Hermitian form

2020 Mathematics Subject Classification: Primary: 15-XX [MSN][ZBL]

An hermitian form on a left $R$-module $X$ is a mapping $\def\phi{\varphi}\phi:X\times X \to R$ that is linear in the first argument and satisfies the condition

$$\phi(y,x) = \phi(x,y)^J,\quad x,y\in X.$$ Here $R$ is a ring with a unit element and equipped with an involutory anti-automorphism $J$. In particular, $\phi$ is a sesquilinear form on $X$. The module $X$ itself is then called a Hermitian space. By analogy with what is done for bilinear forms, equivalence is defined for Hermitian forms (in another terminology, isometry) and, correspondingly, isomorphism (isometry) of Hermitian spaces (in particular, automorphism). All automorphisms of a Hermitian form $\phi$ form a group $U(\phi)$, which is called the unitary group associated with the Hermitian form $\phi$; its structure has been well studied when $R$ is a skew-field (see Unitary group).

A Hermitian form is a special case of an $\def\e{\epsilon}\e$-Hermitian form (where $\e$ is an element in the centre of $R$), that is, a sesquilinear form $\psi$ on $X$ for which

$$\psi(y,x) = \e\psi(x,y)^J,\quad x,y\in X.$$ When $\e = 1$, an $\e$-Hermitian form is Hermitian, and when $\e=-1$ the form is called skew-Hermitian or anti-Hermitian. If $J=1$, a Hermitian form is a symmetric bilinear form, and a skew-Hermitian form is a skew-symmetric or anti-symmetric bilinear form. If the mapping

$$X\to\def\Hom{ {\rm Hom}}\Hom_R(X,R),\quad y\mapsto f_y,$$ where $f_y(x) = \phi(x,y)$ for any $x\in X$, is bijective, then $\phi$ is called a non-degenerate Hermitian form or a Hermitian scalar product on $X$.

If $X$ is a free $R$-module with a basis $e_1,\dots,e_n$, then the matrix $(a_{ij})$, where $a_{ij} = \phi(e_i,e_j)$, is called the matrix of $\phi$ in the given basis; it is a Hermitian matrix (that is, $a_{ji}=a_{ij}^J$). A Hermitian form $\phi$ is non-degenerate if and only if $(a_{ij})$ is invertible. If $R$ is a skew-field, if ${\rm char}\; R \ne 2$, and if $X$ is finite-dimensional over $R$, then $X$ has an orthogonal basis relative to $\phi$ (in which the matrix is diagonal).

If $R$ is a commutative ring with identity, if $R_0 = \{r\in R : r^J = r\}$, and if the matrix of $\phi$ is definite, then its determinant lies in $R_0$. Under a change of basis in $X$ this determinant is multiplied by a non-zero element of $R$ of the form $\def\a{\alpha}\a\a^J$, where $\a$ is an invertible element of $R$. The determinant regarded up to multiplication by such elements is called the determinant of the Hermitian form or of the Hermitian space $X$; it is an important invariant and is used in the classification of Hermitian forms.

Let $R$ be commutative. Then a Hermitian form $\phi$ on $X$ gives rise to a quadratic form $Q(x)=\phi(x,x)$ on $X$ over $R_0$. The analysis of such forms lies at the basis of the construction of the Witt group of $R$ with an involution (see Witt ring; Witt decomposition; Witt theorem). When $R$ is a maximal ordered field, then the law of inertia extends to Hermitian forms (and there arise the corresponding concepts of the signature, the index of inertia, and positive and negative definiteness). If $R$ is a field and $J\ne 1$, then $R$ is a quadratic Galois extension of $R_0$, and isometry of two non-degenerate Hermitian forms over $R$ is equivalent to isometry of the quadratic forms over $R_0$ generated by them; this reduces the classification of non-degenerate Hermitian forms over $R$ to that of non-degenerate quadratic forms over $R_0$.

If $R=\C$ and $J$ is the involution of complex conjugation, then a complete system of invariants of Hermitian forms over a finite-dimensional space is given by the rank and the signature of the corresponding quadratic forms. If $R$ is a local field or the field of functions of a single variable over a finite field, then a complete system of invariants for non-degenerate Hermitian forms is given by the rank and the determinant. If $R$ is a finite field, then there is only one invariant, the rank. For the case when $R$ is an algebraic extension of $\Q$, see [MiHu]. Ch. Hermite was the first, in 1853, to consider the forms that bear his name in connection with certain problems of number theory.

References

[Bo]	N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra", 1, Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207
[Di]	J.A. Dieudonné, "La géométrie des groups classiques", Springer (1955) Zbl 0221.20056
[MiHu]	J. Milnor, D. Husemoller, "Symmetric bilinear forms", Springer (1973) MR0506372 Zbl 0292.10016