^{\mu }(\sigma ),A^{\nu }({\sigma }^{\prime })]=(i\pi /2){\eta }^{\mu \nu }{\partial }_{\sigma }\delta (\sigma -{\sigma }^{\prime })$. The noncentral part of $\{J(\sigma ),J({\sigma }^{\prime })\}$ is $\pi \delta A^2+(i\pi /2){\partial }_{\sigma }\delta {\psi }_1(\sigma )\cdot {\psi }_1({\sigma }^{\prime })$. The distributional derivative identity $f(\sigma ,{\sigma }^{\prime }){\delta }^{\prime }=f(\sigma ,\sigma ){\delta }^{\prime }+({\partial }_{{\sigma }^{\prime }}f)(\sigma ,\sigma )\delta$, with a vanishing coincident Grassmann bilinear, converts this to $\pi \delta {\theta }_{++}$. Use the chiral equation of motion to identify ${\partial }_{\sigma }{\psi }_1={\partial }_{+}{\psi }_1$. Quantum normal ordering adds a central contact term proportional to ${\delta }^{\prime \prime }$, the local form of the N=1 super-Virasoro algebra central extension.

For a current subalgebra $\mathfrak{h}\subset \mathfrak{g}$, its currents have conformal weight one under both Sugawara construction stress tensors. Their difference therefore commutes with all $\mathfrak{h}$ currents and with the $\mathfrak{h}$ stress tensor. It obeys the Virasoro algebra with central charge $c_{\mathfrak{g}}-c_{\mathfrak{h}}$. For a positive-level compact unitary vacuum representation, $\Vert L_{-2}^{\mathfrak{g}/\mathfrak{h}}|0⟩{\Vert }^2=(c_{\mathfrak{g}}-c_{\mathfrak{h}})/2\ge 0$. A Cartan subalgebra has $c_{\mathfrak{h}}=\mathrm{rank}\mathfrak{g}$, giving the lower bound on $c_{\mathfrak{g}}$ under these hypotheses.

A sublevel set which is compact. A good rate function has this property for every finite level.

Normalize the Abelian coupling by $g_1^2=(5/3)g^{\prime 2}$ and match $g_1=g_2=g_3$ in magnitude at scale $M_X$. With common-flavor coefficients $b_3=(33-2n_f)/3$, $b_2=(22-2n_f)/3$ and $b_1=-2n_f/3$, the one-loop running of the strong coupling extends to $1/{\alpha }_i(\mu )=1/{\alpha }_5-b_i\log (M_X^2/{\mu }^2)/(4\pi )$. The differences obey ${\alpha }_1^{-1}-{\alpha }_2^{-1}=2({\alpha }_2^{-1}-{\alpha }_s^{-1})$. Using ${\alpha }_2^{-1}={\sin }^2{\theta }_W/\alpha$ and ${\alpha }_1^{-1}=3{\cos }^2{\theta }_W/(5\alpha )$ gives the displayed prediction and $\log (M_X/\mu )=\frac{\pi }{11}({\alpha }^{-1}-8/(3{\alpha }_s))$. These simplified coefficients omit the full matter and scalar counting of a realistic unified model; threshold changes require piecewise running with the actual coefficients.

When a simply connected $SU(5)$ gauge group undergoes spontaneous symmetry breaking to the connected physical Standard Model subgroupthe vacuum manifold is $G/H$. The long exact sequence of homotopy groups of a fibration gives ${\pi }_2(G/H)\cong {\pi }_1(H)\cong \mathbb{Z}$. Thus the theory admits sectors with nonzero integer magnetic monopole topological charge; a charged configuration cannot relax continuously to the vacuum while its asymptotic field remains on the vacuum manifold. This classifies permitted stable defects. Their actual formation and abundance also depend on the transition and subsequent dynamics, as in the Kibble mechanism.^[ref]

In natural units, let $\eta$ be the magnitude setting the broken-symmetry Higgs field, $g$ its gauge coupling and $\lambda$ a characteristic dimensionless coefficient of the Higgs field potential. Up to normalization-dependent constants, the massive gauge bosons and radial Higgs field excitation have masses $g\eta$ and $\sqrt{\lambda }\eta$. Their inverse masses set the gauge and scalar core radii, respectively:Rescaling a static classical magnetic monopole profile by $g\eta$ gives its energy in the form $M_M=(4\pi \eta /g)f$ for a dimensionless profile factor $f$; its value depends on dimensionless ratios of scalar couplings to $g^2$, the Higgs field potential and the charge sector. Weak gauge coupling therefore makes these defects much heavier than the associated massive gauge bosons, rather than assigning them the elementary-particle scale $g\eta$.^[ref]

// Target: cosmology.bigb

Each root $\alpha$ of the SU(3) Lie algebra defines an SU(2) Lie algebra generated by $J_3=\alpha \cdot H$ and $J_{\pm }=\sqrt{2}{E}_{\pm \alpha }$ in the root-length-one convention. They obey $[J_3,J_{\pm }]=\pm J_{\pm }$ and $[J_{+},J_{-}]=2J_3$. These subalgebras explain root strings, Weyl group reflections and the integer Dynkin labels of highest weights.

// Target: lie-theory.bigb

Lower the output index of the structure constants of a Lie algebra with its Killing form. The Lie bracket makes the result antisymmetric in the first two indices, while invariance gives $\kappa ([t_a,t_b],t_c)=\kappa (t_a,[t_b,t_c])$, making it antisymmetric in the last two indices. These two adjacent transpositions imply total antisymmetry. Nondegeneracy of the Killing form is unnecessary.

// Target: lie-theory.bigb

An oriented three-dimensional Lie algebra has its antisymmetric structure constants encoded by the displayed matrix, whose symmetric part is $n^{bd}$ and whose antisymmetric part uniquely defines the covector $a_e$. In a volume-normalized basis, the inverse is $C_a{}^b{}_c=\frac{1}{2}{ϵ}_{acd}{n}^{bd}+\frac{1}{2}({\delta }_c^b{a}_a-{\delta }_a^b{a}_c)$. The only independent triple in the Jacobi identity has coefficient $-\frac{1}{2}{n}^{be}{a}_e$, so Jacobi holds exactly when $na=0$. This decomposition is useful in classifying three-dimensional Lie algebras.

If $\delta ={\min }_i(|a_{ii}|-{\sum }_{j\ne i}|a_{ij}|)>0$, then $A$ is invertible and its operator norm on ${\ell }^{\infty }$ satisfies the displayed bound. For $Ax=b$, choose $i$ maximizing $|x_i|$. The reverse triangle inequality gives $|b_i|\ge \delta \Vert x{\Vert }_{{\ell }^{\infty }}$. Taking $b=0$ proves injectivity and hence invertibility in finite dimensions; taking arbitrary $b$ proves the bound. The estimate is useful for converting strict diagonal dominance into quantitative matrix inverse stability.

For finite $A\subset \mathbb{Z}$ put $D(\eta ,A)={\prod }_{x\in A}\eta (x)$, with empty product one. Interchanging occupations gives $D({\eta }^{xy},A)=D(\eta ,A^{xy})$, where $A^{xy}$ interchanges membership of the two sites. The edge-swap generators therefore agree on $D$. The stirring representation of symmetric exclusion proves the displayed identity at all times by tracing the ancestors of $A$ backwards. A general asymmetric jump kernel does not satisfy this product self-duality.

The space of ordered orthonormal complex $k$-frames in ${\mathbb{C}}^n$, for $0\le k\le n$. It is a compact smooth manifold of real dimension $2nk-k^2$. Completing a frame to a unitary basis gives the displayed homogeneous-space description; $V_0$ is a point and $V_n({\mathbb{C}}^n)$ is the unitary group $U(n)$.

For a rank-$n$ Euclidean vector bundle, take its ordered orthonormal $k$-frame bundle and quotient by the symmetric group permuting the $k$ vectors. The whole fiber is generally positive-dimensional. A prescribed global framing selects a finite subcover consisting of the unordered $k$-element subsets of its distinguished frame vectors.

At a steady-curve minimum, differentiation and the steady-Hopf merger in vertical-field magnetoconvection give the first displayed wavenumber. At an oscillatory-curve minimum they give the second. For fixed positive Prandtl number $\sigma$ and large $Q_{\kappa }$, the corresponding merger diffusivity deficits are $1-{\zeta }_s\sim [(1+\sigma )/\sigma ]({\pi }^2/(4Q_{\kappa }){)}^{1/3}$ and $1-{\zeta }_o\sim [(1+\sigma ){\pi }^2/(16\sigma Q_{\kappa }){]}^{1/3}$. The two parameter surfaces are distinct: coincidence of neutral curves at one wavenumber does not imply coincidence of their minimizing wavenumbers.

Group the normalizing sum by its odd and even flow counts. The coefficient sums are $h_p({\rho }_1,{\rho }_3)={\sum }_{i=0}^p{\rho }_1^i{\rho }_3^{p-i}$ and the analogous $h_q$ for even routes. If $a$ and $b$ are the two maximum loads, then $a^p\le h_p\le (p+1)a^p$ and similarly for $b$. The generating identity ${\sum }_{p,q}\left(\genfrac{}{}{0ex}{}{p+q}{p}\right)a^p{b}^q=(1-a-b{)}^{-1}$ proves necessity and sufficiency of the displayed strict inequality. It is equivalent to strictly subunit offered load on each of the four resources.