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A principal sl2 subalgebra of a semisimple Lie algebra has semisimple generator $h_{\mathrm{pr}}=2{\rho }^{\vee }$. Restriction to it packages a representation's weights into ordinary ${\mathfrak{sl}}_2$ weight strings.

The infinitesimal embedding is a subalgebra of the semisimple Lie algebra; integrating it to a Lie-group homomorphism is a separate group-level construction.

The quadratic Casimir element is a central element of $U(\mathfrak{g})$. On a highest-weight module of highest weight $\lambda$, one standard normalization makes it act by $|\lambda +\rho {|}^2-|\rho {|}^2$.

A highest-weight representation is generated by a weight vector killed by every positive-root subspace.

The generating vector has a highest Weight relative to the chosen positive root system.

For a complex semisimple Lie algebra, a Cartan subalgebra is a maximal abelian subalgebra consisting of semisimple elements. It is also called a maximal torus in this setting.

A nonabelian Lie algebra is simple when its only ideals are zero and the entire algebra.

For a choice of positive roots, the corresponding Borel subalgebra is $\mathfrak{b}=\mathfrak{t}\oplus {\mathfrak{n}}^{+}$, the sum of a Cartan subalgebra and all positive root spaces.

A semimartingale has a decomposition $X=X_0+M+A$ into a local martingale $M$ and an adapted finite-variation process $A$. Under standard normalizations the decomposition is unique.

A process $A$ has finite variation on compact intervals when every sample path has finite total variation of a function there. Such a process can be integrated pathwise by the Lebesgue-Stieltjes integral.

For the even potentialthe turning points are $\pm 4E$ for $0\le E\le 1$ and $\pm (E+1{)}^2$ for $E\ge 1$. Its action is

The internal effective temperature parametrizes the intrinsic cooling luminosity of a planet through $F_{\mathrm{int}}=\sigma T_{\mathrm{int}}^4$.

The irradiation temperature is defined from the incident stellar flux, commonly by $T_{\mathrm{irr}}=T_{\ast }\sqrt{R_{\ast }/a}$ before redistribution and albedo factors are applied.

Two Seifert forms are algebraically concordant when their difference is metabolic. Stable equivalence classes form the algebraic concordance group.

A Seifert matrix represents the Seifert form in an integral basis of $H_1(F)$. Its skew-symmetrization $A-A^T$ represents the intersection form of the surface and is unimodular.

Let ${\pi }_k(x)$ count primes at most $x$ congruent to $k$ modulo eight. Bounding the prime-power divisibility ofabove and its size below gives, for all sufficiently large $x$,

If $q_1,\dots ,q_r$ listed all primes congruent to seven modulo eight, then for $n=8q_1\cdots q_r$ the odd number $(n^2-2)/2$ is seven modulo eight. Every prime factor is one or seven modulo eight, at least one is seven, and none is among the $q_i$, a contradiction.

For $z=z(x)$, the second derivative transforms as $w_{xx}=z_x^2{w}_{zz}+z_{xx}{w}_z$.

The equation $(1-x^2)y\prime \prime -2xy\prime +\ell (\ell +1)y=0$ has polynomial solutions for nonnegative integer $\ell$.

The equation $(1-x^2)y\prime \prime -2xy\prime +\ell (\ell +1)y=0$ has the Legendre polynomial of degree $\ell$ as its normalized regular polynomial solution when $\ell$ is a nonnegative integer.

Given one nonzero solution of a homogeneous second-order linear equation, reduction of order constructs a second by writing it as a variable multiple of the first.

Variation of parameters replaces the constants in a complementary solution by functions to construct a particular solution.